Wave function - Knowledge

9810: 9405: 1789: 1764: 10286: 9805:{\displaystyle |\Psi \rangle =\overbrace {\sum _{s_{z\,1},\ldots ,s_{z\,N}}} ^{\text{discrete labels}}\overbrace {\int _{R_{N}}d^{3}\mathbf {r} _{N}\cdots \int _{R_{1}}d^{3}\mathbf {r} _{1}} ^{\text{continuous labels}}\;\underbrace {{\Psi }(\mathbf {r} _{1},\ldots ,\mathbf {r} _{N},s_{z\,1},\ldots ,s_{z\,N})} _{\begin{array}{c}{\text{wave function (component of }}\\{\text{ state vector along basis state)}}\end{array}}\;\underbrace {|\mathbf {r} _{1},\ldots ,\mathbf {r} _{N},s_{z\,1},\ldots ,s_{z\,N}\rangle } _{\text{basis state (basis ket)}}\,.} 9826: 14797: 7589: 6186: 8683: 48: 12978: 19236: 10990: 7138: 5884: 10281:{\displaystyle (\Psi _{1},\Psi _{2})=\sum _{s_{z\,N}}\cdots \sum _{s_{z\,2}}\sum _{s_{z\,1}}\int \limits _{\mathrm {all\,space} }d^{3}\mathbf {r} _{1}\int \limits _{\mathrm {all\,space} }d^{3}\mathbf {r} _{2}\cdots \int \limits _{\mathrm {all\,space} }d^{3}\mathbf {r} _{N}\Psi _{1}^{*}\left(\mathbf {r} _{1}\cdots \mathbf {r} _{N},s_{z\,1}\cdots s_{z\,N},t\right)\Psi _{2}\left(\mathbf {r} _{1}\cdots \mathbf {r} _{N},s_{z\,1}\cdots s_{z\,N},t\right)} 6689: 1698: 12708: 6424: 13503: 3924: 9251: 8652:

tensor product state, which essentially means any unentangled state remains unentangled under time evolution. This is said to happen when there is no physical interaction between the states of the tensor products. In the case of non separable Hamiltonians, energy eigenstates are said to be some linear combination of such states, which need not be factorizable; examples include a particle in a

12633: 12192: 10631: 6440: 7584:{\displaystyle \xi ={\begin{bmatrix}\xi (s,t)\\\xi (s-1,t)\\\vdots \\\xi (-(s-1),t)\\\xi (-s,t)\\\end{bmatrix}}=\xi (s,t){\begin{bmatrix}1\\0\\\vdots \\0\\0\\\end{bmatrix}}+\xi (s-1,t){\begin{bmatrix}0\\1\\\vdots \\0\\0\\\end{bmatrix}}+\cdots +\xi (-(s-1),t){\begin{bmatrix}0\\0\\\vdots \\1\\0\\\end{bmatrix}}+\xi (-s,t){\begin{bmatrix}0\\0\\\vdots \\0\\1\\\end{bmatrix}}} 6181:{\displaystyle |s\rangle \leftrightarrow {\begin{bmatrix}1\\0\\\vdots \\0\\0\\\end{bmatrix}}\,,\quad |s-1\rangle \leftrightarrow {\begin{bmatrix}0\\1\\\vdots \\0\\0\\\end{bmatrix}}\,,\ldots \,,\quad |-(s-1)\rangle \leftrightarrow {\begin{bmatrix}0\\0\\\vdots \\1\\0\\\end{bmatrix}}\,,\quad |-s\rangle \leftrightarrow {\begin{bmatrix}0\\0\\\vdots \\0\\1\\\end{bmatrix}}} 12965: 6209: 15006: 13169: 27: 10556: 3680: 9051: 12348: 16136:. The abstract state space is always taken as a Hilbert space. The matching requirement for the function spaces is a natural one. The Hilbert space property of the abstract state space was originally extracted from the observation that the function spaces forming normalizable solutions to the Schrödinger equation are Hilbert spaces. 1392:

the wave function is large. This was shown to be incompatible with the elastic scattering of a wave packet (representing a particle) off a target; it spreads out in all directions. While a scattered particle may scatter in any direction, it does not break up and take off in all directions. In 1926, Born provided the perspective of

4281: 8379: 8021: 8976:. In other words, the wave function is either totally symmetric in the positions of bosons, or totally antisymmetric in the positions of fermions. The physical interchange of particles corresponds to mathematically switching arguments in the wave function. The antisymmetry feature of fermionic wave functions leads to the 16394: 10985:{\displaystyle P_{\mathbf {r} _{1}\in R_{1},s_{z\,1}=m_{1},\ldots ,\mathbf {r} _{N}\in R_{N},s_{z\,N}=m_{N}}(t)=\int _{R_{1}}d^{3}\mathbf {r} _{1}\int _{R_{2}}d^{3}\mathbf {r} _{2}\cdots \int _{R_{N}}d^{3}\mathbf {r} _{N}\left|\Psi \left(\mathbf {r} _{1}\cdots \mathbf {r} _{N},m_{1}\cdots m_{N},t\right)\right|^{2}} 12108: 8958: 11275: 12769: 14004:

Between all these different function spaces and the abstract state space, there are one-to-one correspondences (here disregarding normalization and unobservable phase factors), the common denominator here being a particular abstract state. The relationship between the momentum and position space wave

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This applies to free field equations; interactions are not included. If a Lagrangian density (including interactions) is available, then the Lagrangian formalism will yield an equation of motion at the classical level. This equation may be very complex and not amenable to solution. Any solution would

13838:. Physical observables are represented by linear operators, also called observables, on the vectors space. Maximality means that there can be added to the set no further algebraically independent observables that commute with the ones already present. A choice of such a set may be called a choice of 5693:

of a prepared state in superposition can be determined based on physical meaning of the prepared state and its symmetry. For example, the construction of spin states along x direction as a superposition of spin states along z direction, can done by applying appropriate rotation transformation on the

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conceptions for a single spinless particle. The two processes differ greatly. The classical process (A–B) is represented as the motion of a particle along a trajectory. The quantum process (C–H) has no such trajectory. Rather, it is represented as a wave; here, the vertical axis shows the real part

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one may construct a sequence of functions approximating the true wave function. This sequence will be guaranteed to converge in a larger space, but without the assumption of a full-fledged Hilbert space, it will not be guaranteed that the convergence is to a function in the relevant space and hence

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Each choice of representation should be thought of as specifying a unique function space in which wave functions corresponding to that choice of representation lives. This distinction is best kept, even if one could argue that two such function spaces are mathematically equal, e.g. being the set of

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and the de Broglie relations and the solutions of the equation are the wave functions for the quantum system. However, no one was clear on how to interpret it. At first, Schrödinger and others thought that wave functions represent particles that are spread out with most of the particle being where

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particles. For example, any two electrons are identical and fundamentally indistinguishable from each other; the laws of physics make it impossible to "stamp an identification number" on a certain electron to keep track of it. This translates to a requirement on the wave function for a system of

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underlying the system's dynamics (in other words, the Hamiltonian can be split into the sum of orbital and spin terms). The time dependence can be placed in either factor, and time evolution of each can be studied separately. Under such Hamiltonians, any tensor product state evolves into another

6684:{\displaystyle |\phi \rangle ={\begin{bmatrix}\langle s|\phi \rangle \\\langle s-1|\phi \rangle \\\vdots \\\langle -(s-1)|\phi \rangle \\\langle -s|\phi \rangle \\\end{bmatrix}}={\begin{bmatrix}\varepsilon _{s}\\\varepsilon _{s-1}\\\vdots \\\varepsilon _{-s+1}\\\varepsilon _{-s}\\\end{bmatrix}}} 13958:

description of it is not given. This is the same as saying that no choice of maximal set of commuting observables has been given. This is analogous to a vector space without a specified basis. Wave functions corresponding to a state are accordingly not unique. This non-uniqueness reflects the

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It is possible to relax these conditions somewhat for special purposes. If these requirements are not met, it is not possible to interpret the wave function as a probability amplitude. Note that exceptions can arise to the continuity of derivatives rule at points of infinite discontinuity of

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Not all introductory textbooks take the long route and introduce the full Hilbert space machinery, but the focus is on the non-relativistic Schrödinger equation in position representation for certain standard potentials. The following constraints on the wave function are sometimes explicitly

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While the relative phase has observable effects in experiments, the global phase of the system is experimentally indistinguishable. For example in a particle in superposition of two states, the global phase of the particle cannot be distinguished by finding expectation value of observable or

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enter symmetrically, so there it does not matter which description one uses. The same equation (modulo constants) results. From this, with a little bit of afterthought, it follows that solutions to the wave equation of the harmonic oscillator are eigenfunctions of the Fourier transform in

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and others, argued that the wave function must have an objective, physical existence. Einstein thought that a complete description of physical reality should refer directly to physical space and time, as distinct from the wave function, which refers to an abstract mathematical space.

7767:. The term "spin function" instead of "wave function" is used by some authors. This contrasts the solutions to position space wave functions, the position coordinates being continuous degrees of freedom, because then the Schrödinger equation does take the form of a wave equation. 1545:

All these wave equations are of enduring importance. The Schrödinger equation and the Pauli equation are under many circumstances excellent approximations of the relativistic variants. They are considerably easier to solve in practical problems than the relativistic counterparts.

5374: 6419:{\displaystyle {\frac {1}{\hbar }}{\hat {S}}_{z}={\begin{bmatrix}s&0&\cdots &0&0\\0&s-1&\cdots &0&0\\\vdots &\vdots &\ddots &\vdots &\vdots \\0&0&\cdots &-(s-1)&0\\0&0&\cdots &0&-s\end{bmatrix}}} 4123: 8199: 8165: 7823: 15783: 15685: 3119: 13498:{\displaystyle \Psi _{n\ell m}(r,\theta ,\phi )={\sqrt {{\left({\frac {2}{na_{0}}}\right)}^{3}{\frac {(n-\ell -1)!}{2n}}}}e^{-r/na_{0}}\left({\frac {2r}{na_{0}}}\right)^{\ell }L_{n-\ell -1}^{2\ell +1}\left({\frac {2r}{na_{0}}}\right)\cdot Y_{\ell }^{m}(\theta ,\phi )} 6727:

In the following discussion involving spin, the complete wavefunction is considered as tensor product of spin states from finite dimensional Hilbert spaces and the wavefunction which was previously developed. The basis for this Hilbert space are hence considered:

16263: 13651:, in the lower right of each image. These are the principal quantum number, the orbital angular momentum quantum number, and the magnetic quantum number. Together with one spin-projection quantum number of the electron, this is a complete set of observables. 14449:) in which wave functions of interest can be expressed. There is also the artifact "normalization to a delta function" that is frequently employed for notational convenience, see further down. The delta functions themselves are not square integrable either. 9396: 12170:

evolves with time according to the Heisenberg equation of motion. The Dirac (or interaction) picture is intermediate, time dependence is places in both operators and states which evolve according to equations of motion. It is useful primarily in computing

2068: 11941: 11859: 8830: 3919:{\displaystyle {\begin{aligned}|\Psi \rangle =I|\Psi \rangle &=\int |x\rangle \langle x|\Psi \rangle dx=\int \Psi (x)|x\rangle dx,\\|\Psi \rangle =I|\Psi \rangle &=\int |p\rangle \langle p|\Psi \rangle dp=\int \Phi (p)|p\rangle dp.\end{aligned}}} 11384: 9246:{\displaystyle \Psi \left(\ldots \mathbf {r} _{a},\ldots ,\mathbf {r} _{b},\ldots ,\mathbf {x} _{1},\mathbf {x} _{2},\ldots \right)=\pm \Psi \left(\ldots \mathbf {r} _{b},\ldots ,\mathbf {r} _{a},\ldots ,\mathbf {x} _{1},\mathbf {x} _{2},\ldots \right)} 3656:

For another thing, though they are linearly independent, there are too many of them (they form an uncountable set) for a basis for physical Hilbert space. They can still be used to express all functions in it using Fourier transforms as described next.

14633:. The Legendre polynomials are ingredients in the spherical harmonics. Most problems with rotational symmetry will have "the same" (known) solution with respect to that symmetry, so the original problem is reduced to a problem of lower dimensionality. 13949:

used for the Hydrogen atomic wave functions. This final choice also fixes a basis in abstract Hilbert space. The basic states are labeled by the quantum numbers corresponding to the maximal set of commuting observables and an appropriate coordinate

12628:{\displaystyle \Psi (x)={\begin{cases}A_{\mathrm {r} }e^{ikx}+A_{\mathrm {l} }e^{-ikx}&x<-a,\\B_{\mathrm {r} }e^{\kappa x}+B_{\mathrm {l} }e^{-\kappa x}&|x|\leq a,\\C_{\mathrm {r} }e^{ikx}+C_{\mathrm {l} }e^{-ikx}&x>a.\end{cases}}} 1581:(or just fields where "operator" is understood) on the Hilbert space of states (to be described next section). It turns out that the original relativistic wave equations and their solutions are still needed to build the Hilbert space. Moreover, the 15551: 4831: 6973: 5763:

relating to the position or momentum of the particle. Nonetheless, the techniques developed for finite dimensional Hilbert space are useful since they can either be treated independently or treated in consideration of linearity of tensor product.

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are observed apply. These are usually formulated in the preservation of some quantum numbers. This means that certain processes allowable from some perspectives (e.g. energy and momentum conservation) do not occur because the initial and final

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In practice, the position-space wave function is used much more often than the momentum-space wave function. The potential entering the relevant equation (Schrödinger, Dirac, etc.) determines in which basis the description is easiest. For the

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number of particles and would not account for the term "interaction" as referred to in these theories, which involves the creation and annihilation of particles and not external potentials as in ordinary "first quantized" quantum theory.

5471: 2783:, and then any vector in the vector space can be expressed in this basis. This explains the relationship between a wave function in position space and a wave function in momentum space and suggests that there are other possibilities too. 16165:. This means that inner products, hence norms, are preserved and that the mapping is a bounded, hence continuous, linear bijection. The property of completeness is preserved as well. Thus this is the right concept of isomorphism in the 117:. Since the wave function is complex-valued, only its relative phase and relative magnitude can be measured; its value does not, in isolation, tell anything about the magnitudes or directions of measurable observables. One has to apply 8643: 16106:

The resulting basis may or may not technically be a basis in the mathematical sense of Hilbert spaces. For instance, states of definite position and definite momentum are not square integrable. This may be overcome with the use of

11139: 1557:, while being relativistic, do not represent full reconciliation of quantum mechanics and special relativity. The branch of quantum mechanics where these equations are studied the same way as the Schrödinger equation, often called 13117: 3185:

Finding the identity operator in a basis allows the abstract state to be expressed explicitly in a basis, and more (the inner product between two state vectors, and other operators for observables, can be expressed in the basis).

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For this statement to make sense, the observables need to be elements of a maximal commuting set. To see this, it is a simple matter to note that, for example, the momentum operator of the i'th particle in a n-particle system is

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For systems in time-independent potentials, the wave function can always be written as a function of the degrees of freedom multiplied by a time-dependent phase factor, the form of which is given by the Schrödinger equation. For

1503:. Pauli found the wave function was not described by a single complex function of space and time, but needed two complex numbers, which respectively correspond to the spin +1/2 and −1/2 states of the fermion. Soon after in 1928, 14380:

The above observations encapsulate the essence of the function spaces of which wave functions are elements. However, the description is not yet complete. There is a further technical requirement on the function space, that of

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In the figure of the hydrogen orbitals, the 19 sub-images are images of wave functions in position space (their norm squared). The wave functions represent the abstract state characterized by the triple of quantum numbers

8464: 6803: 2378: 5085: 320: 14768:. It is built from free single-particle states, i.e. wave functions when a representation is chosen, and can accommodate any finite, not necessarily constant in time, number of particles. The interesting (or rather the 5233: 12960:{\displaystyle \Psi _{n}(x)={\sqrt {\frac {1}{2^{n}\,n!}}}\cdot \left({\frac {m\omega }{\pi \hbar }}\right)^{1/4}\cdot e^{-{\frac {m\omega x^{2}}{2\hbar }}}\cdot H_{n}{\left({\sqrt {\frac {m\omega }{\hbar }}}x\right)}} 8033: 3935: 11560: 1684:, the situation remains analogous. For instance, a wave function in momentum space has the role of Fourier expansion coefficient in a general state of a particle (string) with momentum that is not sharply defined. 15695: 15001:{\displaystyle |\Psi \rangle =\sum _{\boldsymbol {\alpha }}\int d^{m}\!{\boldsymbol {\omega }}\,\,\Psi ({\boldsymbol {\alpha }},{\boldsymbol {\omega }},t)\,|{\boldsymbol {\alpha }},{\boldsymbol {\omega }}\rangle } 13764:

The Schrödinger equation is linear. This means that the solutions to it, wave functions, can be added and multiplied by scalars to form a new solution. The set of solutions to the Schrödinger equation is a vector

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is needed. In this theory, the wave equations and the wave functions have their place, but in a somewhat different guise. The main objects of interest are not the wave functions, but rather operators, so called

2631: 4685: 3652: 14166: 11461: 11134: 193:. The inner product between two wave functions is a measure of the overlap between the corresponding physical states and is used in the foundational probabilistic interpretation of quantum mechanics, the 3564: 3497:, since it is an eigenfunction of the momentum operator. These functions are not normalizable to unity (they are not square-integrable), so they are not really elements of physical Hilbert space. The set 2766: 15965:, the space of square integrable functions. The elements of this space are more precisely equivalence classes of square integrable functions, two functions declared equivalent if they differ on a set of 2891: 4521:

Following are the general forms of the wave function for systems in higher dimensions and more particles, as well as including other degrees of freedom than position coordinates or momentum components.

10551:{\displaystyle \rho \left(\mathbf {r} _{1}\cdots \mathbf {r} _{N},s_{z\,1}\cdots s_{z\,N},t\right)=\left|\Psi \left(\mathbf {r} _{1}\cdots \mathbf {r} _{N},s_{z\,1}\cdots s_{z\,N},t\right)\right|^{2}} 9276: 11746: 8686:

Traveling waves of two free particles, with two of three dimensions suppressed. Top is position-space wave function, bottom is momentum-space wave function, with corresponding probability densities.

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respectively) to extract from the tensor product the spaces in which the (total) spin wave functions reside. (Further problems arise in the relativistic case unless the particles are free. See the

11287: 15878:, take the more classical approach and regard the wave function as representing information in the mind of the observer, i.e. a measure of our knowledge of reality. Some, including Schrödinger, 14385:, that allows one to take limits of sequences in the function space, and be ensured that, if the limit exists, it is an element of the function space. A complete inner product space is called a 11739: 3685: 11907: 15466: 5609: 4878: 4736: 3180: 6884: 2768:

and is referred to as a "quantum state vector", or simply "quantum state". There are several advantages to understanding wave functions as representing elements of an abstract vector space:

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enters naturally in the discussion about wave functions. A function space is a set of functions, usually with some defining requirements on the functions (in the present case that they are

4276:{\displaystyle \langle x|p\rangle =p(x)={\frac {1}{\sqrt {2\pi \hbar }}}e^{{\frac {i}{\hbar }}px}\Rightarrow \langle p|x\rangle ={\frac {1}{\sqrt {2\pi \hbar }}}e^{-{\frac {i}{\hbar }}px},} 3291: 12704:

since no particles are coming from the right. By applying the continuity of wave functions and their derivatives at the boundaries, it is hence possible to determine the constants above.

4286: 3487: 11602: 8374:{\displaystyle |\psi (t)\rangle \!\otimes \!|\xi (t)\rangle =\sum _{s_{z}}\int d^{3}\!\mathbf {r} \,\psi (\mathbf {r} ,t)\,\xi (s_{z},t)\,|\mathbf {r} \rangle \!\otimes \!|s_{z}\rangle } 7818: 4389: 2134: 8016:{\displaystyle \Psi (\mathbf {r} ,t)={\begin{bmatrix}\Psi (\mathbf {r} ,s,t)\\\Psi (\mathbf {r} ,s-1,t)\\\vdots \\\Psi (\mathbf {r} ,-(s-1),t)\\\Psi (\mathbf {r} ,-s,t)\\\end{bmatrix}}} 14001:

For each choice of maximal commuting sets of observables for the abstract state space, there is a corresponding representation that is associated to a function space of wave functions.

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appear in the hydrogenic wave function problem after factoring out the spherical harmonics. These span the Hilbert space of square integrable functions on the semi-infinite interval

1318: 1258: 14298: 9037:(not identical with each other, and not identical to the aforementioned identical particles), the wave function is symmetric or antisymmetric in the identical particle coordinates 6848: 5559: 5128: 4927: 4726: 4608: 2645: 11666: 11634: 8566: 16251:

Some functions not being square-integrable, like the plane-wave free particle solutions are necessary for the description as outlined in a previous note and also further below.

6722: 16389:{\displaystyle \sum _{\boldsymbol {\alpha }}\equiv \sum _{\alpha _{1},\alpha _{2},\ldots ,\alpha _{n}}\equiv \sum _{\alpha _{1}}\sum _{\alpha _{2}}\cdots \sum _{\alpha _{n}}} 5691: 5644: 13020: 11498: 138:. The information represented by a wave function that is dependent upon position can be converted into a wave function dependent upon momentum and vice versa, by means of a 13959:

non-uniqueness in the choice of a maximal set of commuting observables. For one spin particle in one dimension, to a particular state there corresponds two wave functions,

7024: 2896: 2231: 158:. When a system has internal degrees of freedom, the wave function at each point in the continuous degrees of freedom (e.g., a point in space) assigns a complex number for 14416:

In summary, the set of all possible normalizable wave functions for a system with a particular choice of basis, together with the null vector, constitute a Hilbert space.

12103:{\displaystyle \Psi (\mathbf {r} _{1},\mathbf {r} _{2},\ldots ,\mathbf {r} _{N},t)=e^{-iEt/\hbar }\,\psi (\mathbf {r} _{1},\mathbf {r} _{2},\ldots ,\mathbf {r} _{N})\,,} 8953:{\displaystyle \Psi \left(\ldots \mathbf {r} _{a},\ldots ,\mathbf {r} _{b},\ldots \right)=\pm \Psi \left(\ldots \mathbf {r} _{b},\ldots ,\mathbf {r} _{a},\ldots \right)} 3231: 12232: 7600: 5226: 4992: 4961: 4490:

of each other. They are two representations of the same state; containing the same information, and either one is sufficient to calculate any property of the particle.

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Hilbert space. Due to the multiple possible choices of representation basis, these Hilbert spaces are not unique. One therefore talks about an abstract Hilbert space,

12204:. The amplitudes and direction of left and right moving waves are indicated. In red, those waves used for the derivation of the reflection and transmission amplitude. 8715: 4728:

spans the entire Hilbert space, thus leaving any vector from Hilbert space unchanged. This is also known as completeness relation of finite dimensional Hilbert space.

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a particle as being at a given place. The integral of a wavefunction's squared modulus over all the system's degrees of freedom must be equal to 1, a condition called

11005: 8468: 5831: 8384: 13166:. This is the only atom for which the Schrödinger equation has been solved exactly. Multi-electron atoms require approximative methods. The family of solutions is: 5148: 2267: 14890:, where the choice of representation and basis is left undetermined. Specifically, each state is represented as an abstract vector in state space. A quantum state 5379: 5003: 1174: 11270:{\textstyle \mathbf {J} (\mathbf {x} ,t)={\frac {\hbar }{2im}}(\psi ^{*}\nabla \psi -\psi \nabla \psi ^{*})={\frac {\hbar }{m}}{\text{Im}}(\psi ^{*}\nabla \psi )} 14452:

The above description of the function space containing the wave functions is mostly mathematically motivated. The function spaces are, due to completeness, very

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3D confined electron wave functions in a quantum dot. Here, rectangular and triangular-shaped quantum dots are shown. Energy states in rectangular dots are more

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The time dependence of the quantum state and the operators can be placed according to unitary transformations on the operators and states. For any quantum state

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In this case, the wave functions are square integrable. One can initially take the function space as the space of square integrable functions, usually denoted

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Then utilizing the known expression for suitably normalized eigenstates of momentum in the position representation solutions of the free Schrödinger equation

1657: 14401:. It is not very important in introductory quantum mechanics, and technical details and links may be found in footnotes like the one that follows. The space 11505: 5789: 5720: 154:, and the wave function for such particles includes spin as an intrinsic, discrete degree of freedom; other discrete variables can also be included, such as 18140: 13744:

on the set. The latter will sparsely be used here, it is only needed to obtain a precise definition of what it means for a subset of a function space to be

2433: 14222:. The explicit appearance of the inner product (usually an integral or a sum of integrals) depends on the choice of representation, but the complex number 2644:, meaning that it is possible to add together different wave functions, and multiply wave functions by complex numbers. Technically, wave functions form a 14397:

relies on the completeness of the space. These projection operators, in turn, are essential for the statement and proof of many useful theorems, e.g. the

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Once a representation is chosen, there is still arbitrariness. It remains to choose a coordinate system. This may, for example, correspond to a choice of

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It is a postulate of quantum mechanics that a physically observable quantity of a system, such as position, momentum, or spin, is represented by a linear

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This does not alter the structure of the Hilbert space that these particular wave functions inhabit, but the subspace of the square-integrable functions

2549: 18480: 13700:, there corresponds a basis wave function. If spin is taken into account, there are two basis functions for each triple. The function space thus has a 12221:

One of the most prominent features of wave mechanics is the possibility for a particle to reach a location with a prohibitive (in classical mechanics)

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The tensor product factorization of energy eigenstates is always possible if the orbital and spin angular momenta of the particle are separable in the

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For identical particles, symmetry requirements apply to both position and spin arguments of the wave function so it has the overall correct symmetry.

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The position-space wave function of a single particle without spin in three spatial dimensions is similar to the case of one spatial dimension above:

16132:, then the aforementioned limits will be in the function space. The inner product space is then called complete. A complete inner product space is a 10345:

The multidimensional Fourier transforms of the position or position–spin space wave functions yields momentum or momentum–spin space wave functions.

6731: 1585:, i.e. when interactions are assumed not to exist, turn out to (formally) satisfy the same equation as do the fields (wave functions) in many cases. 14868:, hence not a Hilbert space in itself. The functions that does not meet the requirements are still needed for both technical and practical reasons. 14205:

on the vector space of abstract quantum states, compatible with the mathematical observations above when passing to a representation. It is denoted

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where the derivative of wavefunction can be discontinuous at the boundary of the box where the potential is known to have infinite discontinuity.

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The formulae for the inner products are integrals over all coordinates or momenta and sums over all spin quantum numbers. For the general case of

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in which the spin dependence is placed in indexing the entries, and the wave function is a complex vector-valued function of space and time only.

19033: 9000:, i.e. no two having the same set of quantum numbers), there is no requirement for the wave function to be either symmetric or antisymmetric. 18681: 5198: 4964: 2661: 586: 18441: 3577: 13861:

The physical interpretation is that such a set represents what can – in theory – simultaneously be measured with arbitrary precision. The

3570:. This "basis" is not a basis in the usual mathematical sense. For one thing, since the functions are not normalizable, they are instead 1087: 16450: 2831: 14389:. The property of completeness is crucial in advanced treatments and applications of quantum mechanics. For instance, the existence of 8690:

If there are many particles, in general there is only one wave function, not a separate wave function for each particle. The fact that

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independent spin vector components, it is usually preferable to denote spin components using matrix/column/row notation as applicable.

19204: 14689:. The inner product is the standard inner product on these spaces. In it, the "spin part" of a single particle wave function resides. 6977:

All the previous remarks on inner products, momentum space wave functions, Fourier transforms, and so on extend to higher dimensions.

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which specify state of the quantum mechanical system, have magnitudes whose square gives the probability of measuring the respective

5369:{\displaystyle P_{\psi }(\lambda )=\sum _{j}|\langle \lambda ^{(j)}|\psi \rangle |^{2}=|{\widehat {P}}_{\lambda }|\psi \rangle |^{2}} 1396:. This relates calculations of quantum mechanics directly to probabilistic experimental observations. It is accepted as part of the 18530: 16075:

for a description of the Fourier transform as a unitary transformation. For eigenvalues and eigenvalues, refer to Problem 27 Ch. 9.

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While the space of solutions as a whole is a Hilbert space there are many other Hilbert spaces that commonly occur as ingredients.

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of quantum mechanics, wave functions can be added together and multiplied by complex numbers to form new wave functions and form a

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are not synonymous or equal to the column vectors. Column vectors simply provide a convenient way to express the spin components.

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The standard interpretation of this is as a stream of particles being fired at the step from the left (the direction of negative

8160:{\displaystyle |\Psi (t)\rangle =\sum _{s_{z}}\int d^{3}\!\mathbf {r} \,\Psi (\mathbf {r} ,s_{z},t)\,|\mathbf {r} ,s_{z}\rangle } 7773: 4111:{\displaystyle \int \Psi (x)\langle p|x\rangle dx=\int \Phi (p')\langle p|p'\rangle dp'=\int \Phi (p')\delta (p-p')dp'=\Phi (p).} 794: 18900: 18473: 15778:{\displaystyle 1=\sum _{{\boldsymbol {\alpha }}\in A}\int _{\Omega }\rho _{\alpha ,\omega }(t)\,d^{m}\!{\boldsymbol {\omega }}} 7047:

axis is an arbitrary choice; other axes can be used instead if the wave function is transformed appropriately, see below.) The

567: 217:. However, the wave function in quantum mechanics describes a kind of physical phenomenon, as of 2023 still open to different 43:. Panels (G–H) further show two different wave functions that are solutions of the Schrödinger equation but not standing waves. 17694: 15680:{\displaystyle P(t)=\sum _{{\boldsymbol {\alpha }}\in D}\int _{C}\rho _{\alpha ,\omega }(t)\,\,d^{m}\!{\boldsymbol {\omega }}} 14882:

As has been demonstrated, the set of all possible wave functions in some representation for a system constitute an in general

3114:{\displaystyle |\Psi \rangle =\int |x\rangle \langle x|\Psi \rangle dx=\left(\int |x\rangle \langle x|dx\right)|\Psi \rangle } 1499:

phenomenologically found a non-relativistic equation to describe spin-1/2 particles in electromagnetic fields, now called the

18834: 18428: 18401: 18349: 18330: 18311: 18277: 18252: 18231: 18212: 18105: 18046: 18023: 18000: 17969: 17942: 17918: 17881: 17851: 17744: 17676: 17651: 17493: 17467: 17444: 17205: 16962: 14456:

in a certain sense. Not all functions are realistic descriptions of any physical system. For instance, in the function space

750: 14319:

describes a state in the "distant future" (an "out state") after interactions between scattering particles have ceased, and

14245: 13760:

A wave function is an element of a function space partly characterized by the following concrete and abstract descriptions.

2686:

are components of a vector. There are uncountably infinitely many of them and integration is used in place of summation. In

18757: 18126: 15837: 11000:

In non-relativistic quantum mechanics, it can be shown using Schrodinger's time dependent wave equation that the equation:

6816: 2640:

For a given system, the set of all possible normalizable wave functions (at any given time) forms an abstract mathematical

1401: 673: 218: 13558: 13011: 6432:

of z-component spin operator are the above column vectors, with eigenvalues being the corresponding spin quantum numbers.

1533:. In the non-relativistic limit, the Dirac wave function resembles the Pauli wave function for the electron. Later, other 1488:. De Broglie also arrived at the same equation in 1928. This relativistic wave equation is now most commonly known as the 18193: 17723: 17704: 17227: 15843: 14431:. These are plane wave solutions of the Schrödinger equation for a free particle, but are not normalizable, hence not in 14407:

is a Hilbert space, with inner product presented later. The function space of the example of the figure is a subspace of

13017:

It is convenient to use spherical coordinates, and the wave function can be separated into functions of each coordinate,

5755:. However, the general wavefunction of a particle that fully describes its state, is always from an infinite dimensional 18369: 11675: 1334:, and this can be viewed as the starting point for the modern development of quantum mechanics. The equations represent 17646:. The Library of Living Philosophers. Vol. VII (3rd ed.). La Salle Publishing Company, Illinois: Open Court. 16609: 11874: 11672:. Substituting the form of wavefunction in Schrodinger's time dependent wave equation, and taking the classical limit, 2802:, whereas the idea that quantum states are complex-valued "wave" functions of space is only true in certain situations. 39:(blue) and imaginary part (red) of the wave function. Panels (C–F) show four different standing-wave solutions of the 18503: 18466: 17552: 16994: 5567: 4836: 1745: 333: 162:

possible value of the discrete degrees of freedom (e.g., z-component of spin). These values are often displayed in a

1727: 18613: 17396: 11916: 8170: 6987: 2395: 2197: 422: 110: 14549:. The latter space is a Hilbert space and the Fourier transform is an isomorphism of Hilbert spaces. Its basis is 9391:{\displaystyle \Psi (\mathbf {r} _{1},\mathbf {r} _{2}\cdots \mathbf {r} _{N},s_{z\,1},s_{z\,2}\cdots s_{z\,N},t)} 7770:

More generally, for a particle in 3d with any spin, the wave function can be written in "position–spin space" as:

5655:

probabilities of observing different states but relative phases can affect the expectation values of observables.

1572:

Relativity makes it inevitable that the number of particles in a system is not constant. For full reconciliation,

19171: 18880: 18875: 18598: 18173: 16989:

Physics for Scientists and Engineers – with Modern Physics (6th Edition), P. A. Tipler, G. Mosca, Freeman, 2008,

14784:

Due to the infinite-dimensional nature of the system, the appropriate mathematical tools are objects of study in

13862: 2088: 2063:{\displaystyle (\Psi _{1},\Psi _{2})=\int _{-\infty }^{\infty }\,\Psi _{1}^{*}(x,t)\Psi _{2}(x,t)\,dx<\infty } 1872: 1080: 529: 509: 377: 15984:. This is essential for completeness of the space, thus yielding a complete inner product space = Hilbert space. 14022:

There is an additional algebraic structure on the vector spaces of wave functions and the abstract state space.

11854:{\displaystyle {\frac {1}{2m}}|\nabla S(\mathbf {x} ,t)|^{2}+V(\mathbf {x} )+{\frac {\partial S}{\partial t}}=0} 11574: 6984:, ignoring the position degrees of freedom, the wave function is a function of spin only (time is a parameter); 3201: 19183: 18855: 13832: 13819:

This similarity is of course not accidental. There are also a distinctions between the spaces to keep in mind.

1723: 1719: 1558: 769: 499: 18423: 16019: 14609:

The most basic example of spanning polynomials is in the space of square integrable functions on the interval

12719:. However, in a triangular dot the wave functions are mixed due to confinement symmetry. (Click for animation) 11379:{\displaystyle \psi (\mathbf {x} ,t)={\sqrt {\rho (\mathbf {x} ,t)}}\exp {\frac {iS(\mathbf {x} ,t)}{\hbar }}} 7720:

is a solution of the Schrödinger equation (with a suitable Hamiltonian), which unfolds to a coupled system of

6435:

Corresponding to the notation, a vector from such a finite dimensional Hilbert space is hence represented as:

19161: 18938: 18860: 18643: 18513: 14754:. In this case, as well, the part of the wave functions corresponding to the inner symmetries reside in some 14665: 13855: 5476: 5153: 1629: 1613: 1290: 1230: 809: 547: 447: 14840:. This is motivated by the appearance of the Schrödinger equation for most physically reasonable potentials. 18895: 18829: 18824: 18795: 18508: 16006: 14014:

square integrable functions. One can then think of the function spaces as two distinct copies of that set.

13851: 11864: 2381: 1656:. Their solutions must transform under Lorentz transformation in a prescribed way, i.e. under a particular 745: 740: 711: 562: 343: 118: 106: 18963: 18870: 17167: 15546:{\displaystyle \rho _{\alpha ,\omega }(t)=|\Psi ({\boldsymbol {\alpha }},{\boldsymbol {\omega }},t)|^{2}} 15057: 14837: 14735: 12751: 4826:{\displaystyle |\psi \rangle =I|\psi \rangle =\sum _{i}|\phi _{i}\rangle \langle \phi _{i}|\psi \rangle } 1534: 779: 524: 514: 35: 16600: 16197:. This relaxation is necessary for potentials that are not functions but are distributions, such as the 14437:. But they are nonetheless fundamental for the description. One can, using them, express functions that 6968:{\displaystyle |\Psi (t)\rangle =\int d^{3}\!\mathbf {r} \,\Psi (\mathbf {r} ,t)\,|\mathbf {r} \rangle } 5523: 5092: 4891: 4690: 4572: 19239: 19001: 18809: 18780: 17843: 17790:

Hanle, P.A. (1977), "Erwin Schrodinger's Reaction to Louis de Broglie's Thesis on the Quantum Theory",

16215: 16186: 15902: 14649: 13752:. This observation is the foundation of the predominant mathematical formulation of quantum mechanics. 11868: 11639: 11607: 5658:

While the overall phase of the system is considered to be arbitrary, the relative phase for each state

3414:{\displaystyle (\Phi _{1},\Phi _{2})=\int _{-\infty }^{\infty }\,\Phi _{1}^{*}(p,t)\Phi _{2}(p,t)dp\,.} 2183: 2179: 1073: 725: 696: 14090:

Mathematically, it turns out that solutions to the Schrödinger equation for particular potentials are

6694: 2798:

are vectors in an abstract vector space is completely general in all aspects of quantum mechanics and

19023: 18890: 18814: 18775: 18732: 18706: 18663: 18556: 17617: 15859: 14829:. This is motivated by the Copenhagen interpretation of the wave function as a probability amplitude. 14648:

More generally, one may consider a unified treatment of all second order polynomial solutions to the

14178:

are (sets of) indices (quantum numbers) labeling different solutions, the strictly positive function

13654:

The figure can serve to illustrate some further properties of the function spaces of wave functions.

13014:(these are defined differently by different authors—see main article on them and the hydrogen atom). 8977: 8664: 4880:, is a set of complex numbers which can be used to construct a wavefunction using the above formula. 4377:{\displaystyle \Phi (p)={\frac {1}{\sqrt {2\pi \hbar }}}\int \Psi (x)e^{-{\frac {i}{\hbar }}px}dx\,.} 1550: 1489: 1474: 1397: 1335: 730: 691: 644: 619: 542: 402: 214: 201:

determines how wave functions evolve over time, and a wave function behaves qualitatively like other

93:. For example, a wave function might assign a complex number to each point in a region of space. The 14026:

Physically, different wave functions are interpreted to overlap to some degree. A system in a state

13669:

The displayed functions are solutions to the Schrödinger equation. Obviously, not every function in

12372: 12256: 12183:

The following are solutions to the Schrödinger equation for one non-relativistic spinless particle.

5661: 5614: 4477:{\displaystyle \Psi (x)={\frac {1}{\sqrt {2\pi \hbar }}}\int \Phi (p)e^{{\frac {i}{\hbar }}px}dp\,.} 19090: 19070: 19060: 19050: 19006: 18581: 14883: 14369: 13675:

satisfies the Schrödinger equation for the hydrogen atom. The function space is thus a subspace of

13595: 13580: 12735: 11466: 10625:

is the integral of the probability density over these regions and evaluated at these spin numbers:

4930: 2649: 2399: 2175: 1801: 1708: 814: 1652:) for the field operators. All of them are essentially a direct consequence of the requirement of 19260: 18785: 18711: 18438: 18418: 15922: 13915: 13621: 8657: 3182:

which is analogous to completeness relation of orthonormal basis in N-dimensional Hilbert space.

2242: 1712: 1481: 1439: 1431: 983: 701: 609: 186: 121:, whose eigenvalues correspond to sets of possible results of measurements, to the wave function 19199: 18747: 18433: 17770: 15927: 15842:

Whether the wave function exists in reality, and what it represents, are major questions in the

14652:

in the setting of Hilbert space. These include the Legendre and Laguerre polynomials as well as

4610:

is orthonormal, then the projection operator for the space spanned by these states is given by:

1377: 659: 557: 323: 198: 40: 19110: 18885: 18865: 18790: 18658: 18286: 17567:"Über einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen Gesichtspunkt" 17350: 15932: 15907: 15871: 13954:

The abstract states are "abstract" only in that an arbitrary choice necessary for a particular

12707: 2780: 1525:

represented by four complex-valued components: two for the electron and two for the electron's

1384: 988: 706: 534: 504: 467: 131: 31: 18701: 17734: 15994: 13827:

Basic states are characterized by a set of quantum numbers. This is a set of eigenvalues of a

13807:

counts as a valid state ("no system present") is a matter of definition. The null vector does

7122:). Inserting each quantum number gives a complex valued function of space and time, there are 5466:{\textstyle {\widehat {P}}_{\lambda }=\sum _{j}|\lambda ^{(j)}\rangle \langle \lambda ^{(j)}|} 5204: 4970: 4939: 2171: 1208: 614: 19135: 18648: 18628: 17547:. The international series on monographs on physics (4th ed.). Oxford University Press. 16447: 15875: 14653: 14382: 13946: 13919: 11669: 4542: 4535: 2257: 1788: 1661: 1478: 1393: 457: 442: 98: 16210:

It is easy to visualize a sequence of functions meeting the requirement that converges to a

12993:

for the wave function of the electron. Different orbitals are depicted with different scale.

5794: 19166: 19095: 19040: 18770: 18593: 18551: 18157: 18068: 17626: 17580: 17512: 17327: 17283: 17246: 17176: 16579: 16198: 15798: 14637: 14614: 14210: 13868:

The set is non-unique. It may for a one-particle system, for example, be position and spin

8699: 8648: 8638:{\displaystyle |\mathbf {r} ,s_{z}\rangle =|\mathbf {r} \rangle \!\otimes \!|s_{z}\rangle } 7594: 5833: 5694:

spin along z states which provides appropriate phase of the states relative to each other.

2799: 2786: 2687: 2665: 1763: 1573: 1454: 1443: 774: 686: 412: 369: 18136: 17386:

de Broglie, L. (1923). "Radiations—Ondes et quanta" [Radiation—Waves and quanta].

15847: 13854:

of the operator. At a deeper level, most observables, perhaps all, arise as generators of

5702:

An example of finite dimensional Hilbert space can be constructed using spin eigenkets of

5133: 1354: 1150: 1018: 8: 19151: 19120: 19065: 19045: 18953: 18910: 18765: 18691: 18618: 18608: 18520: 17639: 17399: 16884: 16771:

takes the standpoint that quantum field theory appears the way it does because it is the

16238: 16121: 16064: 15426: 14833: 14785: 14661: 14630: 14390: 14219: 13729: 13007: 11565: 11278: 8997: 8981: 8818: 4545: 4538: 4495: 873: 681: 599: 427: 407: 359: 18161: 18072: 17630: 17584: 17516: 17331: 17287: 17250: 17180: 17142: 19211: 19080: 18978: 18686: 18633: 18525: 18450: 18084: 18012: 17823: 17807: 17759: 17571: 17528: 17436: 17410: 17299: 17262: 15432: 14777: 14657: 13847: 13625: 13151: 13112:{\displaystyle \Psi _{n\ell m}(r,\theta ,\phi )=R(r)\,\,Y_{\ell }^{m}\!(\theta ,\phi )} 12755: 11928: 11668:

as velocity since velocity and position cannot be simultaneously determined as per the

4557: 2162: 1882: 1844:

dimension. The colour opacity of the particles corresponds to the probability density (

1653: 1508: 1485: 1435: 1388: 1331: 1265: 1184: 1128: 1107: 594: 519: 452: 364: 222: 12335:

and the steady-state solutions to the wave equation have the form (for some constants

11938:

particles, considering their positions only and suppressing other degrees of freedom,

11636:

is analogous with velocity. Note that this does not imply a literal interpretation of

5839: 5725: 3238: 2991:{\displaystyle \langle x'|\Psi \rangle =\int \Psi (x)\langle x'|x\rangle dx=\Psi (x')} 19221: 19130: 19100: 19028: 18991: 18986: 18968: 18933: 18923: 18638: 18603: 18586: 18489: 18397: 18389: 18345: 18326: 18307: 18290: 18273: 18248: 18242: 18227: 18208: 18189: 18122: 18101: 18088: 18042: 18019: 17996: 17975: 17965: 17938: 17932: 17914: 17877: 17847: 17818: 17811: 17774: 17740: 17719: 17700: 17672: 17666: 17647: 17548: 17532: 17489: 17463: 17440: 17430: 17374: 17303: 17266: 17223: 17201: 16990: 16958: 16502: 16194: 16146: 16120:

In technical terms, this is formulated the following way. The inner product yields a

16068: 15981: 15846:. Many famous physicists of a previous generation puzzled over this problem, such as 15165: 14846: 14826: 14669: 14521: 14364:

the theory at hand, at least as far as predictions go. Measurable quantities such as

14309: 14006: 13741: 13725: 12990: 12742:

model in which the energy of different states is dependent on the length of the box.

12739: 12638: 12226: 9255:

Again, there is no symmetry requirement for the distinguishable particle coordinates

4553: 4487: 3122: 2669: 2391: 2256:

For one spinless particle in one dimension, if the wave function is interpreted as a

1773: 1645: 1413: 1381: 1362: 1028: 1003: 943: 938: 838: 804: 784: 382: 241: 139: 26: 17958: 17460:

Heisenberg and the Interpretation of Quantum Mechanics: the Physicist as Philosopher

14352:

varying over a complete set of in states and out states respectively, is called the

13684:

The displayed functions form part of a basis for the function space. To each triple

18948: 18943: 18800: 18696: 18377:. National Institute of Standards and Technology. pp. 1 (55 s). Archived from 18258: 18165: 18076: 18029: 17983: 17948: 17906: 17888: 17830: 17799: 17782: 17682: 17588: 17520: 17450: 17426: 17417: 17405: 17366: 17362: 17335: 17291: 17254: 17184: 16602:

C/CS Pys C191:Representations and Wave Functions 》 1. Planck-Einstein Relation E=hv

16150: 15966: 15883: 15867: 14398: 12222: 12119: 5774: 5705: 1777: 1637: 1370: 1350: 1033: 1023: 1013: 913: 893: 878: 848: 716: 604: 209:

or waves on a string, because the Schrödinger equation is mathematically a type of

17155: 15302:, neglecting other degrees of freedom, using Cartesian coordinates, we could take 14226:

does not. Much of the physical interpretation of quantum mechanics stems from the

12328:{\displaystyle V(x)={\begin{cases}V_{0}&|x|<a\\0&|x|\geq a\end{cases}}} 8980:. Generally, bosonic and fermionic symmetry requirements are the manifestation of 8663:

The preceding discussion is not limited to spin as a discrete variable, the total

7705:{\displaystyle |\xi (t)\rangle =\sum _{s_{z}=-s}^{s}\xi (s_{z},t)\,|s_{z}\rangle } 2828:

that are normalizable to unity, can only be normalized to a Dirac delta function.

2398:, its location cannot be determined from the wave function, but is described by a 19178: 19105: 19085: 19055: 19018: 19013: 18918: 18742: 18445: 18265: 18148: 17910: 17865: 17562: 17485: 17370: 16454: 16166: 16129: 15998: 15851: 14194: 13850:

on the state space. The possible outcomes of measurement of the quantity are the

13701: 10293: 8779:{\displaystyle \Psi (\mathbf {r} _{1},\mathbf {r} _{2}\cdots \mathbf {r} _{N},t)} 6857: 1900: 1864:

For now, consider the simple case of a non-relativistic single particle, without

1461:

he published the non-relativistic one, but discarded it as it predicted negative

1405: 1376:

In 1926, Schrödinger published the famous wave equation now named after him, the

1283: 1058: 928: 908: 654: 494: 102: 56: 16888: 14796: 11052:{\displaystyle {\frac {\partial \rho }{\partial t}}+\nabla \cdot \mathbf {J} =0} 8557:{\displaystyle \Psi (\mathbf {r} ,s_{z},t)=\psi (\mathbf {r} ,t)\,\xi (s_{z},t)} 3932:

onto eigenfunctions of momentum using the last expression in the two equations,

19265: 19156: 19125: 19115: 18737: 18727: 18561: 18056: 17928: 17873: 17766: 17690: 17662: 17477: 16002: 14773: 14727: 14723: 14715: 14701: 14218:. It yields a complex number. With the inner product, the function space is an 14079: 13721: 12986: 8653: 8459:{\displaystyle |\Psi (t)\rangle =|\psi (t)\rangle \!\otimes \!|\xi (t)\rangle } 7036: 6981: 6879: 6798:{\displaystyle |\mathbf {r} ,s_{z}\rangle =|\mathbf {r} \rangle |s_{z}\rangle } 6206:

Corresponding to the notation, the z-component spin operator can be written as:

3195: 2773: 2633:

because if the particle is measured, there is 100% probability that it will be

2261: 1903: 1865: 1605: 1554: 1516: 1500: 1496: 1447: 1346: 993: 953: 933: 903: 883: 833: 799: 649: 639: 432: 151: 90: 72: 17524: 13918:

in the position representation) and the operator corresponding to momentum (a

2373:{\displaystyle \left|\Psi (x,t)\right|^{2}=\Psi ^{*}(x,t)\Psi (x,t)=\rho (x),} 19254: 19075: 18928: 18819: 18623: 18576: 18378: 17754: 17593: 17566: 16179: 16133: 15863: 14877: 14747: 14731: 14386: 14202: 13749: 13737: 12982: 7131: 5768: 5760: 5756: 5752: 5080:{\displaystyle P_{\psi }(\lambda _{i})=|\langle \phi _{i}|\psi \rangle |^{2}} 4531: 2795: 2182:

which can be added together in various combinations to create every possible

1910: 1906: 1876: 1769: 1681: 1409: 1053: 1048: 978: 948: 918: 789: 735: 437: 315:{\displaystyle i\hbar {\frac {d}{dt}}|\Psi \rangle ={\hat {H}}|\Psi \rangle } 210: 190: 163: 68: 20: 18169: 17895: 17220:

Niels Bohr - Collected Works: Foundations of Quantum Physics I (1926 - 1932)

14664:. All of these actually appear in physical problems, the latter ones in the 13748:. It will be concluded below that the function space of wave functions is a 8682: 1373:". Schrödinger subsequently showed that the two approaches were equivalent. 18958: 18571: 18566: 17378: 16125: 15912: 15365:

is the set of all possible particle positions throughout 3d position space.

14821:

formulated for the calculations and physical interpretation to make sense:

14746:. The models of the nuclear forces of the sixties (still useful today, see 14722:

With more particles, the situations is more complicated. One has to employ

13865:

prohibits simultaneous exact measurements of two non-commuting observables.

13812: 13733: 3493:, which can be used in the description of a particle with momentum exactly 2641: 2194:

The state of such a particle is completely described by its wave function,

1526: 1520: 1453:

Schrödinger did encounter an equation for the wave function that satisfied

1043: 1038: 973: 958: 923: 417: 86: 16145:

As is explained in a later footnote, the integral must be taken to be the

15152:

These quantum numbers index the components of the state vector. More, all

4486:

The position-space and momentum-space wave functions are thus found to be

3660: 2806:

The time parameter is often suppressed, and will be in the following. The

47: 18996: 16814: 16108: 15937: 15272: 14800:

Continuity of the wave function and its first spatial derivative (in the

14621: 14446: 14442: 13804: 13708: 13533: 12977: 12731: 12724: 8703: 6429: 1470: 1462: 1323: 1008: 963: 898: 853: 13815:

in quantum field theory.) The set of allowable states is a vector space.

3423:

One particular solution to the time-independent Schrödinger equation is

3256:

Analogous to the position case, the inner product of two wave functions

1604:) in this guise remain in the theory. Higher spin analogues include the 18080: 17861: 17540: 17295: 17258: 17215: 16957:(3rd ed.). Cambridge: Cambridge University Press. pp. 94–97. 16644: 15879: 15855: 15316:

for the spin quantum number of the particle along the z direction, and

14765: 14365: 13835: 13745: 12114:

is the energy eigenvalue of the system corresponding to the eigenstate

11555:{\displaystyle \mathbf {J} (\mathbf {x} ,t)={\frac {\rho \nabla S}{m}}} 11281:

in accordance with the continuity equation form of the above equation.

4933: 3490: 1562: 1540: 1504: 1202: 1101: 998: 968: 888: 863: 858: 843: 206: 127:

and calculate the statistical distributions for measurable quantities.

17188: 14445:. They are, in a sense, a basis (but not a Hilbert space basis, nor a 14419:

Not all functions of interest are elements of some Hilbert space, say

10361:

is interpreted as a probability amplitude, the probability density is

6878:

is a complex-valued function of real variables. As a single vector in

2776:

can be used to manipulate and understand wave functions. For example:

2531:{\displaystyle P_{a\leq x\leq b}(t)=\int _{a}^{b}\,|\Psi (x,t)|^{2}dx} 1687: 18850: 18546: 17340: 17315: 14227: 4995: 1665: 1566: 1427: 1424: 489: 194: 94: 18458: 14780:

is the most common choice (constant states, time varying operators).

14413:. A subspace of a Hilbert space is a Hilbert space if it is closed. 14062:

degree, there is a chance that measurement of a system described by

1868:, in one spatial dimension. More general cases are discussed below. 1697: 75:. The most common symbols for a wave function are the Greek letters 17995:. Manchester Physics Series (3rd ed.). John Wiley & Sons. 17803: 17311: 16466: 16190: 15959: 14726:

and use representation theory of the symmetry groups involved (the

14353: 13660: 13631:

This solution does not take into account the spin of the electron.

12172: 7071: 2626:{\displaystyle \int _{-\infty }^{\infty }\,|\Psi (x,t)|^{2}dx=1\,,} 1530: 1512: 1366: 1342: 1179: 868: 143: 135: 17905:. Berlin, Heidelberg: Springer Berlin Heidelberg. pp. 64–70. 16787:

See especially chapter 5, where some of these results are derived.

14871: 12153:

is constant. In the Heisenberg picture it is the other way round,

2542:

is the time at which the particle was measured. This leads to the

17668:

Quantum Physics of Atoms, Molecules, Solids, Nuclei and Particles

17156:"Einstein's proposal of the photon concept: A translation of the 16178:

One such relaxation is that the wave function must belong to the

16111:

or by enclosing the system in a "box". See further remarks below.

16005:

rather than an orthonormal basis in the sense of linear algebra (

15917: 14739: 12728: 8671: 4680:{\displaystyle P=\sum _{i}|\phi _{i}\rangle \langle \phi _{i}|=I} 2674:

At a particular instant of time, all values of the wave function

1286:. In 1923, De Broglie was the first to suggest that the relation 155: 17979: 17778: 17503:

Dirac, P. A. M. (1939). "A new notation for quantum mechanics".

16668: 16097:

a generator of a symmetry in nature; the translational symmetry.

16050:

The Fourier transform viewed as a unitary operator on the space

8816:

In quantum mechanics there is a fundamental distinction between

101:

into actual probabilities. In one common form, it says that the

17505:

Mathematical Proceedings of the Cambridge Philosophical Society

16571: 15268: 14679:

There occurs also finite-dimensional Hilbert spaces. The space

9003:

For a collection of particles, some identical with coordinates

1466: 147: 51:

The wave function of an initially very localized free particle.

18141:"An Undulatory Theory of the Mechanics of Atoms and Molecules" 17149:. Department of Quantum Nanoscience studies at TU Delft. 2022. 16680: 16089:

a generator of any symmetry in nature. On the other hand, the

14858:, which is a Hilbert space, satisfying the second requirement 14477:

square integrable, but can hardly represent a physical state.

14161:{\displaystyle \int \Psi _{m}^{*}\Psi _{n}w\,dV=\delta _{nm},} 12147:

changes with time according to the Schrödinger equation while

11456:{\textstyle \rho (\mathbf {x} ,t)=|\psi (\mathbf {x} ,t)|^{2}} 11129:{\textstyle \rho (\mathbf {x} ,t)=|\psi (\mathbf {x} ,t)|^{2}} 4883: 4687:

where the projection is equivalent to identity operator since

1660:

and that together with few other reasonable demands, e.g. the

1341:

In the 1920s and 1930s, quantum mechanics was developed using

17141: 17023: 16820: 16709: 15897: 14751: 14743: 14629:

is a Hilbert space. The basis functions in this case are the

12658:

corresponds to firing particles singly; the terms containing

3559:{\displaystyle \{\Psi _{p}(x,t),-\infty \leq p\leq \infty \}} 2761:{\displaystyle |\Psi (t)\rangle =\int \Psi (x,t)|x\rangle dx} 2384:

for a measurement of the particle's position at a given time

213:. This explains the name "wave function", and gives rise to 17047: 15577:

in some or all possible continuous-variable configurations,

14668:, and what is otherwise a bewildering maze of properties of 12191: 8677: 1361:". Those who applied the methods of linear algebra included 197:, relating transition probabilities to inner products. The 16775:

way to reconcile quantum mechanics with special relativity.

15785:

must hold at all times during the evolution of the system.

14693:

In the non-relativistic description of an electron one has

14489:

Square integrable complex valued functions on the interval

12621: 12321: 8026:

All values of the wave function, not only for discrete but

6808: 1507:

found an equation from the first successful unification of

1358: 202: 18205:

Physics for Scientists and Engineers – with Modern Physics

18121:. Vol. 2 (5th ed.). Taylor & Francis Group. 18059:(1927). "Zur Quantenmechanik des magnetischen Elektrons". 17237:

Born, M. (1926a). "Zur Quantenmechanik der Stoßvorgange".

17059: 16999: 16922: 16910: 16540: 15561:

in some or all possible discrete-variable configurations,

14197:. The integration is taken over all of the relevant space. 14005:

functions, for instance, describing the same state is the

13914:. In this case, the operator corresponding to position (a 8805:

is time. Altogether, this is a complex-valued function of

8670:

may also be used. Other discrete degrees of freedom, like

5649: 5473:

is a projection operator of states to subspace spanned by

2779:

Linear algebra explains how a vector space can be given a

1178:

and in 1916 the corresponding relation between a photon's

17615:

Einstein, A. (1917). "Zur Quantentheorie der Strahlung".

17602:

Einstein, A. (1916). "Zur Quantentheorie der Strahlung".

16865: 16750: 15972:. This is necessary to obtain an inner product (that is, 14896:

in any representation is generally expressed as a vector

14517:

is a Hilbert space basis, i.e. a maximal orthonormal set.

14326:

an "in state" in the "distant past", then the quantities

14094:

in some manner, this is usually described by an integral

4552:. For every finite dimensional Hilbert space there exist 1430:

to approximate the solution. Now it is also known as the

17825:

Physics and Philosophy: the Revolution in Modern Science

16855: 16853: 16656: 16484: 16018:

As, technically, they are not in the Hilbert space. See

14764:

In quantum field theory the underlying Hilbert space is

11867:

from classical mechanics. This interpretation fits with

11500:

is the phase of the wavefunction, it can be shown that:

9400:

Accumulating all these components into a single vector,

9273:

particles each with spin is the complex-valued function

7597:, these easily arrange into the components of a vector: 170:

column vector for a non-relativistic electron with spin

17119: 16826: 16699: 16697: 16695: 16552: 16528: 12637:

Note that these wave functions are not normalized; see

11564:

Hence the spacial variation of phase characterizes the

6190:

but it is a common abuse of notation, because the kets

3661:

Relations between position and momentum representations

2075: 1848:

the wave function) of finding the particle at position

1561:, while very successful, has its limitations (see e.g. 18439:

Quantum Mechanics and Quantum Computation at BerkeleyX

17274:

Born, M. (1926b). "Quantenmechanik der Stoßvorgange".

16934: 16838: 16738: 16472: 15382:

for the spin quantum number along the y direction and

11877: 11678: 11642: 11610: 11577: 11469: 11392: 11142: 11065: 10300:

sums over the spins. The differential volume elements

9678: 7855: 7539: 7468: 7379: 7305: 7153: 6697: 6593: 6463: 6250: 6136: 6067: 5979: 5907: 5842: 5797: 5777: 5728: 5708: 5664: 5617: 5570: 5526: 5479: 5382: 5207: 5156: 5136: 5095: 4973: 4942: 4894: 4839: 4693: 4575: 18302:

Young, H. D.; Freedman, R. A. (2008). Pearson (ed.).

18095: 17351:"The statistical interpretation of quantum mechanics" 17083: 16850: 16728: 16726: 16724: 16266: 16185:. It means that it is differentiable in the sense of 15698: 15590: 15469: 15435: 15410:

for the particle's momentum components. In this case

14902: 14772:) dynamics lies not in the wave functions but in the 14248: 14100: 13768:

The superposition principle of quantum mechanics. If

13715: 13172: 13023: 12772: 12351: 12235: 11944: 11749: 11508: 11290: 11008: 10634: 10367: 9829: 9408: 9279: 9054: 8833: 8718: 8674:, can expressed similarly to the case of spin above. 8569: 8471: 8387: 8202: 8036: 7826: 7776: 7603: 7141: 6990: 6887: 6819: 6734: 6443: 6212: 5887: 5236: 5006: 4739: 4618: 4392: 4289: 4126: 3938: 3683: 3647:{\displaystyle (\Psi _{p},\Psi _{p'})=\delta (p-p').} 3580: 3503: 3429: 3294: 3204: 3130: 3004: 2899: 2834: 2696: 2552: 2436: 2270: 2200: 2091: 1940: 1885: 1349:. Those who used the techniques of calculus included 1293: 1268: 1233: 1211: 1187: 1153: 1131: 1110: 1104:

postulated the proportionality between the frequency

253: 105:

of a wave function that depends upon position is the

18224:

Topological Vector Spaces, Distributions and Kernels

18096:

Peleg, Y.; Pnini, R.; Zaarur, E.; Hecht, E. (2010).

17011: 16898: 16802: 16692: 16516: 15148:

continuous variables (not necessarily dimensionless)

14617:

is a Hilbert space basis (complete orthonormal set).

7820:

and these can also be arranged into a column vector

5201:, the probability of measuring the observable to be 4967:, the probability of measuring the observable to be 1541:

Wave functions and wave equations in modern theories

17604:

Mitteilungen der Physikalischen Gesellschaft Zürich

14425:. The most glaring example is the set of functions 12981:The electron probability density for the first few 12694:– to the left. Under this beam interpretation, put 12195:Scattering at a finite potential barrier of height 11734:{\textstyle \hbar |\nabla ^{2}S|\ll |\nabla S|^{2}} 8984:and are present in other quantum state formalisms. 5520:. The equality follows due to orthogonal nature of 4548:as well. In physics, they are often referred to as 4525: 2430:is the integral of the density over this interval: 1688:

Definition (one spinless particle in one dimension)

18202: 18011: 17957: 17822: 17758: 17718:(2nd ed.). Essex England: Pearson Education. 17412:Non-linear Wave Mechanics: a Causal Interpretation 17409: 17107: 17035: 16721: 16686: 16388: 15777: 15689:Since the sum of all probabilities must be 1, the 15679: 15545: 15441: 15000: 14776:that are operators acting on Fock space. Thus the 14672:becomes an organized body of facts. For this, see 14524:takes functions in the above space to elements of 14462:one can find the function that takes on the value 14292: 14160: 13497: 13111: 12959: 12738:. The energy levels can then be modeled using the 12627: 12327: 12102: 11902:{\textstyle \mathbf {P} _{\text{class.}}=\nabla S} 11901: 11853: 11733: 11660: 11628: 11596: 11554: 11492: 11455: 11378: 11269: 11128: 11051: 10984: 10550: 10280: 9804: 9390: 9245: 8952: 8778: 8637: 8556: 8458: 8373: 8159: 8015: 7812: 7704: 7583: 7018: 6967: 6842: 6797: 6716: 6683: 6418: 6180: 5857: 5825: 5783: 5743: 5714: 5685: 5638: 5603: 5553: 5512: 5465: 5368: 5220: 5189: 5142: 5122: 5079: 4986: 4955: 4921: 4872: 4825: 4720: 4679: 4602: 4476: 4376: 4275: 4110: 3918: 3646: 3558: 3481: 3413: 3225: 3174: 3113: 2990: 2886:{\displaystyle \langle x'|x\rangle =\delta (x'-x)} 2885: 2760: 2625: 2530: 2372: 2225: 2128: 2062: 1891: 1312: 1274: 1252: 1217: 1193: 1168: 1137: 1116: 314: 18100:. Schaum's outlines (2nd ed.). McGraw Hill. 17095: 16952: 16633:, pp. 47–62, and a nearly identical version 16063:. The eigenvectors are "Hermite functions", i.e. 16034:Also called "Dirac orthonormality", according to 15957:The functions are here assumed to be elements of 15769: 15671: 14940: 14738:.) Corresponding remarks apply to the concept of 14700:and the total wave function is a solution of the 13093: 10995: 10560:and the probability that particle 1 is in region 8616: 8612: 8435: 8431: 8352: 8348: 8281: 8227: 8223: 8090: 7091:, and not any other value. (In general, for spin 6924: 2078:. However, the inner product of a wave function 19252: 17689: 17053: 15858:. Some advocate formulations or variants of the 14760:or subspaces of tensor products of such spaces. 12989:shown as cross-sections. These orbitals form an 11284:Using the following expression for wavefunction: 5604:{\textstyle \{\langle \phi _{i}|\psi \rangle \}} 4873:{\textstyle \{\langle \phi _{i}|\psi \rangle \}} 4541:they, by definition, include finite dimensional 3189: 3175:{\displaystyle I=\int |x\rangle \langle x|dx\,.} 2189: 17484:. Graduate Texts in Mathematics. Vol. 96. 17071: 16214:function. For this, modify an example given in 15358:is the set of allowed spin quantum numbers and 14872:More on wave functions and abstract state space 13922:in the position representation) do not commute. 8706:possible. The position-space wave function for 8196:gives the composite position-spin state vector 7728:ordinary differential equations with solutions 5791:-spin particles can be represented as a finite 2264:of the wave function, the positive real number 1899:, is given by the wave function belonging to a 1412:made the first step in an attempt to solve the 17860: 17660: 17065: 16928: 16608:. EESC Instructional and Electronics Support, 16546: 16505:(1927/1985/2009). Heisenberg is translated by 15335:for the particle's position coordinates. Here 15025:the basis vectors of the chosen representation 12745: 12229:", the one-dimensional case has the potential 5697: 3241:in one dimension, which can be any value from 1927:can be defined as the complex number (at time 18474: 18429:Identical Particles Revisited, Michael Fowler 18301: 17435:. Dover Books on Physics (revised ed.). 16715: 16435: 16433: 16416: 16414: 16412: 14707:In the corresponding relativistic treatment, 12727:whose radius is smaller than the size of its 10348: 8801:-th particle in three-dimensional space, and 2662:Mathematical formulation of quantum mechanics 1477:and Fock also found it, but incorporated the 1081: 18323:Quantum Mechanics: Concepts and Applications 18285: 18039:Quantum Physics: a Text for Graduate Student 18018:. Cambridge UK: Cambridge University Press. 17462:. Cambridge UK: Cambridge University Press. 17432:Mathematics of Classical and Quantum Physics 17153: 16424: 15584:, is the sum and integral over the density, 14995: 14911: 12178: 9783: 9417: 8632: 8609: 8593: 8453: 8428: 8405: 8368: 8345: 8245: 8220: 8154: 8054: 7699: 7621: 6962: 6905: 6792: 6774: 6758: 6711: 6698: 6574: 6557: 6550: 6521: 6507: 6487: 6480: 6466: 6452: 6125: 6056: 5968: 5896: 5680: 5633: 5598: 5595: 5574: 5571: 5548: 5545: 5527: 5507: 5504: 5480: 5439: 5436: 5351: 5301: 5274: 5184: 5181: 5157: 5117: 5114: 5096: 5062: 5041: 4916: 4913: 4895: 4867: 4864: 4843: 4840: 4820: 4799: 4796: 4765: 4748: 4715: 4712: 4694: 4653: 4650: 4597: 4594: 4576: 4516: 4386:Likewise, using eigenfunctions of position, 4220: 4206: 4141: 4127: 4019: 4000: 3968: 3954: 3900: 3865: 3851: 3848: 3827: 3810: 3786: 3751: 3737: 3734: 3713: 3696: 3553: 3504: 3482:{\displaystyle \Psi _{p}(x)=e^{ipx/\hbar },} 3151: 3148: 3108: 3078: 3075: 3047: 3033: 3030: 3013: 2959: 2940: 2919: 2900: 2854: 2835: 2749: 2714: 2655: 2117: 2110: 1636:free fields two examples are the free field 309: 283: 18297:. Princeton NJ: Princeton University Press. 18247:, vol. 1, Cambridge University Press, 18135: 17927: 17424: 16953:Sakurai, Jun John; Napolitano, Jim (2021). 16674: 16572:"Planck - A very short biography of Planck" 16072: 14673: 14473:for the irrationals in the interval . This 12186: 11597:{\textstyle \mathbf {J} =\rho \mathbf {v} } 7813:{\displaystyle \Psi (\mathbf {r} ,s_{z},t)} 4884:Probability interpretation of inner product 2129:{\displaystyle (\Psi ,\Psi )=\|\Psi \|^{2}} 1726:. Unsourced material may be challenged and 1400:of quantum mechanics. There are many other 221:, which fundamentally differs from that of 18481: 18467: 18203:Tipler, P. A.; Mosca, G.; Freeman (2008). 17990: 17870:Quantum Mechanics: Non-Relativistic Theory 17817: 17404: 17385: 16744: 16650: 16534: 16490: 16430: 16409: 14360:. Knowledge of it is, effectively, having 9700: 9582: 2405: 1820:, and corresponding probability densities 1088: 1074: 134:of variables other than position, such as 18434:The Nature of Many-Electron Wavefunctions 18394:Quantum Theory, A Very Short Introduction 17829:. New York: Harper & Row – via 17732: 17713: 17592: 17457: 17339: 17222:. Vol. 6. Amsterdam: North Holland. 17005: 16859: 16832: 16506: 16001:, implies it admits a countably infinite 15758: 15660: 15659: 14976: 14947: 14946: 14791: 14132: 13755: 13077: 13076: 12812: 12118:. Wave functions of this form are called 12096: 12038: 10740: 10676: 10520: 10503: 10431: 10414: 10261: 10244: 10174: 10157: 10058: 9999: 9943: 9920: 9899: 9875: 9798: 9777: 9754: 9662: 9639: 9465: 9442: 9373: 9356: 9339: 8678:Many-particle states in 3d position space 8528: 8334: 8308: 8287: 8130: 8096: 7683: 6951: 6930: 6109: 6028: 6021: 5949: 5881:is usually identified as a column vector: 4534:originally refer to infinite dimensional 4470: 4370: 3407: 3345: 3194:The particle also has a wave function in 3168: 2789:can be used to manipulate wave functions. 2619: 2571: 2486: 2219: 2047: 1991: 1746:Learn how and when to remove this message 1338:for both massless and massive particles. 97:provides the means to turn these complex 16:Mathematical description of quantum state 18339: 18304:Sears' and Zemansky's University Physics 18264: 18240: 17893: 17840:Probability Theory: The Logic of Science 17753: 17637: 17614: 17601: 17561: 17125: 17029: 17017: 16977: 16904: 16883: 16808: 16796: 16784: 16768: 16703: 16638: 16634: 16630: 14795: 14480: 13776:are two states in the abstract space of 12976: 12734:, the excitons are squeezed, leading to 12706: 12190: 8681: 6809:One-particle states in 3d position space 5759:since it involves a tensor product with 5513:{\textstyle \{|\lambda ^{(j)}\rangle \}} 5190:{\textstyle \{|\lambda ^{(j)}\rangle \}} 5150:have subset of eigenvectors labelled as 2178:, which means there is no finite set of 228: 46: 25: 18320: 18183: 18009: 17838:Jaynes, E. T. (2003). Larry, G. (ed.). 17671:(2nd ed.). John Wiley & Sons. 17316:"Physical aspects of quantum mechanics" 16940: 16916: 16844: 16522: 16272: 15771: 15711: 15673: 15612: 15555:The probability of finding system with 15518: 15510: 14991: 14983: 14963: 14955: 14942: 14922: 14620:The square integrable functions on the 13803:is a valid state as well. (Whether the 13002:The wave functions of an electron in a 5650:Physical significance of relative phase 1313:{\displaystyle \lambda ={\frac {h}{p}}} 1253:{\displaystyle \lambda ={\frac {h}{p}}} 67:) is a mathematical description of the 19253: 18221: 18036: 17955: 17837: 17644:Albert Einstein: Philosopher-Scientist 17476: 17195: 17113: 17089: 17041: 16871: 16732: 16558: 16443: 16162: 15298:For a single particle in 3d with spin 15118:dimensionless discrete quantum numbers 15060:" in the continuous degrees of freedom 14293:{\displaystyle p=|(\Phi ,\Psi )|^{2},} 14234:of finding upon measurement the state 14201:This motivates the introduction of an 7130:of them. These can be arranged into a 4731:The wavefunction is instead given by: 2810:coordinate is a continuous index. The 2652:rather than an ordinary vector space. 1446:) was part of the method, provided by 18488: 18462: 18055: 17789: 17539: 17502: 17273: 17236: 17154:Arons, A. B.; Peppard, M. B. (1965). 17101: 16756: 16662: 16439: 16420: 16035: 16030: 16028: 15788:The normalization condition requires 14812:coordinates not shown), at some time 9692: state vector along basis state) 6843:{\displaystyle \Psi (\mathbf {r} ,t)} 3928:Now take the projection of the state 1588:Thus the Klein–Gordon equation (spin 1511:and quantum mechanics applied to the 18226:. Mineola, NY: Courier Corporation. 17736:Introduction to elementary particles 17348: 17310: 17214: 16510: 16478: 15838:Interpretations of quantum mechanics 13780:of a quantum mechanical system, and 1879:of a physical system, at fixed time 1794:Travelling waves of a free particle. 1724:adding citations to reliable sources 1691: 1565:) and conceptual problems (see e.g. 1402:interpretations of quantum mechanics 89:, respectively). Wave functions are 18114: 17545:The principles of quantum mechanics 17147:Lecture notes for the course AP3303 17143:"Applications of Quantum Mechanics" 17077: 15844:interpretation of quantum mechanics 14308:are assumed normalized. Consider a 14038:cannot be found to be in the state 12676:signify motion to the right, while 9430: 8030:also, collect into a single vector 6724:are corresponding complex numbers. 5554:{\textstyle \{|\phi _{i}\rangle \}} 5130:of some observable, if eigenvalues 5123:{\textstyle \{|\phi _{i}\rangle \}} 4922:{\textstyle \{|\phi _{i}\rangle \}} 4721:{\textstyle \{|\phi _{i}\rangle \}} 4603:{\textstyle \{|\phi _{i}\rangle \}} 2144:a positive real number. The number 1658:representation of the Lorentz group 13: 18360: 18014:Niels Bohr's Philosophy of Physics 17896:"Born Rule and its Interpretation" 17416:. Amsterdam: Elsevier – via 17134: 16877: 16610:University of California, Berkeley 16025: 15980:. The integral is taken to be the 15728: 15503: 14948: 14908: 14742:, for which the symmetry group is 14372:are calculable from the S-matrix. 14269: 14263: 14120: 14105: 13822: 13792:are any two complex numbers, then 13732:on the set (in the present case a 13716:Wave functions and function spaces 13174: 13025: 12774: 12579: 12548: 12492: 12464: 12412: 12381: 12352: 11945: 11922: 11893: 11836: 11828: 11770: 11713: 11688: 11661:{\textstyle {\frac {\nabla S}{m}}} 11646: 11629:{\textstyle {\frac {\nabla S}{m}}} 11614: 11540: 11258: 11211: 11199: 11032: 11020: 11012: 10899: 10457: 10192: 10100: 10071: 10068: 10065: 10062: 10059: 10055: 10052: 10049: 10012: 10009: 10006: 10003: 10000: 9996: 9993: 9990: 9956: 9953: 9950: 9947: 9944: 9940: 9937: 9934: 9847: 9834: 9588: 9414: 9280: 9152: 9055: 8895: 8834: 8719: 8472: 8393: 8097: 8042: 7976: 7931: 7888: 7858: 7827: 7777: 6931: 6893: 6820: 4429: 4393: 4326: 4290: 4093: 4039: 3983: 3942: 3880: 3862: 3824: 3807: 3766: 3748: 3710: 3693: 3598: 3585: 3550: 3538: 3508: 3431: 3377: 3347: 3340: 3335: 3312: 3299: 3205: 3105: 3044: 3010: 2971: 2928: 2916: 2723: 2702: 2577: 2566: 2561: 2492: 2410:The probability that its position 2334: 2310: 2277: 2201: 2113: 2101: 2095: 2057: 2023: 1993: 1986: 1981: 1958: 1945: 1519:. In this, the wave function is a 620:Sum-over-histories (path integral) 306: 280: 236:Part of a series of articles about 14: 19277: 18412: 18368:Kim, Yong-Ki (2 September 2000). 18306:(12th ed.). Addison-Wesley. 17716:Introduction to Quantum Mechanics 16889:"Hilbert Space Quantum Mechanics" 16821:Applications of Quantum Mechanics 16637:, pp. 121–128 translated in 16216:Inner product space#Some examples 16038:Introduction to Quantum Mechanics 15258:is the set of allowed values for 15203:is the set of allowed values for 14845:potential field. For example, in 14714:and the wave function solves the 14238:given the system is in the state 14230:. It states that the probability 14182:is called a weight function, and 13891:, or it may be momentum and spin 13707:The basis functions are mutually 12942: 12904: 12845: 12033: 11371: 9683:wave function (component of 8964:sign occurs if the particles are 6717:{\textstyle \{\varepsilon _{i}\}} 6218: 4550:finite dimensional Hilbert spaces 4451: 4420: 4351: 4317: 4257: 4238: 4190: 4174: 3471: 1836:, for one spin-0 particle in one 1419:wave function, and developed the 257: 19235: 19234: 17937:(2nd ed.). VHC Publishers. 16687:Tipler, Mosca & Freeman 2008 16124:. This norm, in turn, induces a 14685:is a Hilbert space of dimension 14375: 14017: 13559:generalized Laguerre polynomials 13012:generalized Laguerre polynomials 12997: 12083: 12062: 12047: 11989: 11968: 11953: 11880: 11815: 11780: 11590: 11579: 11518: 11510: 11477: 11428: 11400: 11357: 11323: 11298: 11152: 11144: 11101: 11073: 11039: 10924: 10909: 10883: 10841: 10802: 10706: 10642: 10482: 10467: 10393: 10378: 10223: 10208: 10136: 10121: 10089: 10030: 9974: 9733: 9712: 9618: 9597: 9560: 9518: 9318: 9303: 9288: 9222: 9207: 9186: 9165: 9125: 9110: 9089: 9068: 8929: 8908: 8868: 8847: 8757: 8742: 8727: 8605: 8576: 8515: 8479: 8341: 8295: 8283: 8137: 8104: 8092: 7983: 7938: 7895: 7865: 7834: 7784: 6958: 6938: 6926: 6860:in three-dimensional space, and 6827: 6770: 6741: 4526:Finite dimensional Hilbert space 2394:. If the particle's position is 1787: 1762: 1696: 1668:is enough to fix the equations. 1330:particles, the chief clue being 18453:The quantum theory of radiation 18419:Quantum Mechanics for Engineers 18342:A First Course in String Theory 18207:(6th ed.). W. H. Freeman. 18186:Principles of Quantum Mechanics 17991:Martin, B.R.; Shaw, G. (2008). 17482:A Course in Functional Analysis 16983: 16971: 16946: 16790: 16778: 16762: 16624: 16593: 16564: 16254: 16245: 16231: 16204: 16172: 16156: 16139: 16114: 16100: 16078: 16044: 16012: 15987: 15951: 14638:associated Laguerre polynomials 13863:Heisenberg uncertainty relation 11463:is the probability density and 11136:is the probability density and 6113: 6032: 5953: 5686:{\textstyle |\phi _{i}\rangle } 5639:{\textstyle |\phi _{i}\rangle } 5199:postulates of quantum mechanics 4965:postulates of quantum mechanics 1873:postulates of quantum mechanics 1592:) and the Dirac equation (spin 19184:Relativistic quantum mechanics 18344:. Cambridge University Press. 18295:Quantum Theory and Measurement 18272:, Cambridge University Press, 17872:. Vol. 3 (3rd ed.). 17198:Quanta: A Handbook of Concepts 16612:. 30 September 2008. p. 1 16496: 16460: 15755: 15749: 15656: 15650: 15600: 15594: 15533: 15528: 15506: 15499: 15492: 15486: 15429:of finding the system at time 14978: 14973: 14951: 14904: 14312:. In quantum field theory, if 14277: 14272: 14260: 14256: 14087:wave functions do not overlap. 13492: 13480: 13310: 13304: 13292: 13289: 13275: 13257: 13207: 13189: 13106: 13094: 13073: 13067: 13058: 13040: 12789: 12783: 12525: 12517: 12361: 12355: 12308: 12300: 12280: 12272: 12245: 12239: 12093: 12042: 12005: 11948: 11819: 11811: 11795: 11790: 11776: 11766: 11721: 11709: 11701: 11683: 11528: 11514: 11493:{\textstyle S(\mathbf {x} ,t)} 11487: 11473: 11443: 11438: 11424: 11417: 11410: 11396: 11367: 11353: 11333: 11319: 11308: 11294: 11264: 11245: 11224: 11186: 11162: 11148: 11116: 11111: 11097: 11090: 11083: 11069: 10996:Physical significance of phase 10767: 10761: 10357:particles with spin in 3d, if 9856: 9830: 9706: 9668: 9592: 9410: 9385: 9283: 8773: 8722: 8618: 8600: 8571: 8551: 8532: 8525: 8511: 8502: 8475: 8450: 8444: 8437: 8425: 8419: 8412: 8402: 8396: 8389: 8354: 8336: 8331: 8312: 8305: 8291: 8242: 8236: 8229: 8217: 8211: 8204: 8132: 8127: 8100: 8051: 8045: 8038: 8002: 7979: 7969: 7960: 7948: 7934: 7917: 7891: 7881: 7861: 7844: 7830: 7807: 7780: 7685: 7680: 7661: 7618: 7612: 7605: 7531: 7516: 7460: 7451: 7439: 7433: 7371: 7353: 7297: 7285: 7268: 7253: 7243: 7234: 7222: 7216: 7199: 7181: 7171: 7159: 7037:spin projection quantum number 7013: 6994: 6953: 6948: 6934: 6902: 6896: 6889: 6837: 6823: 6778: 6765: 6736: 6567: 6543: 6539: 6527: 6500: 6473: 6445: 6370: 6358: 6230: 6128: 6115: 6059: 6053: 6041: 6034: 5971: 5955: 5899: 5889: 5814: 5798: 5722:-spin particles which forms a 5666: 5619: 5588: 5531: 5499: 5493: 5484: 5459: 5453: 5447: 5431: 5425: 5416: 5356: 5344: 5320: 5306: 5294: 5288: 5282: 5270: 5253: 5247: 5176: 5170: 5161: 5100: 5067: 5055: 5037: 5030: 5017: 4899: 4857: 4813: 4782: 4758: 4741: 4698: 4667: 4636: 4580: 4438: 4432: 4402: 4396: 4335: 4329: 4299: 4293: 4213: 4203: 4156: 4150: 4134: 4102: 4096: 4076: 4059: 4053: 4042: 4007: 3997: 3986: 3961: 3951: 3945: 3893: 3889: 3883: 3858: 3841: 3820: 3803: 3779: 3775: 3769: 3744: 3727: 3706: 3689: 3638: 3621: 3612: 3581: 3572:normalized to a delta function 3529: 3517: 3446: 3440: 3398: 3386: 3373: 3361: 3321: 3295: 3220: 3208: 3158: 3141: 3101: 3085: 3068: 3040: 3023: 3006: 2985: 2974: 2952: 2937: 2931: 2912: 2880: 2863: 2847: 2742: 2738: 2726: 2711: 2705: 2698: 2597: 2592: 2580: 2573: 2512: 2507: 2495: 2488: 2465: 2459: 2364: 2358: 2349: 2337: 2331: 2319: 2292: 2280: 2216: 2204: 2104: 2092: 2044: 2032: 2019: 2007: 1967: 1941: 1662:cluster decomposition property 1559:relativistic quantum mechanics 770:Relativistic quantum mechanics 302: 295: 276: 1: 19162:Quantum statistical mechanics 18939:Quantum differential calculus 18861:Delayed-choice quantum eraser 18644:Symmetry in quantum mechanics 18270:Lectures in Quantum Mechanics 17964:. Oxford UK: Pergamon Press. 17903:Compendium of Quantum Physics 17739:. Wiley-VCH. pp. 162ff. 17349:Born, M. (11 December 1954). 16980:Chapter 3, Scattering matrix. 16653:, pp. 507–510, 548, 630. 16242:solving the original problem. 14466:for all rational numbers and 13628:have very similar solutions. 12754:can be expressed in terms of 12137:, in the Schrödinger picture 11917:Hamilton's principal function 8176:of its position state vector 7019:{\displaystyle \xi (s_{z},t)} 3190:Momentum-space wave functions 2390:. The asterisk indicates the 2226:{\displaystyle \Psi (x,t)\,,} 2190:Position-space wave functions 2174:being considered is infinite- 1380:. This equation was based on 810:Quantum statistical mechanics 18244:The Quantum Theory of Fields 17911:10.1007/978-3-540-70626-7_20 17371:10.1126/science.122.3172.675 17054:Greiner & Reinhardt 2008 16403: 15997:are considered, which using 15804:must have the same units as 9823:particles with spin in 3-d, 1628:), and, more generally, the 7: 18964:Quantum stochastic calculus 18954:Quantum measurement problem 18876:Mach–Zehnder interferometer 18396:. Oxford University Press. 17218:(1985). Kalckar, J. (ed.). 17168:American Journal of Physics 15993:In quantum mechanics, only 15890: 15831: 15288:are not necessarily equal. 15058:differential volume element 14838:continuously differentiable 12752:quantum harmonic oscillator 12750:The wave functions for the 12746:Quantum harmonic oscillator 9020:and others distinguishable 8169:For a single particle, the 5698:Application to include spin 2180:square integrable functions 1812:and momentum wave function 1535:relativistic wave equations 1124:of a photon and its energy 780:Quantum information science 36:quantum harmonic oscillator 10: 19282: 17844:Cambridge University Press 17699:(4th ed.). springer. 17066:Eisberg & Resnick 1985 16929:Landau & Lifshitz 1977 16547:Landau & Lifshitz 1977 15944: 15835: 15078:a component of the vector 14875: 14825:The wave function must be 14750:) used the symmetry group 13006:are expressed in terms of 11926: 11571:In classical analogy, for 10349:Probability interpretation 4560:the entire Hilbert space. 3226:{\displaystyle \Phi (p,t)} 2772:All the powerful tools of 2659: 2184:square integrable function 1804:of position wave function 1369:, and others, developing " 1357:, and others, developing " 18: 19230: 19192: 19144: 19024:Quantum complexity theory 19002:Quantum cellular automata 18977: 18909: 18843: 18756: 18720: 18707:Path integral formulation 18674: 18539: 18496: 18340:Zwiebach, Barton (2009). 18222:Treves, Francois (2006). 17733:Griffiths, David (2008). 17714:Griffiths, D. J. (2004). 17618:Physikalische Zeitschrift 17525:10.1017/S0305004100021162 16716:Young & Freedman 2008 16446:, pp. 206–225. Also 16073:Byron & Fuller (1992) 15860:Copenhagen interpretation 15368:An alternative choice is 14650:Sturm–Liouville equations 14370:scattering cross sections 14042:upon measurement. But if 13811:at any rate describe the 13125:are radial functions and 12225:. A common model is the " 12179:Non-relativistic examples 8381:with the identifications 5221:{\textstyle \lambda _{i}} 4987:{\textstyle \lambda _{i}} 4956:{\textstyle \lambda _{i}} 4517:Definitions (other cases) 3566:forms what is called the 2690:, this vector is written 2656:Quantum states as vectors 1630:Bargmann–Wigner equations 1614:Rarita–Schwinger equation 1398:Copenhagen interpretation 19091:Quantum machine learning 19071:Quantum key distribution 19061:Quantum image processing 19051:Quantum error correction 18901:Wheeler's delayed choice 18371:Practical Atomic Physics 17934:Encyclopaedia of Physics 17894:Landsman, N. P. (2009). 17693:; Reinhardt, J. (2008). 17661:Eisberg, Robert Martin; 17594:10.1002/andp.19053220607 16955:Modern quantum mechanics 16425:Wheeler & Zurek 1983 15995:separable Hilbert spaces 15870:) while others, such as 15797:to be dimensionless, by 14674:Byron & Fuller (1992 14215:⟨Ψ|Φ⟩ 14066:will be found in states 13596:azimuthal quantum number 13581:principal quantum number 12187:Finite potential barrier 11865:Hamilton-Jacobi equation 10591:particle 2 is in region 10353:For the general case of 9427: 8996:particles (no two being 8698:particles is what makes 8694:wave function describes 2650:projective Hilbert space 2416:will be in the interval 2400:probability distribution 1664:, with implications for 1218:{\displaystyle \lambda } 815:Quantum machine learning 568:Wheeler's delayed-choice 85:(lower-case and capital 19:Not to be confused with 19007:Quantum finite automata 18424:Spin wave functions NYU 18325:(2nd ed.). Wiley. 18170:10.1103/PhysRev.28.1049 17696:Quantum Electrodynamics 15923:Phase-space formulation 15691:normalization condition 15420:are the same as before. 14736:Bethe–Salpeter equation 13943:curvilinear coordinates 13916:multiplication operator 13622:magnetic quantum number 9794:basis state (basis ket) 8797:is the position of the 5826:{\textstyle (2s+1)^{2}} 4929:are eigenkets of a non- 2544:normalization condition 2406:Normalization condition 2243:complex-valued function 2172:separable Hilbert space 2074:More details are given 1484:and proved that it was 525:Leggett–Garg inequality 187:superposition principle 142:. Some particles, like 19111:Quantum neural network 18061:Zeitschrift für Physik 18041:. New York: Springer. 17931:; Trigg, G.L. (1991). 17761:The Old Quantum Theory 17458:Camilleri, K. (2009). 17196:Atkins, P. W. (1974). 16745:Martin & Shaw 2008 16390: 16218:. This element though 15933:Wave function collapse 15908:Double-slit experiment 15903:De Broglie–Bohm theory 15872:John Archibald Wheeler 15779: 15681: 15547: 15443: 15224:-dimensional "volume" 15002: 14832:It must be everywhere 14817: 14792:Simplified description 14395:orthogonal projections 14294: 14162: 13993:, both describing the 13945:as exemplified by the 13941:-axis, or a choice of 13756:Vector space structure 13499: 13113: 12994: 12961: 12720: 12629: 12329: 12218: 12217:for this illustration. 12104: 11903: 11869:Hamilton–Jacobi theory 11863:Which is analogous to 11855: 11735: 11662: 11630: 11598: 11556: 11494: 11457: 11380: 11271: 11130: 11053: 10986: 10552: 10282: 9806: 9392: 9269:The wave function for 9247: 8954: 8780: 8712:particles is written: 8687: 8639: 8558: 8460: 8375: 8186:and spin state vector 8161: 8017: 7814: 7706: 7657: 7585: 7020: 6969: 6844: 6799: 6718: 6685: 6420: 6182: 5859: 5827: 5785: 5745: 5716: 5687: 5640: 5605: 5555: 5514: 5467: 5370: 5222: 5191: 5144: 5124: 5081: 4994:is given according to 4988: 4957: 4923: 4874: 4827: 4722: 4681: 4604: 4478: 4378: 4277: 4112: 3920: 3648: 3560: 3483: 3415: 3227: 3176: 3121:which illuminates the 3115: 2992: 2887: 2762: 2627: 2532: 2380:is interpreted as the 2374: 2245:of two real variables 2227: 2130: 2064: 1913:of two wave functions 1893: 1421:self-consistency cycle 1385:conservation of energy 1314: 1276: 1254: 1219: 1195: 1170: 1139: 1118: 316: 130:Wave functions can be 99:probability amplitudes 52: 44: 19136:Quantum teleportation 18664:Wave–particle duality 18241:Weinberg, S. (2002), 18037:Newton, R.G. (2002). 17638:Einstein, A. (1998). 17400:Online copy (English) 16677:, pp. 1049–1070. 16391: 15876:Edwin Thompson Jaynes 15780: 15682: 15548: 15444: 15003: 14799: 14654:Chebyshev polynomials 14481:Common Hilbert spaces 14310:scattering experiment 14295: 14163: 14034:overlap with a state 13947:spherical coordinates 13920:differential operator 13728:), sometimes with an 13500: 13114: 12980: 12962: 12710: 12630: 12330: 12194: 12105: 11904: 11856: 11736: 11670:uncertainty principle 11663: 11631: 11599: 11557: 11495: 11458: 11381: 11272: 11131: 11054: 10987: 10553: 10283: 9807: 9393: 9248: 8955: 8827:identical particles: 8781: 8685: 8640: 8559: 8461: 8376: 8162: 8018: 7815: 7707: 7627: 7586: 7070:. For example, for a 7021: 6970: 6845: 6800: 6719: 6686: 6421: 6183: 5860: 5828: 5786: 5746: 5717: 5688: 5641: 5606: 5556: 5515: 5468: 5371: 5223: 5192: 5145: 5143:{\textstyle \lambda } 5125: 5082: 4989: 4958: 4924: 4875: 4828: 4723: 4682: 4605: 4479: 4379: 4278: 4113: 3921: 3649: 3561: 3484: 3416: 3228: 3177: 3116: 2993: 2888: 2763: 2628: 2533: 2375: 2258:probability amplitude 2228: 2166:of the wave function 2131: 2065: 1894: 1644:) and the free field 1583:free fields operators 1551:Klein–Gordon equation 1490:Klein–Gordon equation 1394:probability amplitude 1336:wave–particle duality 1315: 1277: 1255: 1220: 1196: 1171: 1140: 1119: 510:Elitzur–Vaidman 500:Davisson–Germer 317: 229:Historical background 215:wave–particle duality 50: 29: 19167:Quantum field theory 19096:Quantum metamaterial 19041:Quantum cryptography 18771:Consistent histories 18321:Zettili, N. (2009). 18184:Shankar, R. (1994). 18179:on 17 December 2008. 18115:Rae, A.I.M. (2008). 18010:Murdoch, D. (1987). 17397:Online copy (French) 17394:: 507–510, 548, 630. 16885:B. Griffiths, Robert 16759:, pp. 601–623.. 16580:Institute of Physics 16509:, p. 71, (from 16264: 16199:Dirac delta function 16128:. If this metric is 16036:Griffiths, David J. 15928:Schrödinger equation 15799:dimensional analysis 15696: 15588: 15467: 15433: 14900: 14884:infinite-dimensional 14615:Legendre polynomials 14391:projection operators 14246: 14098: 13170: 13021: 12770: 12349: 12233: 11942: 11875: 11747: 11676: 11640: 11608: 11575: 11506: 11467: 11390: 11288: 11140: 11063: 11059:is satisfied, where 11006: 10632: 10365: 9827: 9406: 9277: 9052: 8831: 8716: 8700:quantum entanglement 8649:Hamiltonian operator 8567: 8469: 8385: 8200: 8034: 8028:continuous variables 7824: 7774: 7601: 7139: 6988: 6980:For a particle with 6885: 6817: 6732: 6695: 6441: 6210: 5885: 5840: 5795: 5775: 5726: 5706: 5662: 5615: 5568: 5524: 5477: 5380: 5234: 5205: 5154: 5134: 5093: 5004: 4971: 4940: 4892: 4837: 4737: 4691: 4616: 4573: 4546:inner product spaces 4539:inner product spaces 4390: 4287: 4124: 3936: 3681: 3677:representations are 3578: 3501: 3427: 3292: 3202: 3128: 3002: 2897: 2832: 2800:quantum field theory 2694: 2550: 2434: 2268: 2198: 2089: 1938: 1883: 1720:improve this section 1574:quantum field theory 1457:energy conservation 1378:Schrödinger equation 1291: 1266: 1231: 1209: 1185: 1169:{\displaystyle E=hf} 1151: 1129: 1108: 775:Quantum field theory 687:Consistent histories 324:Schrödinger equation 251: 199:Schrödinger equation 41:Schrödinger equation 19152:Quantum fluctuation 19121:Quantum programming 19081:Quantum logic gates 19066:Quantum information 19046:Quantum electronics 18521:Classical mechanics 18162:1926PhRv...28.1049S 18073:1927ZPhy...43..601P 17956:Ludwig, G. (1968). 17631:1917PhyZ...18..121E 17585:1905AnP...322..132E 17517:1939PCPS...35..416D 17332:1927Natur.119..354B 17288:1926ZPhy...38..803B 17251:1926ZPhy...37..863B 17200:. Clarendon Press. 17181:1965AmJPh..33..367A 16919:, pp. 378–379. 16665:, pp. 606–609. 16641:, pp. 167–183. 16481:, pp. 354–357. 16239:perturbation theory 16065:Hermite polynomials 15427:probability density 14786:functional analysis 14666:harmonic oscillator 14662:Hermite polynomials 14631:spherical harmonics 14441:normalizable using 14220:inner product space 14118: 13740:), together with a 13730:algebraic structure 13626:Hydrogen-like atoms 13479: 13426: 13152:spherical harmonics 13092: 13008:spherical harmonics 12756:Hermite polynomials 12736:quantum confinement 12723:In a semiconductor 10288:this is altogether 10113: 8982:particle statistics 8819:identical particles 8658:spin–orbit coupling 6866:is time. As always 5089:For non-degenerate 4496:harmonic oscillator 3360: 3344: 3288:can be defined as: 2570: 2485: 2382:probability density 2241:is time. This is a 2006: 1990: 1432:Hartree–Fock method 1324:De Broglie relation 563:Stern–Gerlach 360:Classical mechanics 107:probability density 19205:in popular culture 18987:Quantum algorithms 18835:Von Neumann–Wigner 18815:Objective collapse 18526:Old quantum theory 18444:2013-05-13 at the 18390:Polkinghorne, John 18081:10.1007/bf01397326 17572:Annalen der Physik 17437:Dover Publications 17296:10.1007/bf01397184 17282:(11–12): 803–827. 17259:10.1007/bf01397477 17158:Annalen der Physik 16874:, p. 112-125. 16453:2020-12-01 at the 16396:is a multiple sum. 16386: 16385: 16365: 16348: 16328: 16276: 16169:of Hilbert spaces. 16153:is not sufficient. 15978:semi-inner product 15976:) as opposed to a 15974:(Ψ, Ψ) = 0 ⇒ Ψ ≡ 0 15775: 15722: 15677: 15623: 15543: 15439: 14998: 14926: 14818: 14778:Heisenberg picture 14658:Jacobi polynomials 14290: 14158: 14104: 13848:Hermitian operator 13736:structure with an 13495: 13465: 13391: 13109: 13078: 12995: 12957: 12721: 12625: 12620: 12325: 12320: 12219: 12159:is constant while 12100: 11929:Dynamical pictures 11899: 11851: 11731: 11658: 11626: 11594: 11552: 11490: 11453: 11376: 11277:, is known as the 11267: 11126: 11049: 10982: 10548: 10314:are also written " 10292:three-dimensional 10278: 10099: 10076: 10017: 9961: 9927: 9906: 9882: 9802: 9797: 9790: 9699: 9697: 9675: 9472: 9388: 9243: 8950: 8776: 8688: 8635: 8554: 8456: 8371: 8267: 8157: 8076: 8013: 8007: 7810: 7714:The entire vector 7702: 7581: 7575: 7504: 7415: 7341: 7273: 7056:parameter, unlike 7016: 6965: 6840: 6795: 6714: 6681: 6675: 6579: 6416: 6410: 6178: 6172: 6103: 6015: 5943: 5868:For example, each 5855: 5823: 5781: 5741: 5712: 5683: 5636: 5601: 5551: 5510: 5463: 5414: 5366: 5268: 5218: 5187: 5140: 5120: 5077: 4984: 4953: 4919: 4870: 4823: 4780: 4718: 4677: 4634: 4600: 4488:Fourier transforms 4474: 4374: 4273: 4108: 3916: 3914: 3644: 3556: 3479: 3411: 3346: 3327: 3223: 3172: 3111: 2988: 2883: 2758: 2623: 2553: 2528: 2471: 2370: 2223: 2126: 2060: 1992: 1973: 1889: 1654:Lorentz invariance 1509:special relativity 1436:Slater determinant 1332:Lorentz invariance 1310: 1272: 1250: 1215: 1191: 1166: 1135: 1114: 751:Von Neumann–Wigner 731:Objective-collapse 530:Mach–Zehnder 520:Leggett inequality 515:Franck–Hertz 365:Old quantum theory 312: 223:classic mechanical 53: 45: 19248: 19247: 19222:Quantum mysticism 19200:Schrödinger's cat 19131:Quantum simulator 19101:Quantum metrology 19029:Quantum computing 18992:Quantum amplifier 18969:Quantum spacetime 18934:Quantum cosmology 18924:Quantum chemistry 18639:Scattering theory 18587:Zero-point energy 18582:Degenerate levels 18490:Quantum mechanics 18403:978-0-19-280252-1 18351:978-0-521-88032-9 18332:978-0-470-02679-3 18313:978-0-321-50130-1 18279:978-1-107-02872-2 18254:978-0-521-55001-7 18233:978-0-486-45352-1 18214:978-0-7167-8964-2 18118:Quantum Mechanics 18107:978-0-07-162358-2 18098:Quantum mechanics 18067:(9–10): 601–623. 18048:978-0-387-95473-8 18025:978-0-521-33320-7 18002:978-0-470-03294-7 17971:978-0-08-203204-5 17944:978-0-89573-752-6 17920:978-3-540-70622-9 17883:978-0-08-020940-1 17853:978-0-521 59271-0 17746:978-3-527-40601-2 17678:978-0-471-87373-0 17653:978-0-87548-133-3 17495:978-0-387-97245-9 17469:978-0-521-88484-6 17446:978-0-486-67164-2 17326:(2992): 354–357. 17207:978-0-19-855494-3 17189:10.1119/1.1971542 17008:, pp. 162ff. 16964:978-1-108-47322-4 16561:, pp. 19–21. 16369: 16349: 16332: 16280: 16267: 16237:For instance, in 16195:square-integrable 16147:Lebesgue integral 16069:Gaussian function 16022:for more details. 15982:Lebesgue integral 15848:Erwin Schrödinger 15705: 15606: 15442:{\displaystyle t} 15280:. For generality 14917: 14847:particle in a box 14827:square integrable 14670:special functions 14522:Fourier transform 14358:scattering matrix 14007:Fourier transform 13726:square integrable 13456: 13379: 13316: 13314: 13241: 12991:orthonormal basis 12946: 12945: 12908: 12849: 12821: 12820: 12740:particle in a box 12639:scattering theory 12227:potential barrier 12173:S-matrix elements 12120:stationary states 11887: 11843: 11763: 11656: 11624: 11550: 11374: 11336: 11243: 11238: 11184: 11027: 10043: 9984: 9928: 9907: 9886: 9862: 9795: 9703: 9701: 9693: 9684: 9585: 9583: 9581: 9579: 9578:continuous labels 9574: 9484: 9482: 9477: 9429: 8972:sign if they are 8251: 8060: 7068:discrete variable 6233: 6221: 5858:{\textstyle 2s+1} 5744:{\textstyle 2s+1} 5405: 5393: 5334: 5259: 4936:with eigenvalues 4771: 4625: 4569:-dimensional set 4554:orthonormal basis 4454: 4424: 4423: 4354: 4321: 4320: 4260: 4242: 4241: 4193: 4178: 4177: 3123:identity operator 2670:Position operator 2392:complex conjugate 1892:{\displaystyle t} 1871:According to the 1778:stationary states 1774:particle in a box 1756: 1755: 1748: 1646:Einstein equation 1515:, now called the 1486:Lorentz invariant 1389:quantum operators 1363:Werner Heisenberg 1355:Erwin Schrödinger 1308: 1275:{\displaystyle h} 1248: 1194:{\displaystyle p} 1138:{\displaystyle E} 1117:{\displaystyle f} 1098: 1097: 805:Scattering theory 785:Quantum computing 558:Schrödinger's cat 490:Bell's inequality 298: 273: 242:Quantum mechanics 185:According to the 140:Fourier transform 119:quantum operators 19273: 19238: 19237: 18949:Quantum geometry 18944:Quantum dynamics 18801:Superdeterminism 18697:Matrix mechanics 18552:Bra–ket notation 18483: 18476: 18469: 18460: 18459: 18407: 18385: 18384:on 22 July 2011. 18383: 18376: 18355: 18336: 18317: 18298: 18282: 18261: 18259:Internet Archive 18237: 18218: 18199: 18188:(2nd ed.). 18180: 18178: 18172:. Archived from 18156:(6): 1049–1070. 18145: 18132: 18128:978-1-5848-89700 18111: 18092: 18052: 18033: 18030:Internet Archive 18017: 18006: 17993:Particle Physics 17987: 17984:Internet Archive 17963: 17952: 17949:Internet Archive 17924: 17900: 17887: 17857: 17834: 17831:Internet Archive 17828: 17814: 17786: 17783:Internet Archive 17764: 17750: 17729: 17710: 17686: 17683:Internet Archive 17657: 17634: 17611: 17598: 17596: 17558: 17536: 17499: 17473: 17454: 17451:Internet Archive 17421: 17418:Internet Archive 17415: 17395: 17382: 17363:Nobel Foundation 17345: 17343: 17341:10.1038/119354a0 17307: 17270: 17233: 17211: 17192: 17164: 17150: 17129: 17123: 17117: 17111: 17105: 17099: 17093: 17087: 17081: 17075: 17069: 17063: 17057: 17051: 17045: 17039: 17033: 17027: 17021: 17015: 17009: 17003: 16997: 16987: 16981: 16975: 16969: 16968: 16950: 16944: 16938: 16932: 16926: 16920: 16914: 16908: 16902: 16896: 16895: 16893: 16881: 16875: 16869: 16863: 16857: 16848: 16842: 16836: 16830: 16824: 16818: 16812: 16806: 16800: 16794: 16788: 16782: 16776: 16766: 16760: 16754: 16748: 16742: 16736: 16730: 16719: 16713: 16707: 16701: 16690: 16684: 16678: 16675:Schrödinger 1926 16672: 16666: 16660: 16654: 16648: 16642: 16628: 16622: 16621: 16619: 16617: 16607: 16597: 16591: 16590: 16588: 16586: 16568: 16562: 16556: 16550: 16544: 16538: 16532: 16526: 16520: 16514: 16500: 16494: 16488: 16482: 16476: 16470: 16464: 16458: 16442:, translated in 16437: 16428: 16423:, translated in 16418: 16397: 16395: 16393: 16392: 16387: 16384: 16383: 16382: 16364: 16363: 16362: 16347: 16346: 16345: 16327: 16326: 16325: 16307: 16306: 16294: 16293: 16275: 16258: 16252: 16249: 16243: 16235: 16229: 16227: 16208: 16202: 16176: 16170: 16160: 16154: 16151:Riemann integral 16143: 16137: 16118: 16112: 16104: 16098: 16082: 16076: 16067:multiplied by a 16062: 16056:has eigenvalues 16055: 16048: 16042: 16041: 16032: 16023: 16020:Spectral theorem 16016: 16010: 15991: 15985: 15975: 15971: 15967:Lebesgue measure 15964: 15955: 15884:Hugh Everett III 15868:John von Neumann 15827: 15803: 15796: 15784: 15782: 15781: 15776: 15774: 15768: 15767: 15748: 15747: 15732: 15731: 15721: 15714: 15686: 15684: 15683: 15678: 15676: 15670: 15669: 15649: 15648: 15633: 15632: 15622: 15615: 15583: 15576: 15570: 15560: 15552: 15550: 15549: 15544: 15542: 15541: 15536: 15521: 15513: 15502: 15485: 15484: 15462: 15461: 15448: 15446: 15445: 15440: 15419: 15415: 15409: 15381: 15364: 15357: 15334: 15315: 15287: 15283: 15279: 15266: 15257: 15244: 15227: 15223: 15217: 15211: 15202: 15193: 15163: 15157: 15147: 15117: 15083: 15082: 15077: 15055: 15024: 15023: 15007: 15005: 15004: 14999: 14994: 14986: 14981: 14966: 14958: 14945: 14939: 14938: 14925: 14907: 14895: 14894: 14867: 14857: 14759: 14713: 14699: 14688: 14684: 14643: 14628: 14612: 14605: 14569: 14548: 14534: 14516: 14496: 14472: 14465: 14461: 14436: 14430: 14424: 14412: 14406: 14399:spectral theorem 14351: 14344: 14337: 14325: 14318: 14307: 14303: 14299: 14297: 14296: 14291: 14286: 14285: 14280: 14259: 14241: 14237: 14233: 14225: 14217: 14216: 14211:Bra–ket notation 14208: 14192: 14181: 14177: 14167: 14165: 14164: 14159: 14154: 14153: 14128: 14127: 14117: 14112: 14077: 14065: 14057: 14053: 14041: 14037: 14029: 13992: 13975: 13940: 13934: 13913: 13896: 13890: 13873: 13802: 13791: 13785: 13775: 13771: 13699: 13680: 13674: 13665: 13650: 13619: 13593: 13578: 13571: 13556: 13555: 13554: 13531: 13504: 13502: 13501: 13496: 13478: 13473: 13461: 13457: 13455: 13454: 13453: 13440: 13432: 13425: 13411: 13390: 13389: 13384: 13380: 13378: 13377: 13376: 13363: 13355: 13348: 13347: 13346: 13345: 13333: 13317: 13315: 13313: 13281: 13255: 13253: 13252: 13247: 13246: 13242: 13240: 13239: 13238: 13222: 13214: 13188: 13187: 13165: 13159: 13149: 13139: 13138: 13124: 13118: 13116: 13115: 13110: 13091: 13086: 13039: 13038: 12973: 12966: 12964: 12963: 12958: 12956: 12955: 12951: 12947: 12941: 12933: 12932: 12924: 12923: 12911: 12910: 12909: 12907: 12899: 12898: 12897: 12881: 12868: 12867: 12863: 12854: 12850: 12848: 12840: 12832: 12822: 12819: 12811: 12810: 12797: 12796: 12782: 12781: 12765: 12703: 12693: 12684: 12675: 12666: 12657: 12647: 12641:for discussion. 12634: 12632: 12631: 12626: 12624: 12623: 12603: 12602: 12584: 12583: 12582: 12569: 12568: 12553: 12552: 12551: 12528: 12520: 12513: 12512: 12497: 12496: 12495: 12482: 12481: 12469: 12468: 12467: 12436: 12435: 12417: 12416: 12415: 12402: 12401: 12386: 12385: 12384: 12344: 12334: 12332: 12331: 12326: 12324: 12323: 12311: 12303: 12283: 12275: 12268: 12267: 12216: 12203: 12169: 12158: 12157: 12152: 12146: 12145: 12136: 12130: 12129: 12117: 12113: 12109: 12107: 12106: 12101: 12092: 12091: 12086: 12071: 12070: 12065: 12056: 12055: 12050: 12037: 12036: 12032: 11998: 11997: 11992: 11977: 11976: 11971: 11962: 11961: 11956: 11937: 11913: 11908: 11906: 11905: 11900: 11889: 11888: 11885: 11883: 11860: 11858: 11857: 11852: 11844: 11842: 11834: 11826: 11818: 11804: 11803: 11798: 11783: 11769: 11764: 11762: 11751: 11740: 11738: 11737: 11732: 11730: 11729: 11724: 11712: 11704: 11696: 11695: 11686: 11667: 11665: 11664: 11659: 11657: 11652: 11644: 11635: 11633: 11632: 11627: 11625: 11620: 11612: 11603: 11601: 11600: 11595: 11593: 11582: 11566:probability flux 11561: 11559: 11558: 11553: 11551: 11546: 11535: 11521: 11513: 11499: 11497: 11496: 11491: 11480: 11462: 11460: 11459: 11454: 11452: 11451: 11446: 11431: 11420: 11403: 11385: 11383: 11382: 11377: 11375: 11370: 11360: 11345: 11337: 11326: 11315: 11301: 11279:probability flux 11276: 11274: 11273: 11268: 11257: 11256: 11244: 11241: 11239: 11231: 11223: 11222: 11198: 11197: 11185: 11183: 11169: 11155: 11147: 11135: 11133: 11132: 11127: 11125: 11124: 11119: 11104: 11093: 11076: 11058: 11056: 11055: 11050: 11042: 11028: 11026: 11018: 11010: 10991: 10989: 10988: 10983: 10981: 10980: 10975: 10971: 10970: 10966: 10959: 10958: 10946: 10945: 10933: 10932: 10927: 10918: 10917: 10912: 10892: 10891: 10886: 10880: 10879: 10870: 10869: 10868: 10867: 10850: 10849: 10844: 10838: 10837: 10828: 10827: 10826: 10825: 10811: 10810: 10805: 10799: 10798: 10789: 10788: 10787: 10786: 10760: 10759: 10758: 10757: 10745: 10744: 10728: 10727: 10715: 10714: 10709: 10694: 10693: 10681: 10680: 10664: 10663: 10651: 10650: 10645: 10624: 10618: 10599: 10587: 10568: 10557: 10555: 10554: 10549: 10547: 10546: 10541: 10537: 10536: 10532: 10525: 10524: 10508: 10507: 10491: 10490: 10485: 10476: 10475: 10470: 10447: 10443: 10436: 10435: 10419: 10418: 10402: 10401: 10396: 10387: 10386: 10381: 10360: 10356: 10341: 10324: 10313: 10299: 10294:volume integrals 10291: 10287: 10285: 10284: 10279: 10277: 10273: 10266: 10265: 10249: 10248: 10232: 10231: 10226: 10217: 10216: 10211: 10200: 10199: 10190: 10186: 10179: 10178: 10162: 10161: 10145: 10144: 10139: 10130: 10129: 10124: 10112: 10107: 10098: 10097: 10092: 10086: 10085: 10075: 10074: 10039: 10038: 10033: 10027: 10026: 10016: 10015: 9983: 9982: 9977: 9971: 9970: 9960: 9959: 9926: 9925: 9924: 9905: 9904: 9903: 9881: 9880: 9879: 9855: 9854: 9842: 9841: 9822: 9811: 9809: 9808: 9803: 9796: 9793: 9791: 9786: 9782: 9781: 9759: 9758: 9742: 9741: 9736: 9721: 9720: 9715: 9709: 9698: 9694: 9691: 9685: 9682: 9676: 9671: 9667: 9666: 9644: 9643: 9627: 9626: 9621: 9606: 9605: 9600: 9591: 9580: 9577: 9575: 9570: 9569: 9568: 9563: 9557: 9556: 9547: 9546: 9545: 9544: 9527: 9526: 9521: 9515: 9514: 9505: 9504: 9503: 9502: 9487: 9485: 9483: 9480: 9478: 9473: 9471: 9470: 9469: 9447: 9446: 9425: 9423: 9413: 9397: 9395: 9394: 9389: 9378: 9377: 9361: 9360: 9344: 9343: 9327: 9326: 9321: 9312: 9311: 9306: 9297: 9296: 9291: 9265: 9252: 9250: 9249: 9244: 9242: 9238: 9231: 9230: 9225: 9216: 9215: 9210: 9195: 9194: 9189: 9174: 9173: 9168: 9145: 9141: 9134: 9133: 9128: 9119: 9118: 9113: 9098: 9097: 9092: 9077: 9076: 9071: 9047: 9036: 9019: 8992: 8971: 8963: 8959: 8957: 8956: 8951: 8949: 8945: 8938: 8937: 8932: 8917: 8916: 8911: 8888: 8884: 8877: 8876: 8871: 8856: 8855: 8850: 8813:real variables. 8812: 8804: 8800: 8796: 8785: 8783: 8782: 8777: 8766: 8765: 8760: 8751: 8750: 8745: 8736: 8735: 8730: 8711: 8665:angular momentum 8644: 8642: 8641: 8636: 8631: 8630: 8621: 8608: 8603: 8592: 8591: 8579: 8574: 8563: 8561: 8560: 8555: 8544: 8543: 8518: 8495: 8494: 8482: 8465: 8463: 8462: 8457: 8440: 8415: 8392: 8380: 8378: 8377: 8372: 8367: 8366: 8357: 8344: 8339: 8324: 8323: 8298: 8286: 8280: 8279: 8266: 8265: 8264: 8232: 8207: 8195: 8194: 8185: 8184: 8175: 8166: 8164: 8163: 8158: 8153: 8152: 8140: 8135: 8120: 8119: 8107: 8095: 8089: 8088: 8075: 8074: 8073: 8041: 8022: 8020: 8019: 8014: 8012: 8011: 7986: 7941: 7898: 7868: 7837: 7819: 7817: 7816: 7811: 7800: 7799: 7787: 7766: 7727: 7719: 7711: 7709: 7708: 7703: 7698: 7697: 7688: 7673: 7672: 7656: 7651: 7641: 7640: 7608: 7595:bra–ket notation 7590: 7588: 7587: 7582: 7580: 7579: 7509: 7508: 7420: 7419: 7346: 7345: 7278: 7277: 7129: 7121: 7103: 7094: 7090: 7086: 7082: 7065: 7061: 7055: 7046: 7042: 7034: 7025: 7023: 7022: 7017: 7006: 7005: 6974: 6972: 6971: 6966: 6961: 6956: 6941: 6929: 6923: 6922: 6892: 6877: 6865: 6855: 6849: 6847: 6846: 6841: 6830: 6804: 6802: 6801: 6796: 6791: 6790: 6781: 6773: 6768: 6757: 6756: 6744: 6739: 6723: 6721: 6720: 6715: 6710: 6709: 6690: 6688: 6687: 6682: 6680: 6679: 6672: 6671: 6655: 6654: 6625: 6624: 6605: 6604: 6584: 6583: 6570: 6546: 6503: 6476: 6448: 6425: 6423: 6422: 6417: 6415: 6414: 6241: 6240: 6235: 6234: 6226: 6222: 6214: 6202: 6201: 6187: 6185: 6184: 6179: 6177: 6176: 6118: 6108: 6107: 6037: 6020: 6019: 5958: 5948: 5947: 5892: 5880: 5879: 5864: 5862: 5861: 5856: 5832: 5830: 5829: 5824: 5822: 5821: 5790: 5788: 5787: 5782: 5750: 5748: 5747: 5742: 5721: 5719: 5718: 5713: 5692: 5690: 5689: 5684: 5679: 5678: 5669: 5645: 5643: 5642: 5637: 5632: 5631: 5622: 5610: 5608: 5607: 5602: 5591: 5586: 5585: 5560: 5558: 5557: 5552: 5544: 5543: 5534: 5519: 5517: 5516: 5511: 5503: 5502: 5487: 5472: 5470: 5469: 5464: 5462: 5457: 5456: 5435: 5434: 5419: 5413: 5401: 5400: 5395: 5394: 5386: 5375: 5373: 5372: 5367: 5365: 5364: 5359: 5347: 5342: 5341: 5336: 5335: 5327: 5323: 5315: 5314: 5309: 5297: 5292: 5291: 5273: 5267: 5246: 5245: 5227: 5225: 5224: 5219: 5217: 5216: 5196: 5194: 5193: 5188: 5180: 5179: 5164: 5149: 5147: 5146: 5141: 5129: 5127: 5126: 5121: 5113: 5112: 5103: 5086: 5084: 5083: 5078: 5076: 5075: 5070: 5058: 5053: 5052: 5040: 5029: 5028: 5016: 5015: 4993: 4991: 4990: 4985: 4983: 4982: 4962: 4960: 4959: 4954: 4952: 4951: 4928: 4926: 4925: 4920: 4912: 4911: 4902: 4879: 4877: 4876: 4871: 4860: 4855: 4854: 4832: 4830: 4829: 4824: 4816: 4811: 4810: 4795: 4794: 4785: 4779: 4761: 4744: 4727: 4725: 4724: 4719: 4711: 4710: 4701: 4686: 4684: 4683: 4678: 4670: 4665: 4664: 4649: 4648: 4639: 4633: 4609: 4607: 4606: 4601: 4593: 4592: 4583: 4568: 4512: 4505: 4501: 4483: 4481: 4480: 4475: 4463: 4462: 4455: 4447: 4425: 4413: 4409: 4383: 4381: 4380: 4375: 4363: 4362: 4355: 4347: 4322: 4310: 4306: 4282: 4280: 4279: 4274: 4269: 4268: 4261: 4253: 4243: 4231: 4227: 4216: 4202: 4201: 4194: 4186: 4179: 4167: 4163: 4137: 4117: 4115: 4114: 4109: 4089: 4075: 4052: 4032: 4018: 4010: 3996: 3964: 3931: 3925: 3923: 3922: 3917: 3915: 3896: 3861: 3844: 3823: 3806: 3782: 3747: 3730: 3709: 3692: 3676: 3670: 3653: 3651: 3650: 3645: 3637: 3611: 3610: 3609: 3593: 3592: 3565: 3563: 3562: 3557: 3516: 3515: 3496: 3488: 3486: 3485: 3480: 3475: 3474: 3470: 3439: 3438: 3420: 3418: 3417: 3412: 3385: 3384: 3359: 3354: 3343: 3338: 3320: 3319: 3307: 3306: 3287: 3271: 3252: 3248: 3244: 3236: 3232: 3230: 3229: 3224: 3181: 3179: 3178: 3173: 3161: 3144: 3120: 3118: 3117: 3112: 3104: 3099: 3095: 3088: 3071: 3043: 3026: 3009: 2997: 2995: 2994: 2989: 2984: 2955: 2950: 2915: 2910: 2892: 2890: 2889: 2884: 2873: 2850: 2845: 2822:improper vectors 2819: 2818: 2809: 2787:Bra–ket notation 2767: 2765: 2764: 2759: 2745: 2701: 2688:Bra–ket notation 2685: 2666:Bra–ket notation 2632: 2630: 2629: 2624: 2606: 2605: 2600: 2576: 2569: 2564: 2541: 2537: 2535: 2534: 2529: 2521: 2520: 2515: 2491: 2484: 2479: 2458: 2457: 2429: 2415: 2389: 2379: 2377: 2376: 2371: 2318: 2317: 2305: 2304: 2299: 2295: 2252: 2248: 2240: 2237:is position and 2236: 2232: 2230: 2229: 2224: 2169: 2160:) is called the 2159: 2157: 2151: 2149: 2135: 2133: 2132: 2127: 2125: 2124: 2081: 2069: 2067: 2066: 2061: 2031: 2030: 2005: 2000: 1989: 1984: 1966: 1965: 1953: 1952: 1930: 1926: 1919: 1898: 1896: 1895: 1890: 1857: 1851: 1843: 1839: 1835: 1827: 1819: 1811: 1791: 1766: 1751: 1744: 1740: 1737: 1731: 1700: 1692: 1651: 1643: 1638:Maxwell equation 1627: 1626: 1625: 1621: 1611: 1603: 1602: 1601: 1597: 1591: 1371:matrix mechanics 1351:Louis de Broglie 1321: 1319: 1317: 1316: 1311: 1309: 1301: 1281: 1279: 1278: 1273: 1261: 1259: 1257: 1256: 1251: 1249: 1241: 1226: 1224: 1222: 1221: 1216: 1200: 1198: 1197: 1192: 1177: 1175: 1173: 1172: 1167: 1146: 1144: 1142: 1141: 1136: 1123: 1121: 1120: 1115: 1090: 1083: 1076: 717:Superdeterminism 370:Bra–ket notation 321: 319: 318: 313: 305: 300: 299: 291: 279: 274: 272: 261: 233: 232: 181: 180: 179: 175: 169: 126: 84: 80: 19281: 19280: 19276: 19275: 19274: 19272: 19271: 19270: 19251: 19250: 19249: 19244: 19226: 19212:Wigner's friend 19188: 19179:Quantum gravity 19140: 19126:Quantum sensing 19106:Quantum network 19086:Quantum machine 19056:Quantum imaging 19019:Quantum circuit 19014:Quantum channel 18973: 18919:Quantum biology 18905: 18881:Elitzur–Vaidman 18856:Davisson–Germer 18839: 18791:Hidden-variable 18781:de Broglie–Bohm 18758:Interpretations 18752: 18716: 18670: 18557:Complementarity 18535: 18492: 18487: 18446:Wayback Machine 18415: 18410: 18404: 18388: 18381: 18374: 18367: 18363: 18361:Further reading 18358: 18352: 18333: 18314: 18280: 18255: 18234: 18215: 18196: 18195:978-030644790-7 18176: 18149:Physical Review 18143: 18137:Schrödinger, E. 18129: 18108: 18057:Pauli, Wolfgang 18049: 18026: 18003: 17972: 17945: 17921: 17898: 17884: 17866:Lifshitz, E. M. 17854: 17747: 17726: 17725:978-013111892-8 17707: 17706:978-354087560-4 17679: 17663:Resnick, Robert 17654: 17555: 17541:Dirac, P. A. M. 17496: 17486:Springer Verlag 17470: 17447: 17245:(12): 863–867. 17230: 17229:978-044453289-3 17208: 17162: 17137: 17135:General sources 17132: 17124: 17120: 17112: 17108: 17100: 17096: 17088: 17084: 17076: 17072: 17064: 17060: 17052: 17048: 17040: 17036: 17028: 17024: 17016: 17012: 17004: 17000: 16988: 16984: 16976: 16972: 16965: 16951: 16947: 16939: 16935: 16927: 16923: 16915: 16911: 16903: 16899: 16891: 16882: 16878: 16870: 16866: 16858: 16851: 16843: 16839: 16831: 16827: 16819: 16815: 16807: 16803: 16795: 16791: 16785:Weinberg (2002) 16783: 16779: 16769:Weinberg (2002) 16767: 16763: 16755: 16751: 16743: 16739: 16731: 16722: 16718:, p. 1333. 16714: 16710: 16702: 16693: 16685: 16681: 16673: 16669: 16661: 16657: 16651:de Broglie 1923 16649: 16645: 16629: 16625: 16615: 16613: 16605: 16599: 16598: 16594: 16584: 16582: 16570: 16569: 16565: 16557: 16553: 16545: 16541: 16535:de Broglie 1960 16533: 16529: 16521: 16517: 16513:, p. 142). 16501: 16497: 16491:Heisenberg 1958 16489: 16485: 16477: 16473: 16465: 16461: 16455:Wayback Machine 16438: 16431: 16427:at pages 52–55. 16419: 16410: 16406: 16401: 16400: 16378: 16374: 16373: 16358: 16354: 16353: 16341: 16337: 16336: 16321: 16317: 16302: 16298: 16289: 16285: 16284: 16271: 16265: 16262: 16261: 16259: 16255: 16250: 16246: 16236: 16232: 16223: 16209: 16205: 16177: 16173: 16161: 16157: 16144: 16140: 16119: 16115: 16105: 16101: 16083: 16079: 16057: 16051: 16049: 16045: 16040:(3rd ed.). 16033: 16026: 16017: 16013: 15992: 15988: 15973: 15969: 15958: 15956: 15952: 15947: 15942: 15893: 15852:Albert Einstein 15840: 15834: 15824: 15818: 15812: 15805: 15801: 15789: 15770: 15763: 15759: 15737: 15733: 15727: 15723: 15710: 15709: 15697: 15694: 15693: 15672: 15665: 15661: 15638: 15634: 15628: 15624: 15611: 15610: 15589: 15586: 15585: 15578: 15572: 15562: 15556: 15537: 15532: 15531: 15517: 15509: 15498: 15474: 15470: 15468: 15465: 15464: 15451: 15450: 15434: 15431: 15430: 15423: 15417: 15411: 15406: 15399: 15392: 15383: 15378: 15369: 15359: 15336: 15317: 15312: 15303: 15285: 15281: 15275: 15264: 15259: 15252: 15246: 15243: 15237: 15233: 15229: 15225: 15219: 15213: 15209: 15204: 15200: 15195: 15191: 15185: 15178: 15168: 15159: 15153: 15144: 15138: 15131: 15121: 15114: 15108: 15101: 15091: 15081:|Ψ⟩ 15080: 15079: 15063: 15053: 15047: 15041: 15028: 15013: 15012: 14990: 14982: 14977: 14962: 14954: 14941: 14934: 14930: 14921: 14903: 14901: 14898: 14897: 14893:|Ψ⟩ 14892: 14891: 14880: 14874: 14863: 14853: 14836:and everywhere 14794: 14774:field operators 14755: 14724:tensor products 14708: 14694: 14686: 14680: 14641: 14624: 14611:[–1, 1] 14610: 14592: 14579: 14571: 14559: 14550: 14540: 14537:square summable 14535:, the space of 14525: 14498: 14490: 14483: 14467: 14463: 14457: 14432: 14426: 14420: 14408: 14402: 14378: 14350: 14346: 14343: 14339: 14335: 14331: 14327: 14324: 14320: 14317: 14313: 14305: 14301: 14281: 14276: 14275: 14255: 14247: 14244: 14243: 14239: 14235: 14231: 14223: 14214: 14213: 14206: 14195:Kronecker delta 14191: 14183: 14179: 14169: 14146: 14142: 14123: 14119: 14113: 14108: 14099: 14096: 14095: 14080:selection rules 14075: 14071: 14067: 14063: 14055: 14051: 14047: 14043: 14039: 14035: 14027: 14020: 13990: 13977: 13973: 13960: 13936: 13926: 13911: 13898: 13892: 13888: 13875: 13869: 13825: 13823:Representations 13793: 13787: 13781: 13773: 13769: 13758: 13722:function spaces 13720:The concept of 13718: 13702:countable basis 13685: 13676: 13670: 13659: 13636: 13599: 13584: 13573: 13562: 13553: 13543: 13542: 13541: 13537: 13528: 13519: 13512: 13506: 13474: 13469: 13449: 13445: 13441: 13433: 13431: 13427: 13412: 13395: 13385: 13372: 13368: 13364: 13356: 13354: 13350: 13349: 13341: 13337: 13329: 13322: 13318: 13282: 13256: 13254: 13248: 13234: 13230: 13226: 13221: 13217: 13216: 13215: 13213: 13177: 13173: 13171: 13168: 13167: 13161: 13155: 13137: 13132: 13131: 13130: 13126: 13120: 13087: 13082: 13028: 13024: 13022: 13019: 13018: 13000: 12968: 12934: 12931: 12930: 12926: 12925: 12919: 12915: 12900: 12893: 12889: 12882: 12880: 12876: 12872: 12859: 12855: 12841: 12833: 12831: 12827: 12826: 12806: 12802: 12801: 12795: 12777: 12773: 12771: 12768: 12767: 12763: 12758: 12748: 12701: 12695: 12692: 12686: 12683: 12677: 12674: 12668: 12665: 12659: 12655: 12649: 12645: 12619: 12618: 12604: 12589: 12585: 12578: 12577: 12573: 12558: 12554: 12547: 12546: 12542: 12539: 12538: 12524: 12516: 12514: 12502: 12498: 12491: 12490: 12486: 12474: 12470: 12463: 12462: 12458: 12455: 12454: 12437: 12422: 12418: 12411: 12410: 12406: 12391: 12387: 12380: 12379: 12375: 12368: 12367: 12350: 12347: 12346: 12336: 12319: 12318: 12307: 12299: 12297: 12291: 12290: 12279: 12271: 12269: 12263: 12259: 12252: 12251: 12234: 12231: 12230: 12223:force potential 12215: 12205: 12202: 12196: 12189: 12181: 12160: 12156:|Ψ⟩ 12155: 12154: 12148: 12139: 12138: 12132: 12128:|Ψ⟩ 12127: 12126: 12115: 12111: 12087: 12082: 12081: 12066: 12061: 12060: 12051: 12046: 12045: 12028: 12015: 12011: 11993: 11988: 11987: 11972: 11967: 11966: 11957: 11952: 11951: 11943: 11940: 11939: 11935: 11931: 11925: 11923:Time dependence 11911: 11884: 11879: 11878: 11876: 11873: 11872: 11835: 11827: 11825: 11814: 11799: 11794: 11793: 11779: 11765: 11755: 11750: 11748: 11745: 11744: 11725: 11720: 11719: 11708: 11700: 11691: 11687: 11682: 11677: 11674: 11673: 11645: 11643: 11641: 11638: 11637: 11613: 11611: 11609: 11606: 11605: 11604:, the quantity 11589: 11578: 11576: 11573: 11572: 11536: 11534: 11517: 11509: 11507: 11504: 11503: 11476: 11468: 11465: 11464: 11447: 11442: 11441: 11427: 11416: 11399: 11391: 11388: 11387: 11356: 11346: 11344: 11322: 11314: 11297: 11289: 11286: 11285: 11252: 11248: 11240: 11230: 11218: 11214: 11193: 11189: 11173: 11168: 11151: 11143: 11141: 11138: 11137: 11120: 11115: 11114: 11100: 11089: 11072: 11064: 11061: 11060: 11038: 11019: 11011: 11009: 11007: 11004: 11003: 10998: 10976: 10954: 10950: 10941: 10937: 10928: 10923: 10922: 10913: 10908: 10907: 10906: 10902: 10898: 10894: 10893: 10887: 10882: 10881: 10875: 10871: 10863: 10859: 10858: 10854: 10845: 10840: 10839: 10833: 10829: 10821: 10817: 10816: 10812: 10806: 10801: 10800: 10794: 10790: 10782: 10778: 10777: 10773: 10753: 10749: 10736: 10732: 10723: 10719: 10710: 10705: 10704: 10689: 10685: 10672: 10668: 10659: 10655: 10646: 10641: 10640: 10639: 10635: 10633: 10630: 10629: 10620: 10617: 10610: 10601: 10598: 10592: 10586: 10579: 10570: 10567: 10561: 10542: 10516: 10512: 10499: 10495: 10486: 10481: 10480: 10471: 10466: 10465: 10464: 10460: 10456: 10452: 10451: 10427: 10423: 10410: 10406: 10397: 10392: 10391: 10382: 10377: 10376: 10375: 10371: 10366: 10363: 10362: 10358: 10354: 10351: 10339: 10335: 10331: 10326: 10323: 10315: 10312: 10301: 10297: 10289: 10257: 10253: 10240: 10236: 10227: 10222: 10221: 10212: 10207: 10206: 10205: 10201: 10195: 10191: 10170: 10166: 10153: 10149: 10140: 10135: 10134: 10125: 10120: 10119: 10118: 10114: 10108: 10103: 10093: 10088: 10087: 10081: 10077: 10048: 10047: 10034: 10029: 10028: 10022: 10018: 9989: 9988: 9978: 9973: 9972: 9966: 9962: 9933: 9932: 9916: 9912: 9911: 9895: 9891: 9890: 9871: 9867: 9866: 9850: 9846: 9837: 9833: 9828: 9825: 9824: 9818: 9792: 9773: 9769: 9750: 9746: 9737: 9732: 9731: 9716: 9711: 9710: 9705: 9704: 9702: 9696: 9695: 9690: 9687: 9686: 9681: 9677: 9658: 9654: 9635: 9631: 9622: 9617: 9616: 9601: 9596: 9595: 9587: 9586: 9584: 9576: 9564: 9559: 9558: 9552: 9548: 9540: 9536: 9535: 9531: 9522: 9517: 9516: 9510: 9506: 9498: 9494: 9493: 9489: 9488: 9486: 9481:discrete labels 9479: 9461: 9457: 9438: 9434: 9433: 9426: 9424: 9409: 9407: 9404: 9403: 9369: 9365: 9352: 9348: 9335: 9331: 9322: 9317: 9316: 9307: 9302: 9301: 9292: 9287: 9286: 9278: 9275: 9274: 9264: 9256: 9226: 9221: 9220: 9211: 9206: 9205: 9190: 9185: 9184: 9169: 9164: 9163: 9159: 9155: 9129: 9124: 9123: 9114: 9109: 9108: 9093: 9088: 9087: 9072: 9067: 9066: 9062: 9058: 9053: 9050: 9049: 9046: 9038: 9034: 9027: 9021: 9017: 9010: 9004: 8994:distinguishable 8988: 8978:Pauli principle 8969: 8961: 8933: 8928: 8927: 8912: 8907: 8906: 8902: 8898: 8872: 8867: 8866: 8851: 8846: 8845: 8841: 8837: 8832: 8829: 8828: 8824:distinguishable 8806: 8802: 8798: 8795: 8787: 8761: 8756: 8755: 8746: 8741: 8740: 8731: 8726: 8725: 8717: 8714: 8713: 8707: 8680: 8626: 8622: 8617: 8604: 8599: 8587: 8583: 8575: 8570: 8568: 8565: 8564: 8539: 8535: 8514: 8490: 8486: 8478: 8470: 8467: 8466: 8436: 8411: 8388: 8386: 8383: 8382: 8362: 8358: 8353: 8340: 8335: 8319: 8315: 8294: 8282: 8275: 8271: 8260: 8256: 8255: 8228: 8203: 8201: 8198: 8197: 8188: 8187: 8178: 8177: 8173: 8148: 8144: 8136: 8131: 8115: 8111: 8103: 8091: 8084: 8080: 8069: 8065: 8064: 8037: 8035: 8032: 8031: 8006: 8005: 7982: 7973: 7972: 7937: 7928: 7927: 7921: 7920: 7894: 7885: 7884: 7864: 7851: 7850: 7833: 7825: 7822: 7821: 7795: 7791: 7783: 7775: 7772: 7771: 7729: 7721: 7715: 7693: 7689: 7684: 7668: 7664: 7652: 7636: 7632: 7631: 7604: 7602: 7599: 7598: 7574: 7573: 7567: 7566: 7560: 7559: 7553: 7552: 7546: 7545: 7535: 7534: 7503: 7502: 7496: 7495: 7489: 7488: 7482: 7481: 7475: 7474: 7464: 7463: 7414: 7413: 7407: 7406: 7400: 7399: 7393: 7392: 7386: 7385: 7375: 7374: 7340: 7339: 7333: 7332: 7326: 7325: 7319: 7318: 7312: 7311: 7301: 7300: 7272: 7271: 7247: 7246: 7210: 7209: 7203: 7202: 7175: 7174: 7149: 7148: 7140: 7137: 7136: 7123: 7105: 7101: 7096: 7092: 7088: 7084: 7081: 7075: 7063: 7057: 7053: 7048: 7044: 7040: 7033: 7027: 7001: 6997: 6989: 6986: 6985: 6957: 6952: 6937: 6925: 6918: 6914: 6888: 6886: 6883: 6882: 6867: 6861: 6858:position vector 6851: 6826: 6818: 6815: 6814: 6811: 6786: 6782: 6777: 6769: 6764: 6752: 6748: 6740: 6735: 6733: 6730: 6729: 6705: 6701: 6696: 6693: 6692: 6674: 6673: 6664: 6660: 6657: 6656: 6641: 6637: 6634: 6633: 6627: 6626: 6614: 6610: 6607: 6606: 6600: 6596: 6589: 6588: 6578: 6577: 6566: 6554: 6553: 6542: 6518: 6517: 6511: 6510: 6499: 6484: 6483: 6472: 6459: 6458: 6444: 6442: 6439: 6438: 6409: 6408: 6400: 6395: 6390: 6385: 6379: 6378: 6373: 6353: 6348: 6343: 6337: 6336: 6331: 6326: 6321: 6316: 6310: 6309: 6304: 6299: 6294: 6283: 6277: 6276: 6271: 6266: 6261: 6256: 6246: 6245: 6236: 6225: 6224: 6223: 6213: 6211: 6208: 6207: 6198: 6192: 6191: 6171: 6170: 6164: 6163: 6157: 6156: 6150: 6149: 6143: 6142: 6132: 6131: 6114: 6102: 6101: 6095: 6094: 6088: 6087: 6081: 6080: 6074: 6073: 6063: 6062: 6033: 6014: 6013: 6007: 6006: 6000: 5999: 5993: 5992: 5986: 5985: 5975: 5974: 5954: 5942: 5941: 5935: 5934: 5928: 5927: 5921: 5920: 5914: 5913: 5903: 5902: 5888: 5886: 5883: 5882: 5876: 5870: 5869: 5841: 5838: 5837: 5817: 5813: 5796: 5793: 5792: 5776: 5773: 5772: 5727: 5724: 5723: 5707: 5704: 5703: 5700: 5674: 5670: 5665: 5663: 5660: 5659: 5652: 5627: 5623: 5618: 5616: 5613: 5612: 5587: 5581: 5577: 5569: 5566: 5565: 5539: 5535: 5530: 5525: 5522: 5521: 5492: 5488: 5483: 5478: 5475: 5474: 5458: 5446: 5442: 5424: 5420: 5415: 5409: 5396: 5385: 5384: 5383: 5381: 5378: 5377: 5360: 5355: 5354: 5343: 5337: 5326: 5325: 5324: 5319: 5310: 5305: 5304: 5293: 5281: 5277: 5269: 5263: 5241: 5237: 5235: 5232: 5231: 5212: 5208: 5206: 5203: 5202: 5169: 5165: 5160: 5155: 5152: 5151: 5135: 5132: 5131: 5108: 5104: 5099: 5094: 5091: 5090: 5071: 5066: 5065: 5054: 5048: 5044: 5036: 5024: 5020: 5011: 5007: 5005: 5002: 5001: 4978: 4974: 4972: 4969: 4968: 4947: 4943: 4941: 4938: 4937: 4907: 4903: 4898: 4893: 4890: 4889: 4886: 4856: 4850: 4846: 4838: 4835: 4834: 4812: 4806: 4802: 4790: 4786: 4781: 4775: 4757: 4740: 4738: 4735: 4734: 4706: 4702: 4697: 4692: 4689: 4688: 4666: 4660: 4656: 4644: 4640: 4635: 4629: 4617: 4614: 4613: 4588: 4584: 4579: 4574: 4571: 4570: 4564: 4528: 4519: 4508: 4503: 4499: 4446: 4445: 4441: 4408: 4391: 4388: 4387: 4346: 4342: 4338: 4305: 4288: 4285: 4284: 4252: 4248: 4244: 4226: 4212: 4185: 4184: 4180: 4162: 4133: 4125: 4122: 4121: 4082: 4068: 4045: 4025: 4011: 4006: 3989: 3960: 3937: 3934: 3933: 3929: 3913: 3912: 3892: 3857: 3840: 3830: 3819: 3802: 3799: 3798: 3778: 3743: 3726: 3716: 3705: 3688: 3684: 3682: 3679: 3678: 3672: 3666: 3663: 3630: 3602: 3601: 3597: 3588: 3584: 3579: 3576: 3575: 3511: 3507: 3502: 3499: 3498: 3494: 3466: 3456: 3452: 3434: 3430: 3428: 3425: 3424: 3380: 3376: 3355: 3350: 3339: 3331: 3315: 3311: 3302: 3298: 3293: 3290: 3289: 3277: 3273: 3261: 3257: 3250: 3246: 3242: 3234: 3203: 3200: 3199: 3192: 3157: 3140: 3129: 3126: 3125: 3100: 3084: 3067: 3063: 3059: 3039: 3022: 3005: 3003: 3000: 2999: 2977: 2951: 2943: 2911: 2903: 2898: 2895: 2894: 2866: 2846: 2838: 2833: 2830: 2829: 2812: 2811: 2807: 2741: 2697: 2695: 2692: 2691: 2675: 2672: 2658: 2601: 2596: 2595: 2572: 2565: 2557: 2551: 2548: 2547: 2539: 2516: 2511: 2510: 2487: 2480: 2475: 2441: 2437: 2435: 2432: 2431: 2417: 2411: 2408: 2385: 2313: 2309: 2300: 2276: 2272: 2271: 2269: 2266: 2265: 2250: 2246: 2238: 2234: 2199: 2196: 2195: 2192: 2167: 2155: 2153: 2147: 2145: 2120: 2116: 2090: 2087: 2086: 2079: 2026: 2022: 2001: 1996: 1985: 1977: 1961: 1957: 1948: 1944: 1939: 1936: 1935: 1928: 1925: 1921: 1918: 1914: 1909:. As such, the 1884: 1881: 1880: 1862: 1861: 1860: 1859: 1853: 1849: 1841: 1837: 1829: 1821: 1813: 1805: 1797: 1796: 1795: 1792: 1783: 1782: 1781: 1767: 1752: 1741: 1735: 1732: 1717: 1701: 1690: 1649: 1641: 1623: 1619: 1618: 1617: 1609: 1599: 1595: 1594: 1593: 1589: 1579:field operators 1543: 1479:electromagnetic 1322:now called the 1300: 1292: 1289: 1288: 1287: 1284:Planck constant 1267: 1264: 1263: 1240: 1232: 1229: 1228: 1227: 1210: 1207: 1206: 1205: 1186: 1183: 1182: 1152: 1149: 1148: 1147: 1130: 1127: 1126: 1125: 1109: 1106: 1105: 1094: 1065: 1064: 1063: 828: 820: 819: 765: 764:Advanced topics 757: 756: 755: 707:Hidden-variable 697:de Broglie–Bohm 676: 674:Interpretations 666: 665: 664: 634: 626: 625: 624: 582: 574: 573: 572: 539: 495:CHSH inequality 484: 476: 475: 474: 403:Complementarity 397: 389: 388: 387: 355: 326: 301: 290: 289: 275: 265: 260: 252: 249: 248: 231: 219:interpretations 177: 173: 172: 171: 167: 150:, have nonzero 122: 103:squared modulus 82: 76: 71:of an isolated 57:quantum physics 24: 17: 12: 11: 5: 19279: 19269: 19268: 19263: 19261:Quantum states 19246: 19245: 19243: 19242: 19231: 19228: 19227: 19225: 19224: 19219: 19214: 19209: 19208: 19207: 19196: 19194: 19190: 19189: 19187: 19186: 19181: 19176: 19175: 19174: 19164: 19159: 19157:Casimir effect 19154: 19148: 19146: 19142: 19141: 19139: 19138: 19133: 19128: 19123: 19118: 19116:Quantum optics 19113: 19108: 19103: 19098: 19093: 19088: 19083: 19078: 19073: 19068: 19063: 19058: 19053: 19048: 19043: 19038: 19037: 19036: 19026: 19021: 19016: 19011: 19010: 19009: 18999: 18994: 18989: 18983: 18981: 18975: 18974: 18972: 18971: 18966: 18961: 18956: 18951: 18946: 18941: 18936: 18931: 18926: 18921: 18915: 18913: 18907: 18906: 18904: 18903: 18898: 18893: 18891:Quantum eraser 18888: 18883: 18878: 18873: 18868: 18863: 18858: 18853: 18847: 18845: 18841: 18840: 18838: 18837: 18832: 18827: 18822: 18817: 18812: 18807: 18806: 18805: 18804: 18803: 18788: 18783: 18778: 18773: 18768: 18762: 18760: 18754: 18753: 18751: 18750: 18745: 18740: 18735: 18730: 18724: 18722: 18718: 18717: 18715: 18714: 18709: 18704: 18699: 18694: 18689: 18684: 18678: 18676: 18672: 18671: 18669: 18668: 18667: 18666: 18661: 18651: 18646: 18641: 18636: 18631: 18626: 18621: 18616: 18611: 18606: 18601: 18596: 18591: 18590: 18589: 18584: 18579: 18574: 18564: 18562:Density matrix 18559: 18554: 18549: 18543: 18541: 18537: 18536: 18534: 18533: 18528: 18523: 18518: 18517: 18516: 18506: 18500: 18498: 18494: 18493: 18486: 18485: 18478: 18471: 18463: 18457: 18456: 18448: 18436: 18431: 18426: 18421: 18414: 18413:External links 18411: 18409: 18408: 18402: 18386: 18364: 18362: 18359: 18357: 18356: 18350: 18337: 18331: 18318: 18312: 18299: 18283: 18278: 18262: 18253: 18238: 18232: 18219: 18213: 18200: 18194: 18181: 18133: 18127: 18112: 18106: 18093: 18053: 18047: 18034: 18024: 18007: 18001: 17988: 17970: 17960:Wave Mechanics 17953: 17943: 17925: 17919: 17891: 17882: 17874:Pergamon Press 17858: 17852: 17835: 17819:Heisenberg, W. 17815: 17804:10.1086/351880 17798:(4): 606–609, 17787: 17767:Pergamon Press 17751: 17745: 17730: 17724: 17711: 17705: 17687: 17677: 17658: 17652: 17640:Schilpp, P. A. 17635: 17612: 17599: 17579:(6): 132–148. 17559: 17553: 17537: 17511:(3): 416–418. 17500: 17494: 17474: 17468: 17455: 17445: 17425:Byron, F. W.; 17422: 17406:de Broglie, L. 17402: 17388:Comptes Rendus 17383: 17346: 17308: 17271: 17234: 17228: 17212: 17206: 17193: 17160:paper of 1905" 17151: 17138: 17136: 17133: 17131: 17130: 17128:, p. 682. 17118: 17106: 17094: 17092:, p. 258. 17082: 17070: 17058: 17046: 17034: 17022: 17010: 17006:Griffiths 2008 16998: 16982: 16970: 16963: 16945: 16943:, p. 463. 16933: 16921: 16909: 16897: 16876: 16864: 16860:Griffiths 2004 16849: 16847:, p. 117. 16837: 16833:Griffiths 2004 16825: 16813: 16801: 16789: 16777: 16761: 16749: 16737: 16720: 16708: 16691: 16679: 16667: 16655: 16643: 16623: 16592: 16563: 16551: 16539: 16527: 16515: 16507:Camilleri 2009 16503:Heisenberg, W. 16495: 16493:, p. 143. 16483: 16471: 16459: 16429: 16407: 16405: 16402: 16399: 16398: 16381: 16377: 16372: 16368: 16361: 16357: 16352: 16344: 16340: 16335: 16331: 16324: 16320: 16316: 16313: 16310: 16305: 16301: 16297: 16292: 16288: 16283: 16279: 16274: 16270: 16253: 16244: 16230: 16222:an element of 16203: 16171: 16155: 16138: 16113: 16099: 16077: 16043: 16024: 16011: 16003:Schauder basis 15986: 15949: 15948: 15946: 15943: 15941: 15940: 15935: 15930: 15925: 15920: 15915: 15910: 15905: 15900: 15894: 15892: 15889: 15836:Main article: 15833: 15830: 15822: 15816: 15810: 15773: 15766: 15762: 15757: 15754: 15751: 15746: 15743: 15740: 15736: 15730: 15726: 15720: 15717: 15713: 15708: 15704: 15701: 15675: 15668: 15664: 15658: 15655: 15652: 15647: 15644: 15641: 15637: 15631: 15627: 15621: 15618: 15614: 15609: 15605: 15602: 15599: 15596: 15593: 15540: 15535: 15530: 15527: 15524: 15520: 15516: 15512: 15508: 15505: 15501: 15497: 15494: 15491: 15488: 15483: 15480: 15477: 15473: 15438: 15422: 15421: 15404: 15397: 15390: 15376: 15366: 15310: 15295: 15262: 15248: 15239: 15235: 15231: 15207: 15198: 15189: 15183: 15176: 15150: 15149: 15142: 15136: 15129: 15119: 15112: 15106: 15099: 15089: 15061: 15051: 15045: 15039: 15026: 14997: 14993: 14989: 14985: 14980: 14975: 14972: 14969: 14965: 14961: 14957: 14953: 14950: 14944: 14937: 14933: 14929: 14924: 14920: 14916: 14913: 14910: 14906: 14876:Main article: 14873: 14870: 14842: 14841: 14830: 14793: 14790: 14782: 14781: 14728:rotation group 14720: 14719: 14716:Dirac equation 14705: 14702:Pauli equation 14676:, Chapter 5). 14646: 14645: 14634: 14618: 14613:for which the 14607: 14588: 14575: 14555: 14518: 14482: 14479: 14377: 14374: 14348: 14341: 14333: 14329: 14322: 14315: 14289: 14284: 14279: 14274: 14271: 14268: 14265: 14262: 14258: 14254: 14251: 14199: 14198: 14187: 14157: 14152: 14149: 14145: 14141: 14138: 14135: 14131: 14126: 14122: 14116: 14111: 14107: 14103: 14088: 14073: 14069: 14049: 14045: 14019: 14016: 14011: 14010: 14002: 13986: 13969: 13952: 13951: 13923: 13907: 13884: 13866: 13859: 13840:representation 13824: 13821: 13817: 13816: 13766: 13757: 13754: 13717: 13714: 13713: 13712: 13705: 13682: 13667: 13544: 13526: 13517: 13510: 13494: 13491: 13488: 13485: 13482: 13477: 13472: 13468: 13464: 13460: 13452: 13448: 13444: 13439: 13436: 13430: 13424: 13421: 13418: 13415: 13410: 13407: 13404: 13401: 13398: 13394: 13388: 13383: 13375: 13371: 13367: 13362: 13359: 13353: 13344: 13340: 13336: 13332: 13328: 13325: 13321: 13312: 13309: 13306: 13303: 13300: 13297: 13294: 13291: 13288: 13285: 13280: 13277: 13274: 13271: 13268: 13265: 13262: 13259: 13251: 13245: 13237: 13233: 13229: 13225: 13220: 13212: 13209: 13206: 13203: 13200: 13197: 13194: 13191: 13186: 13183: 13180: 13176: 13133: 13108: 13105: 13102: 13099: 13096: 13090: 13085: 13081: 13075: 13072: 13069: 13066: 13063: 13060: 13057: 13054: 13051: 13048: 13045: 13042: 13037: 13034: 13031: 13027: 12999: 12996: 12972:= 0, 1, 2, ... 12954: 12950: 12944: 12940: 12937: 12929: 12922: 12918: 12914: 12906: 12903: 12896: 12892: 12888: 12885: 12879: 12875: 12871: 12866: 12862: 12858: 12853: 12847: 12844: 12839: 12836: 12830: 12825: 12818: 12815: 12809: 12805: 12800: 12794: 12791: 12788: 12785: 12780: 12776: 12761: 12747: 12744: 12699: 12690: 12681: 12672: 12663: 12653: 12622: 12617: 12614: 12611: 12608: 12605: 12601: 12598: 12595: 12592: 12588: 12581: 12576: 12572: 12567: 12564: 12561: 12557: 12550: 12545: 12541: 12540: 12537: 12534: 12531: 12527: 12523: 12519: 12515: 12511: 12508: 12505: 12501: 12494: 12489: 12485: 12480: 12477: 12473: 12466: 12461: 12457: 12456: 12453: 12450: 12447: 12444: 12441: 12438: 12434: 12431: 12428: 12425: 12421: 12414: 12409: 12405: 12400: 12397: 12394: 12390: 12383: 12378: 12374: 12373: 12371: 12366: 12363: 12360: 12357: 12354: 12322: 12317: 12314: 12310: 12306: 12302: 12298: 12296: 12293: 12292: 12289: 12286: 12282: 12278: 12274: 12270: 12266: 12262: 12258: 12257: 12255: 12250: 12247: 12244: 12241: 12238: 12213: 12200: 12188: 12185: 12180: 12177: 12099: 12095: 12090: 12085: 12080: 12077: 12074: 12069: 12064: 12059: 12054: 12049: 12044: 12041: 12035: 12031: 12027: 12024: 12021: 12018: 12014: 12010: 12007: 12004: 12001: 11996: 11991: 11986: 11983: 11980: 11975: 11970: 11965: 11960: 11955: 11950: 11947: 11927:Main article: 11924: 11921: 11898: 11895: 11892: 11882: 11850: 11847: 11841: 11838: 11833: 11830: 11824: 11821: 11817: 11813: 11810: 11807: 11802: 11797: 11792: 11789: 11786: 11782: 11778: 11775: 11772: 11768: 11761: 11758: 11754: 11728: 11723: 11718: 11715: 11711: 11707: 11703: 11699: 11694: 11690: 11685: 11681: 11655: 11651: 11648: 11623: 11619: 11616: 11592: 11588: 11585: 11581: 11549: 11545: 11542: 11539: 11533: 11530: 11527: 11524: 11520: 11516: 11512: 11489: 11486: 11483: 11479: 11475: 11472: 11450: 11445: 11440: 11437: 11434: 11430: 11426: 11423: 11419: 11415: 11412: 11409: 11406: 11402: 11398: 11395: 11373: 11369: 11366: 11363: 11359: 11355: 11352: 11349: 11343: 11340: 11335: 11332: 11329: 11325: 11321: 11318: 11313: 11310: 11307: 11304: 11300: 11296: 11293: 11266: 11263: 11260: 11255: 11251: 11247: 11237: 11234: 11229: 11226: 11221: 11217: 11213: 11210: 11207: 11204: 11201: 11196: 11192: 11188: 11182: 11179: 11176: 11172: 11167: 11164: 11161: 11158: 11154: 11150: 11146: 11123: 11118: 11113: 11110: 11107: 11103: 11099: 11096: 11092: 11088: 11085: 11082: 11079: 11075: 11071: 11068: 11048: 11045: 11041: 11037: 11034: 11031: 11025: 11022: 11017: 11014: 10997: 10994: 10993: 10992: 10979: 10974: 10969: 10965: 10962: 10957: 10953: 10949: 10944: 10940: 10936: 10931: 10926: 10921: 10916: 10911: 10905: 10901: 10897: 10890: 10885: 10878: 10874: 10866: 10862: 10857: 10853: 10848: 10843: 10836: 10832: 10824: 10820: 10815: 10809: 10804: 10797: 10793: 10785: 10781: 10776: 10772: 10769: 10766: 10763: 10756: 10752: 10748: 10743: 10739: 10735: 10731: 10726: 10722: 10718: 10713: 10708: 10703: 10700: 10697: 10692: 10688: 10684: 10679: 10675: 10671: 10667: 10662: 10658: 10654: 10649: 10644: 10638: 10615: 10605: 10596: 10584: 10574: 10565: 10545: 10540: 10535: 10531: 10528: 10523: 10519: 10515: 10511: 10506: 10502: 10498: 10494: 10489: 10484: 10479: 10474: 10469: 10463: 10459: 10455: 10450: 10446: 10442: 10439: 10434: 10430: 10426: 10422: 10417: 10413: 10409: 10405: 10400: 10395: 10390: 10385: 10380: 10374: 10370: 10350: 10347: 10337: 10333: 10329: 10319: 10308: 10276: 10272: 10269: 10264: 10260: 10256: 10252: 10247: 10243: 10239: 10235: 10230: 10225: 10220: 10215: 10210: 10204: 10198: 10194: 10189: 10185: 10182: 10177: 10173: 10169: 10165: 10160: 10156: 10152: 10148: 10143: 10138: 10133: 10128: 10123: 10117: 10111: 10106: 10102: 10096: 10091: 10084: 10080: 10073: 10070: 10067: 10064: 10061: 10057: 10054: 10051: 10046: 10042: 10037: 10032: 10025: 10021: 10014: 10011: 10008: 10005: 10002: 9998: 9995: 9992: 9987: 9981: 9976: 9969: 9965: 9958: 9955: 9952: 9949: 9946: 9942: 9939: 9936: 9931: 9923: 9919: 9915: 9910: 9902: 9898: 9894: 9889: 9885: 9878: 9874: 9870: 9865: 9861: 9858: 9853: 9849: 9845: 9840: 9836: 9832: 9801: 9789: 9785: 9780: 9776: 9772: 9768: 9765: 9762: 9757: 9753: 9749: 9745: 9740: 9735: 9730: 9727: 9724: 9719: 9714: 9708: 9689: 9688: 9680: 9679: 9674: 9670: 9665: 9661: 9657: 9653: 9650: 9647: 9642: 9638: 9634: 9630: 9625: 9620: 9615: 9612: 9609: 9604: 9599: 9594: 9590: 9573: 9567: 9562: 9555: 9551: 9543: 9539: 9534: 9530: 9525: 9520: 9513: 9509: 9501: 9497: 9492: 9476: 9468: 9464: 9460: 9456: 9453: 9450: 9445: 9441: 9437: 9432: 9428: 9422: 9419: 9416: 9412: 9387: 9384: 9381: 9376: 9372: 9368: 9364: 9359: 9355: 9351: 9347: 9342: 9338: 9334: 9330: 9325: 9320: 9315: 9310: 9305: 9300: 9295: 9290: 9285: 9282: 9260: 9241: 9237: 9234: 9229: 9224: 9219: 9214: 9209: 9204: 9201: 9198: 9193: 9188: 9183: 9180: 9177: 9172: 9167: 9162: 9158: 9154: 9151: 9148: 9144: 9140: 9137: 9132: 9127: 9122: 9117: 9112: 9107: 9104: 9101: 9096: 9091: 9086: 9083: 9080: 9075: 9070: 9065: 9061: 9057: 9042: 9032: 9025: 9015: 9008: 8948: 8944: 8941: 8936: 8931: 8926: 8923: 8920: 8915: 8910: 8905: 8901: 8897: 8894: 8891: 8887: 8883: 8880: 8875: 8870: 8865: 8862: 8859: 8854: 8849: 8844: 8840: 8836: 8791: 8775: 8772: 8769: 8764: 8759: 8754: 8749: 8744: 8739: 8734: 8729: 8724: 8721: 8679: 8676: 8654:magnetic field 8634: 8629: 8625: 8620: 8615: 8611: 8607: 8602: 8598: 8595: 8590: 8586: 8582: 8578: 8573: 8553: 8550: 8547: 8542: 8538: 8534: 8531: 8527: 8524: 8521: 8517: 8513: 8510: 8507: 8504: 8501: 8498: 8493: 8489: 8485: 8481: 8477: 8474: 8455: 8452: 8449: 8446: 8443: 8439: 8434: 8430: 8427: 8424: 8421: 8418: 8414: 8410: 8407: 8404: 8401: 8398: 8395: 8391: 8370: 8365: 8361: 8356: 8351: 8347: 8343: 8338: 8333: 8330: 8327: 8322: 8318: 8314: 8311: 8307: 8304: 8301: 8297: 8293: 8290: 8285: 8278: 8274: 8270: 8263: 8259: 8254: 8250: 8247: 8244: 8241: 8238: 8235: 8231: 8226: 8222: 8219: 8216: 8213: 8210: 8206: 8171:tensor product 8156: 8151: 8147: 8143: 8139: 8134: 8129: 8126: 8123: 8118: 8114: 8110: 8106: 8102: 8099: 8094: 8087: 8083: 8079: 8072: 8068: 8063: 8059: 8056: 8053: 8050: 8047: 8044: 8040: 8010: 8004: 8001: 7998: 7995: 7992: 7989: 7985: 7981: 7978: 7975: 7974: 7971: 7968: 7965: 7962: 7959: 7956: 7953: 7950: 7947: 7944: 7940: 7936: 7933: 7930: 7929: 7926: 7923: 7922: 7919: 7916: 7913: 7910: 7907: 7904: 7901: 7897: 7893: 7890: 7887: 7886: 7883: 7880: 7877: 7874: 7871: 7867: 7863: 7860: 7857: 7856: 7854: 7849: 7846: 7843: 7840: 7836: 7832: 7829: 7809: 7806: 7803: 7798: 7794: 7790: 7786: 7782: 7779: 7701: 7696: 7692: 7687: 7682: 7679: 7676: 7671: 7667: 7663: 7660: 7655: 7650: 7647: 7644: 7639: 7635: 7630: 7626: 7623: 7620: 7617: 7614: 7611: 7607: 7578: 7572: 7569: 7568: 7565: 7562: 7561: 7558: 7555: 7554: 7551: 7548: 7547: 7544: 7541: 7540: 7538: 7533: 7530: 7527: 7524: 7521: 7518: 7515: 7512: 7507: 7501: 7498: 7497: 7494: 7491: 7490: 7487: 7484: 7483: 7480: 7477: 7476: 7473: 7470: 7469: 7467: 7462: 7459: 7456: 7453: 7450: 7447: 7444: 7441: 7438: 7435: 7432: 7429: 7426: 7423: 7418: 7412: 7409: 7408: 7405: 7402: 7401: 7398: 7395: 7394: 7391: 7388: 7387: 7384: 7381: 7380: 7378: 7373: 7370: 7367: 7364: 7361: 7358: 7355: 7352: 7349: 7344: 7338: 7335: 7334: 7331: 7328: 7327: 7324: 7321: 7320: 7317: 7314: 7313: 7310: 7307: 7306: 7304: 7299: 7296: 7293: 7290: 7287: 7284: 7281: 7276: 7270: 7267: 7264: 7261: 7258: 7255: 7252: 7249: 7248: 7245: 7242: 7239: 7236: 7233: 7230: 7227: 7224: 7221: 7218: 7215: 7212: 7211: 7208: 7205: 7204: 7201: 7198: 7195: 7192: 7189: 7186: 7183: 7180: 7177: 7176: 7173: 7170: 7167: 7164: 7161: 7158: 7155: 7154: 7152: 7147: 7144: 7099: 7079: 7051: 7031: 7015: 7012: 7009: 7004: 7000: 6996: 6993: 6964: 6960: 6955: 6950: 6947: 6944: 6940: 6936: 6933: 6928: 6921: 6917: 6913: 6910: 6907: 6904: 6901: 6898: 6895: 6891: 6880:Dirac notation 6839: 6836: 6833: 6829: 6825: 6822: 6810: 6807: 6794: 6789: 6785: 6780: 6776: 6772: 6767: 6763: 6760: 6755: 6751: 6747: 6743: 6738: 6713: 6708: 6704: 6700: 6678: 6670: 6667: 6663: 6659: 6658: 6653: 6650: 6647: 6644: 6640: 6636: 6635: 6632: 6629: 6628: 6623: 6620: 6617: 6613: 6609: 6608: 6603: 6599: 6595: 6594: 6592: 6587: 6582: 6576: 6573: 6569: 6565: 6562: 6559: 6556: 6555: 6552: 6549: 6545: 6541: 6538: 6535: 6532: 6529: 6526: 6523: 6520: 6519: 6516: 6513: 6512: 6509: 6506: 6502: 6498: 6495: 6492: 6489: 6486: 6485: 6482: 6479: 6475: 6471: 6468: 6465: 6464: 6462: 6457: 6454: 6451: 6447: 6413: 6407: 6404: 6401: 6399: 6396: 6394: 6391: 6389: 6386: 6384: 6381: 6380: 6377: 6374: 6372: 6369: 6366: 6363: 6360: 6357: 6354: 6352: 6349: 6347: 6344: 6342: 6339: 6338: 6335: 6332: 6330: 6327: 6325: 6322: 6320: 6317: 6315: 6312: 6311: 6308: 6305: 6303: 6300: 6298: 6295: 6293: 6290: 6287: 6284: 6282: 6279: 6278: 6275: 6272: 6270: 6267: 6265: 6262: 6260: 6257: 6255: 6252: 6251: 6249: 6244: 6239: 6232: 6229: 6220: 6217: 6196: 6175: 6169: 6166: 6165: 6162: 6159: 6158: 6155: 6152: 6151: 6148: 6145: 6144: 6141: 6138: 6137: 6135: 6130: 6127: 6124: 6121: 6117: 6112: 6106: 6100: 6097: 6096: 6093: 6090: 6089: 6086: 6083: 6082: 6079: 6076: 6075: 6072: 6069: 6068: 6066: 6061: 6058: 6055: 6052: 6049: 6046: 6043: 6040: 6036: 6031: 6027: 6024: 6018: 6012: 6009: 6008: 6005: 6002: 6001: 5998: 5995: 5994: 5991: 5988: 5987: 5984: 5981: 5980: 5978: 5973: 5970: 5967: 5964: 5961: 5957: 5952: 5946: 5940: 5937: 5936: 5933: 5930: 5929: 5926: 5923: 5922: 5919: 5916: 5915: 5912: 5909: 5908: 5906: 5901: 5898: 5895: 5891: 5874: 5854: 5851: 5848: 5845: 5836:which acts on 5820: 5816: 5812: 5809: 5806: 5803: 5800: 5784:{\textstyle s} 5780: 5740: 5737: 5734: 5731: 5715:{\textstyle s} 5711: 5699: 5696: 5682: 5677: 5673: 5668: 5651: 5648: 5635: 5630: 5626: 5621: 5600: 5597: 5594: 5590: 5584: 5580: 5576: 5573: 5550: 5547: 5542: 5538: 5533: 5529: 5509: 5506: 5501: 5498: 5495: 5491: 5486: 5482: 5461: 5455: 5452: 5449: 5445: 5441: 5438: 5433: 5430: 5427: 5423: 5418: 5412: 5408: 5404: 5399: 5392: 5389: 5363: 5358: 5353: 5350: 5346: 5340: 5333: 5330: 5322: 5318: 5313: 5308: 5303: 5300: 5296: 5290: 5287: 5284: 5280: 5276: 5272: 5266: 5262: 5258: 5255: 5252: 5249: 5244: 5240: 5215: 5211: 5186: 5183: 5178: 5175: 5172: 5168: 5163: 5159: 5139: 5119: 5116: 5111: 5107: 5102: 5098: 5074: 5069: 5064: 5061: 5057: 5051: 5047: 5043: 5039: 5035: 5032: 5027: 5023: 5019: 5014: 5010: 4981: 4977: 4950: 4946: 4918: 4915: 4910: 4906: 4901: 4897: 4885: 4882: 4869: 4866: 4863: 4859: 4853: 4849: 4845: 4842: 4822: 4819: 4815: 4809: 4805: 4801: 4798: 4793: 4789: 4784: 4778: 4774: 4770: 4767: 4764: 4760: 4756: 4753: 4750: 4747: 4743: 4717: 4714: 4709: 4705: 4700: 4696: 4676: 4673: 4669: 4663: 4659: 4655: 4652: 4647: 4643: 4638: 4632: 4628: 4624: 4621: 4599: 4596: 4591: 4587: 4582: 4578: 4532:Hilbert spaces 4527: 4524: 4518: 4515: 4473: 4469: 4466: 4461: 4458: 4453: 4450: 4444: 4440: 4437: 4434: 4431: 4428: 4422: 4419: 4416: 4412: 4407: 4404: 4401: 4398: 4395: 4373: 4369: 4366: 4361: 4358: 4353: 4350: 4345: 4341: 4337: 4334: 4331: 4328: 4325: 4319: 4316: 4313: 4309: 4304: 4301: 4298: 4295: 4292: 4272: 4267: 4264: 4259: 4256: 4251: 4247: 4240: 4237: 4234: 4230: 4225: 4222: 4219: 4215: 4211: 4208: 4205: 4200: 4197: 4192: 4189: 4183: 4176: 4173: 4170: 4166: 4161: 4158: 4155: 4152: 4149: 4146: 4143: 4140: 4136: 4132: 4129: 4107: 4104: 4101: 4098: 4095: 4092: 4088: 4085: 4081: 4078: 4074: 4071: 4067: 4064: 4061: 4058: 4055: 4051: 4048: 4044: 4041: 4038: 4035: 4031: 4028: 4024: 4021: 4017: 4014: 4009: 4005: 4002: 3999: 3995: 3992: 3988: 3985: 3982: 3979: 3976: 3973: 3970: 3967: 3963: 3959: 3956: 3953: 3950: 3947: 3944: 3941: 3911: 3908: 3905: 3902: 3899: 3895: 3891: 3888: 3885: 3882: 3879: 3876: 3873: 3870: 3867: 3864: 3860: 3856: 3853: 3850: 3847: 3843: 3839: 3836: 3833: 3831: 3829: 3826: 3822: 3818: 3815: 3812: 3809: 3805: 3801: 3800: 3797: 3794: 3791: 3788: 3785: 3781: 3777: 3774: 3771: 3768: 3765: 3762: 3759: 3756: 3753: 3750: 3746: 3742: 3739: 3736: 3733: 3729: 3725: 3722: 3719: 3717: 3715: 3712: 3708: 3704: 3701: 3698: 3695: 3691: 3687: 3686: 3662: 3659: 3643: 3640: 3636: 3633: 3629: 3626: 3623: 3620: 3617: 3614: 3608: 3605: 3600: 3596: 3591: 3587: 3583: 3568:momentum basis 3555: 3552: 3549: 3546: 3543: 3540: 3537: 3534: 3531: 3528: 3525: 3522: 3519: 3514: 3510: 3506: 3478: 3473: 3469: 3465: 3462: 3459: 3455: 3451: 3448: 3445: 3442: 3437: 3433: 3410: 3406: 3403: 3400: 3397: 3394: 3391: 3388: 3383: 3379: 3375: 3372: 3369: 3366: 3363: 3358: 3353: 3349: 3342: 3337: 3334: 3330: 3326: 3323: 3318: 3314: 3310: 3305: 3301: 3297: 3275: 3259: 3222: 3219: 3216: 3213: 3210: 3207: 3196:momentum space 3191: 3188: 3171: 3167: 3164: 3160: 3156: 3153: 3150: 3147: 3143: 3139: 3136: 3133: 3110: 3107: 3103: 3098: 3094: 3091: 3087: 3083: 3080: 3077: 3074: 3070: 3066: 3062: 3058: 3055: 3052: 3049: 3046: 3042: 3038: 3035: 3032: 3029: 3025: 3021: 3018: 3015: 3012: 3008: 2987: 2983: 2980: 2976: 2973: 2970: 2967: 2964: 2961: 2958: 2954: 2949: 2946: 2942: 2939: 2936: 2933: 2930: 2927: 2924: 2921: 2918: 2914: 2909: 2906: 2902: 2882: 2879: 2876: 2872: 2869: 2865: 2862: 2859: 2856: 2853: 2849: 2844: 2841: 2837: 2826:proper vectors 2824:which, unlike 2804: 2803: 2796:quantum states 2794:The idea that 2792: 2791: 2790: 2784: 2774:linear algebra 2757: 2754: 2751: 2748: 2744: 2740: 2737: 2734: 2731: 2728: 2725: 2722: 2719: 2716: 2713: 2710: 2707: 2704: 2700: 2657: 2654: 2622: 2618: 2615: 2612: 2609: 2604: 2599: 2594: 2591: 2588: 2585: 2582: 2579: 2575: 2568: 2563: 2560: 2556: 2527: 2524: 2519: 2514: 2509: 2506: 2503: 2500: 2497: 2494: 2490: 2483: 2478: 2474: 2470: 2467: 2464: 2461: 2456: 2453: 2450: 2447: 2444: 2440: 2407: 2404: 2369: 2366: 2363: 2360: 2357: 2354: 2351: 2348: 2345: 2342: 2339: 2336: 2333: 2330: 2327: 2324: 2321: 2316: 2312: 2308: 2303: 2298: 2294: 2291: 2288: 2285: 2282: 2279: 2275: 2222: 2218: 2215: 2212: 2209: 2206: 2203: 2191: 2188: 2138: 2137: 2123: 2119: 2115: 2112: 2109: 2106: 2103: 2100: 2097: 2094: 2072: 2071: 2059: 2056: 2053: 2050: 2046: 2043: 2040: 2037: 2034: 2029: 2025: 2021: 2018: 2015: 2012: 2009: 2004: 1999: 1995: 1988: 1983: 1980: 1976: 1972: 1969: 1964: 1960: 1956: 1951: 1947: 1943: 1923: 1916: 1888: 1799: 1798: 1793: 1786: 1785: 1784: 1776:, examples of 1770:Standing waves 1768: 1761: 1760: 1759: 1758: 1757: 1754: 1753: 1704: 1702: 1695: 1689: 1686: 1606:Proca equation 1555:Dirac equation 1542: 1539: 1517:Dirac equation 1501:Pauli equation 1448:John C. Slater 1359:wave mechanics 1347:linear algebra 1307: 1304: 1299: 1296: 1271: 1247: 1244: 1239: 1236: 1214: 1190: 1165: 1162: 1159: 1156: 1134: 1113: 1096: 1095: 1093: 1092: 1085: 1078: 1070: 1067: 1066: 1062: 1061: 1056: 1051: 1046: 1041: 1036: 1031: 1026: 1021: 1016: 1011: 1006: 1001: 996: 991: 986: 981: 976: 971: 966: 961: 956: 951: 946: 941: 936: 931: 926: 921: 916: 911: 906: 901: 896: 891: 886: 881: 876: 871: 866: 861: 856: 851: 846: 841: 836: 830: 829: 826: 825: 822: 821: 818: 817: 812: 807: 802: 800:Density matrix 797: 792: 787: 782: 777: 772: 766: 763: 762: 759: 758: 754: 753: 748: 743: 738: 733: 728: 723: 722: 721: 720: 719: 704: 699: 694: 689: 684: 678: 677: 672: 671: 668: 667: 663: 662: 657: 652: 647: 642: 636: 635: 632: 631: 628: 627: 623: 622: 617: 612: 607: 602: 597: 591: 590: 589: 583: 580: 579: 576: 575: 571: 570: 565: 560: 554: 553: 552: 551: 550: 548:Delayed-choice 543:Quantum eraser 538: 537: 532: 527: 522: 517: 512: 507: 502: 497: 492: 486: 485: 482: 481: 478: 477: 473: 472: 471: 470: 460: 455: 450: 445: 440: 435: 433:Quantum number 430: 425: 420: 415: 410: 405: 399: 398: 395: 394: 391: 390: 386: 385: 380: 374: 373: 372: 367: 362: 356: 353: 352: 349: 348: 347: 346: 341: 336: 328: 327: 322: 311: 308: 304: 297: 294: 288: 285: 282: 278: 271: 268: 264: 259: 256: 245: 244: 238: 237: 230: 227: 91:complex-valued 73:quantum system 30:Comparison of 15: 9: 6: 4: 3: 2: 19278: 19267: 19264: 19262: 19259: 19258: 19256: 19241: 19233: 19232: 19229: 19223: 19220: 19218: 19215: 19213: 19210: 19206: 19203: 19202: 19201: 19198: 19197: 19195: 19191: 19185: 19182: 19180: 19177: 19173: 19170: 19169: 19168: 19165: 19163: 19160: 19158: 19155: 19153: 19150: 19149: 19147: 19143: 19137: 19134: 19132: 19129: 19127: 19124: 19122: 19119: 19117: 19114: 19112: 19109: 19107: 19104: 19102: 19099: 19097: 19094: 19092: 19089: 19087: 19084: 19082: 19079: 19077: 19076:Quantum logic 19074: 19072: 19069: 19067: 19064: 19062: 19059: 19057: 19054: 19052: 19049: 19047: 19044: 19042: 19039: 19035: 19032: 19031: 19030: 19027: 19025: 19022: 19020: 19017: 19015: 19012: 19008: 19005: 19004: 19003: 19000: 18998: 18995: 18993: 18990: 18988: 18985: 18984: 18982: 18980: 18976: 18970: 18967: 18965: 18962: 18960: 18957: 18955: 18952: 18950: 18947: 18945: 18942: 18940: 18937: 18935: 18932: 18930: 18929:Quantum chaos 18927: 18925: 18922: 18920: 18917: 18916: 18914: 18912: 18908: 18902: 18899: 18897: 18896:Stern–Gerlach 18894: 18892: 18889: 18887: 18884: 18882: 18879: 18877: 18874: 18872: 18869: 18867: 18864: 18862: 18859: 18857: 18854: 18852: 18849: 18848: 18846: 18842: 18836: 18833: 18831: 18830:Transactional 18828: 18826: 18823: 18821: 18820:Quantum logic 18818: 18816: 18813: 18811: 18808: 18802: 18799: 18798: 18797: 18794: 18793: 18792: 18789: 18787: 18784: 18782: 18779: 18777: 18774: 18772: 18769: 18767: 18764: 18763: 18761: 18759: 18755: 18749: 18746: 18744: 18741: 18739: 18736: 18734: 18731: 18729: 18726: 18725: 18723: 18719: 18713: 18710: 18708: 18705: 18703: 18700: 18698: 18695: 18693: 18690: 18688: 18685: 18683: 18680: 18679: 18677: 18673: 18665: 18662: 18660: 18657: 18656: 18655: 18654:Wave function 18652: 18650: 18647: 18645: 18642: 18640: 18637: 18635: 18632: 18630: 18629:Superposition 18627: 18625: 18624:Quantum state 18622: 18620: 18617: 18615: 18612: 18610: 18607: 18605: 18602: 18600: 18597: 18595: 18592: 18588: 18585: 18583: 18580: 18578: 18577:Excited state 18575: 18573: 18570: 18569: 18568: 18565: 18563: 18560: 18558: 18555: 18553: 18550: 18548: 18545: 18544: 18542: 18538: 18532: 18529: 18527: 18524: 18522: 18519: 18515: 18512: 18511: 18510: 18507: 18505: 18502: 18501: 18499: 18495: 18491: 18484: 18479: 18477: 18472: 18470: 18465: 18464: 18461: 18455: 18454: 18449: 18447: 18443: 18440: 18437: 18435: 18432: 18430: 18427: 18425: 18422: 18420: 18417: 18416: 18405: 18399: 18395: 18391: 18387: 18380: 18373: 18372: 18366: 18365: 18353: 18347: 18343: 18338: 18334: 18328: 18324: 18319: 18315: 18309: 18305: 18300: 18296: 18292: 18288: 18287:Wheeler, J.A. 18284: 18281: 18275: 18271: 18267: 18263: 18260: 18256: 18250: 18246: 18245: 18239: 18235: 18229: 18225: 18220: 18216: 18210: 18206: 18201: 18197: 18191: 18187: 18182: 18175: 18171: 18167: 18163: 18159: 18155: 18151: 18150: 18142: 18138: 18134: 18130: 18124: 18120: 18119: 18113: 18109: 18103: 18099: 18094: 18090: 18086: 18082: 18078: 18074: 18070: 18066: 18063:(in German). 18062: 18058: 18054: 18050: 18044: 18040: 18035: 18031: 18027: 18021: 18016: 18015: 18008: 18004: 17998: 17994: 17989: 17985: 17981: 17977: 17973: 17967: 17962: 17961: 17954: 17950: 17946: 17940: 17936: 17935: 17930: 17926: 17922: 17916: 17912: 17908: 17904: 17897: 17892: 17890: 17885: 17879: 17875: 17871: 17867: 17863: 17859: 17855: 17849: 17845: 17841: 17836: 17832: 17827: 17826: 17820: 17816: 17813: 17809: 17805: 17801: 17797: 17793: 17788: 17784: 17780: 17776: 17772: 17768: 17763: 17762: 17756: 17752: 17748: 17742: 17738: 17737: 17731: 17727: 17721: 17717: 17712: 17708: 17702: 17698: 17697: 17692: 17688: 17684: 17680: 17674: 17670: 17669: 17664: 17659: 17655: 17649: 17645: 17641: 17636: 17632: 17628: 17624: 17621:(in German). 17620: 17619: 17613: 17609: 17605: 17600: 17595: 17590: 17586: 17582: 17578: 17575:(in German). 17574: 17573: 17568: 17564: 17560: 17556: 17554:0-19-852011-5 17550: 17546: 17542: 17538: 17534: 17530: 17526: 17522: 17518: 17514: 17510: 17506: 17501: 17497: 17491: 17487: 17483: 17479: 17478:Conway, J. B. 17475: 17471: 17465: 17461: 17456: 17452: 17448: 17442: 17438: 17434: 17433: 17428: 17427:Fuller, R. W. 17423: 17419: 17414: 17413: 17407: 17403: 17401: 17398: 17393: 17390:(in French). 17389: 17384: 17380: 17376: 17372: 17368: 17364: 17360: 17356: 17355:Nobel Lecture 17352: 17347: 17342: 17337: 17333: 17329: 17325: 17321: 17317: 17313: 17309: 17305: 17301: 17297: 17293: 17289: 17285: 17281: 17277: 17272: 17268: 17264: 17260: 17256: 17252: 17248: 17244: 17240: 17235: 17231: 17225: 17221: 17217: 17213: 17209: 17203: 17199: 17194: 17190: 17186: 17182: 17178: 17174: 17170: 17169: 17161: 17159: 17152: 17148: 17144: 17140: 17139: 17127: 17126:Einstein 1998 17122: 17115: 17110: 17103: 17098: 17091: 17086: 17079: 17074: 17067: 17062: 17055: 17050: 17043: 17038: 17031: 17030:Weinberg 2002 17026: 17019: 17018:Weinberg 2002 17014: 17007: 17002: 16996: 16995:0-7167-8964-7 16992: 16986: 16979: 16978:Weinberg 2002 16974: 16966: 16960: 16956: 16949: 16942: 16937: 16930: 16925: 16918: 16913: 16906: 16905:Landsman 2009 16901: 16890: 16886: 16880: 16873: 16868: 16861: 16856: 16854: 16846: 16841: 16835:, p. 94. 16834: 16829: 16822: 16817: 16810: 16809:Zwiebach 2009 16805: 16798: 16797:Weinberg 2002 16793: 16786: 16781: 16774: 16770: 16765: 16758: 16753: 16746: 16741: 16734: 16729: 16727: 16725: 16717: 16712: 16705: 16704:Weinberg 2013 16700: 16698: 16696: 16688: 16683: 16676: 16671: 16664: 16659: 16652: 16647: 16640: 16639:ter Haar 1967 16636: 16635:Einstein 1917 16632: 16631:Einstein 1916 16627: 16611: 16604: 16603: 16596: 16581: 16577: 16576:spark.iop.org 16573: 16567: 16560: 16555: 16548: 16543: 16537:, p. 48. 16536: 16531: 16525:, p. 43. 16524: 16519: 16512: 16508: 16504: 16499: 16492: 16487: 16480: 16475: 16468: 16463: 16456: 16452: 16449: 16445: 16441: 16436: 16434: 16426: 16422: 16417: 16415: 16413: 16408: 16379: 16375: 16370: 16366: 16359: 16355: 16350: 16342: 16338: 16333: 16329: 16322: 16318: 16314: 16311: 16308: 16303: 16299: 16295: 16290: 16286: 16281: 16277: 16268: 16257: 16248: 16240: 16234: 16226: 16221: 16217: 16213: 16212:discontinuous 16207: 16200: 16196: 16192: 16188: 16187:distributions 16184: 16181: 16180:Sobolev space 16175: 16168: 16164: 16159: 16152: 16148: 16142: 16135: 16134:Hilbert space 16131: 16127: 16123: 16117: 16110: 16103: 16096: 16092: 16088: 16081: 16074: 16070: 16066: 16061: 16054: 16047: 16039: 16031: 16029: 16021: 16015: 16008: 16004: 16000: 15996: 15990: 15983: 15979: 15968: 15963: 15962: 15954: 15950: 15939: 15936: 15934: 15931: 15929: 15926: 15924: 15921: 15919: 15916: 15914: 15911: 15909: 15906: 15904: 15901: 15899: 15896: 15895: 15888: 15885: 15881: 15877: 15873: 15869: 15865: 15864:Eugene Wigner 15861: 15857: 15853: 15849: 15845: 15839: 15829: 15825: 15815: 15809: 15800: 15795: 15792: 15786: 15764: 15760: 15752: 15744: 15741: 15738: 15734: 15724: 15718: 15715: 15706: 15702: 15699: 15692: 15687: 15666: 15662: 15653: 15645: 15642: 15639: 15635: 15629: 15625: 15619: 15616: 15607: 15603: 15597: 15591: 15581: 15575: 15569: 15565: 15559: 15553: 15538: 15525: 15522: 15514: 15495: 15489: 15481: 15478: 15475: 15471: 15459: 15455: 15436: 15428: 15414: 15407: 15400: 15393: 15386: 15379: 15372: 15367: 15363: 15355: 15351: 15347: 15343: 15339: 15332: 15328: 15324: 15320: 15313: 15306: 15301: 15297: 15296: 15294: 15293: 15289: 15278: 15274: 15270: 15265: 15256: 15251: 15242: 15222: 15216: 15210: 15201: 15192: 15182: 15175: 15171: 15167: 15164:-dimensional 15162: 15156: 15145: 15135: 15128: 15124: 15120: 15115: 15105: 15098: 15094: 15090: 15088:of the system 15087: 15086:wave function 15084:, called the 15075: 15071: 15067: 15062: 15059: 15054: 15044: 15038: 15034: 15031: 15027: 15021: 15017: 15011: 15010: 15009: 14987: 14970: 14967: 14959: 14935: 14931: 14927: 14918: 14914: 14889: 14885: 14879: 14878:Quantum state 14869: 14866: 14861: 14860:is not closed 14856: 14850: 14848: 14839: 14835: 14831: 14828: 14824: 14823: 14822: 14815: 14811: 14807: 14803: 14798: 14789: 14787: 14779: 14775: 14771: 14767: 14763: 14762: 14761: 14758: 14753: 14749: 14748:nuclear force 14745: 14741: 14737: 14733: 14732:Lorentz group 14729: 14725: 14717: 14711: 14706: 14703: 14697: 14692: 14691: 14690: 14683: 14677: 14675: 14671: 14667: 14663: 14659: 14655: 14651: 14639: 14635: 14632: 14627: 14623: 14619: 14616: 14608: 14604: 14600: 14596: 14591: 14587: 14583: 14578: 14574: 14567: 14563: 14558: 14554: 14547: 14543: 14538: 14532: 14528: 14523: 14519: 14514: 14510: 14506: 14502: 14494: 14488: 14487: 14486: 14478: 14476: 14471: 14460: 14455: 14450: 14448: 14444: 14440: 14435: 14429: 14423: 14417: 14414: 14411: 14405: 14400: 14396: 14392: 14388: 14387:Hilbert space 14384: 14376:Hilbert space 14373: 14371: 14367: 14363: 14359: 14355: 14311: 14287: 14282: 14266: 14252: 14249: 14229: 14221: 14212: 14204: 14203:inner product 14196: 14190: 14186: 14176: 14172: 14155: 14150: 14147: 14143: 14139: 14136: 14133: 14129: 14124: 14114: 14109: 14101: 14093: 14089: 14086: 14081: 14061: 14033: 14025: 14024: 14023: 14018:Inner product 14015: 14008: 14003: 14000: 13999: 13998: 13996: 13989: 13985: 13981: 13972: 13968: 13964: 13957: 13948: 13944: 13939: 13933: 13929: 13924: 13921: 13917: 13910: 13906: 13902: 13897:-projection, 13895: 13887: 13883: 13879: 13874:-projection, 13872: 13867: 13864: 13860: 13857: 13853: 13849: 13845: 13844: 13843: 13841: 13837: 13834: 13830: 13820: 13814: 13810: 13806: 13800: 13796: 13790: 13784: 13779: 13767: 13763: 13762: 13761: 13753: 13751: 13750:Hilbert space 13747: 13743: 13739: 13738:inner product 13735: 13731: 13727: 13723: 13710: 13706: 13703: 13697: 13693: 13689: 13683: 13679: 13673: 13668: 13664: 13663: 13657: 13656: 13655: 13652: 13648: 13644: 13640: 13632: 13629: 13627: 13623: 13618: 13614: 13610: 13606: 13602: 13597: 13591: 13588:= 0, 1, ..., 13587: 13582: 13576: 13569: 13565: 13560: 13551: 13547: 13540: 13535: 13530: 13522: 13516: 13509: 13489: 13486: 13483: 13475: 13470: 13466: 13462: 13458: 13450: 13446: 13442: 13437: 13434: 13428: 13422: 13419: 13416: 13413: 13408: 13405: 13402: 13399: 13396: 13392: 13386: 13381: 13373: 13369: 13365: 13360: 13357: 13351: 13342: 13338: 13334: 13330: 13326: 13323: 13319: 13307: 13301: 13298: 13295: 13286: 13283: 13278: 13272: 13269: 13266: 13263: 13260: 13249: 13243: 13235: 13231: 13227: 13223: 13218: 13210: 13204: 13201: 13198: 13195: 13192: 13184: 13181: 13178: 13164: 13158: 13153: 13147: 13143: 13136: 13129: 13123: 13103: 13100: 13097: 13088: 13083: 13079: 13070: 13064: 13061: 13055: 13052: 13049: 13046: 13043: 13035: 13032: 13029: 13015: 13013: 13009: 13005: 13004:Hydrogen atom 12998:Hydrogen atom 12992: 12988: 12984: 12983:hydrogen atom 12979: 12975: 12971: 12952: 12948: 12938: 12935: 12927: 12920: 12916: 12912: 12901: 12894: 12890: 12886: 12883: 12877: 12873: 12869: 12864: 12860: 12856: 12851: 12842: 12837: 12834: 12828: 12823: 12816: 12813: 12807: 12803: 12798: 12792: 12786: 12778: 12764: 12757: 12753: 12743: 12741: 12737: 12733: 12730: 12726: 12718: 12714: 12709: 12705: 12698: 12689: 12680: 12671: 12662: 12652: 12642: 12640: 12635: 12615: 12612: 12609: 12606: 12599: 12596: 12593: 12590: 12586: 12574: 12570: 12565: 12562: 12559: 12555: 12543: 12535: 12532: 12529: 12521: 12509: 12506: 12503: 12499: 12487: 12483: 12478: 12475: 12471: 12459: 12451: 12448: 12445: 12442: 12439: 12432: 12429: 12426: 12423: 12419: 12407: 12403: 12398: 12395: 12392: 12388: 12376: 12369: 12364: 12358: 12343: 12339: 12315: 12312: 12304: 12294: 12287: 12284: 12276: 12264: 12260: 12253: 12248: 12242: 12236: 12228: 12224: 12212: 12208: 12199: 12193: 12184: 12176: 12174: 12167: 12163: 12151: 12143: 12135: 12131:and operator 12123: 12121: 12097: 12088: 12078: 12075: 12072: 12067: 12057: 12052: 12039: 12029: 12025: 12022: 12019: 12016: 12012: 12008: 12002: 11999: 11994: 11984: 11981: 11978: 11973: 11963: 11958: 11930: 11920: 11918: 11914: 11896: 11890: 11870: 11866: 11861: 11848: 11845: 11839: 11831: 11822: 11808: 11805: 11800: 11787: 11784: 11773: 11759: 11756: 11752: 11742: 11726: 11716: 11705: 11697: 11692: 11679: 11671: 11653: 11649: 11621: 11617: 11586: 11583: 11569: 11567: 11562: 11547: 11543: 11537: 11531: 11525: 11522: 11501: 11484: 11481: 11470: 11448: 11435: 11432: 11421: 11413: 11407: 11404: 11393: 11364: 11361: 11350: 11347: 11341: 11338: 11330: 11327: 11316: 11311: 11305: 11302: 11291: 11282: 11280: 11261: 11253: 11249: 11235: 11232: 11227: 11219: 11215: 11208: 11205: 11202: 11194: 11190: 11180: 11177: 11174: 11170: 11165: 11159: 11156: 11121: 11108: 11105: 11094: 11086: 11080: 11077: 11066: 11046: 11043: 11035: 11029: 11023: 11015: 11001: 10977: 10972: 10967: 10963: 10960: 10955: 10951: 10947: 10942: 10938: 10934: 10929: 10919: 10914: 10903: 10895: 10888: 10876: 10872: 10864: 10860: 10855: 10851: 10846: 10834: 10830: 10822: 10818: 10813: 10807: 10795: 10791: 10783: 10779: 10774: 10770: 10764: 10754: 10750: 10746: 10741: 10737: 10733: 10729: 10724: 10720: 10716: 10711: 10701: 10698: 10695: 10690: 10686: 10682: 10677: 10673: 10669: 10665: 10660: 10656: 10652: 10647: 10636: 10628: 10627: 10626: 10623: 10619:etc. at time 10614: 10608: 10604: 10595: 10590: 10583: 10577: 10573: 10564: 10558: 10543: 10538: 10533: 10529: 10526: 10521: 10517: 10513: 10509: 10504: 10500: 10496: 10492: 10487: 10477: 10472: 10461: 10453: 10448: 10444: 10440: 10437: 10432: 10428: 10424: 10420: 10415: 10411: 10407: 10403: 10398: 10388: 10383: 10372: 10368: 10346: 10343: 10340: 10322: 10318: 10311: 10307: 10304: 10295: 10274: 10270: 10267: 10262: 10258: 10254: 10250: 10245: 10241: 10237: 10233: 10228: 10218: 10213: 10202: 10196: 10187: 10183: 10180: 10175: 10171: 10167: 10163: 10158: 10154: 10150: 10146: 10141: 10131: 10126: 10115: 10109: 10104: 10094: 10082: 10078: 10044: 10040: 10035: 10023: 10019: 9985: 9979: 9967: 9963: 9929: 9921: 9917: 9913: 9908: 9900: 9896: 9892: 9887: 9883: 9876: 9872: 9868: 9863: 9859: 9851: 9843: 9838: 9821: 9815: 9812: 9799: 9787: 9778: 9774: 9770: 9766: 9763: 9760: 9755: 9751: 9747: 9743: 9738: 9728: 9725: 9722: 9717: 9672: 9663: 9659: 9655: 9651: 9648: 9645: 9640: 9636: 9632: 9628: 9623: 9613: 9610: 9607: 9602: 9571: 9565: 9553: 9549: 9541: 9537: 9532: 9528: 9523: 9511: 9507: 9499: 9495: 9490: 9474: 9466: 9462: 9458: 9454: 9451: 9448: 9443: 9439: 9435: 9420: 9401: 9398: 9382: 9379: 9374: 9370: 9366: 9362: 9357: 9353: 9349: 9345: 9340: 9336: 9332: 9328: 9323: 9313: 9308: 9298: 9293: 9272: 9267: 9263: 9259: 9253: 9239: 9235: 9232: 9227: 9217: 9212: 9202: 9199: 9196: 9191: 9181: 9178: 9175: 9170: 9160: 9156: 9149: 9146: 9142: 9138: 9135: 9130: 9120: 9115: 9105: 9102: 9099: 9094: 9084: 9081: 9078: 9073: 9063: 9059: 9045: 9041: 9031: 9024: 9014: 9007: 9001: 8999: 8995: 8991: 8985: 8983: 8979: 8975: 8967: 8946: 8942: 8939: 8934: 8924: 8921: 8918: 8913: 8903: 8899: 8892: 8889: 8885: 8881: 8878: 8873: 8863: 8860: 8857: 8852: 8842: 8838: 8825: 8821: 8820: 8814: 8810: 8794: 8790: 8770: 8767: 8762: 8752: 8747: 8737: 8732: 8710: 8705: 8701: 8697: 8693: 8684: 8675: 8673: 8669: 8666: 8661: 8659: 8655: 8650: 8645: 8627: 8623: 8613: 8596: 8588: 8584: 8580: 8548: 8545: 8540: 8536: 8529: 8522: 8519: 8508: 8505: 8499: 8496: 8491: 8487: 8483: 8447: 8441: 8432: 8422: 8416: 8408: 8399: 8363: 8359: 8349: 8328: 8325: 8320: 8316: 8309: 8302: 8299: 8288: 8276: 8272: 8268: 8261: 8257: 8252: 8248: 8239: 8233: 8224: 8214: 8208: 8192: 8182: 8172: 8167: 8149: 8145: 8141: 8124: 8121: 8116: 8112: 8108: 8085: 8081: 8077: 8070: 8066: 8061: 8057: 8048: 8029: 8024: 8008: 7999: 7996: 7993: 7990: 7987: 7966: 7963: 7957: 7954: 7951: 7945: 7942: 7924: 7914: 7911: 7908: 7905: 7902: 7899: 7878: 7875: 7872: 7869: 7852: 7847: 7841: 7838: 7804: 7801: 7796: 7792: 7788: 7768: 7764: 7760: 7756: 7752: 7748: 7744: 7740: 7736: 7732: 7725: 7718: 7712: 7694: 7690: 7677: 7674: 7669: 7665: 7658: 7653: 7648: 7645: 7642: 7637: 7633: 7628: 7624: 7615: 7609: 7596: 7591: 7576: 7570: 7563: 7556: 7549: 7542: 7536: 7528: 7525: 7522: 7519: 7513: 7510: 7505: 7499: 7492: 7485: 7478: 7471: 7465: 7457: 7454: 7448: 7445: 7442: 7436: 7430: 7427: 7424: 7421: 7416: 7410: 7403: 7396: 7389: 7382: 7376: 7368: 7365: 7362: 7359: 7356: 7350: 7347: 7342: 7336: 7329: 7322: 7315: 7308: 7302: 7294: 7291: 7288: 7282: 7279: 7274: 7265: 7262: 7259: 7256: 7250: 7240: 7237: 7231: 7228: 7225: 7219: 7213: 7206: 7196: 7193: 7190: 7187: 7184: 7178: 7168: 7165: 7162: 7156: 7150: 7145: 7142: 7134: 7133: 7132:column vector 7127: 7120: 7116: 7112: 7108: 7102: 7078: 7073: 7069: 7060: 7054: 7038: 7030: 7010: 7007: 7002: 6998: 6991: 6983: 6978: 6975: 6945: 6942: 6919: 6915: 6911: 6908: 6899: 6881: 6875: 6871: 6864: 6859: 6854: 6834: 6831: 6806: 6787: 6783: 6761: 6753: 6749: 6745: 6725: 6706: 6702: 6676: 6668: 6665: 6661: 6651: 6648: 6645: 6642: 6638: 6630: 6621: 6618: 6615: 6611: 6601: 6597: 6590: 6585: 6580: 6571: 6563: 6560: 6547: 6536: 6533: 6530: 6524: 6514: 6504: 6496: 6493: 6490: 6477: 6469: 6460: 6455: 6449: 6436: 6433: 6431: 6426: 6411: 6405: 6402: 6397: 6392: 6387: 6382: 6375: 6367: 6364: 6361: 6355: 6350: 6345: 6340: 6333: 6328: 6323: 6318: 6313: 6306: 6301: 6296: 6291: 6288: 6285: 6280: 6273: 6268: 6263: 6258: 6253: 6247: 6242: 6237: 6227: 6215: 6204: 6199: 6188: 6173: 6167: 6160: 6153: 6146: 6139: 6133: 6122: 6119: 6110: 6104: 6098: 6091: 6084: 6077: 6070: 6064: 6050: 6047: 6044: 6038: 6029: 6025: 6022: 6016: 6010: 6003: 5996: 5989: 5982: 5976: 5965: 5962: 5959: 5950: 5944: 5938: 5931: 5924: 5917: 5910: 5904: 5893: 5877: 5866: 5852: 5849: 5846: 5843: 5835: 5818: 5810: 5807: 5804: 5801: 5778: 5770: 5769:spin operator 5765: 5762: 5761:Hilbert space 5758: 5757:Hilbert space 5754: 5753:Hilbert space 5738: 5735: 5732: 5729: 5709: 5695: 5675: 5671: 5656: 5647: 5628: 5624: 5592: 5582: 5578: 5562: 5540: 5536: 5496: 5489: 5450: 5443: 5428: 5421: 5410: 5406: 5402: 5397: 5390: 5387: 5361: 5348: 5338: 5331: 5328: 5316: 5311: 5298: 5285: 5278: 5264: 5260: 5256: 5250: 5242: 5238: 5229: 5228:is given by: 5213: 5209: 5200: 5173: 5166: 5137: 5109: 5105: 5087: 5072: 5059: 5049: 5045: 5033: 5025: 5021: 5012: 5008: 4999: 4997: 4979: 4975: 4966: 4948: 4944: 4935: 4932: 4908: 4904: 4881: 4861: 4851: 4847: 4817: 4807: 4803: 4791: 4787: 4776: 4772: 4768: 4762: 4754: 4751: 4745: 4732: 4729: 4707: 4703: 4674: 4671: 4661: 4657: 4645: 4641: 4630: 4626: 4622: 4619: 4611: 4589: 4585: 4567: 4561: 4559: 4555: 4551: 4547: 4544: 4540: 4537: 4533: 4523: 4514: 4511: 4497: 4491: 4489: 4484: 4471: 4467: 4464: 4459: 4456: 4448: 4442: 4435: 4426: 4417: 4414: 4410: 4405: 4399: 4384: 4371: 4367: 4364: 4359: 4356: 4348: 4343: 4339: 4332: 4323: 4314: 4311: 4307: 4302: 4296: 4270: 4265: 4262: 4254: 4249: 4245: 4235: 4232: 4228: 4223: 4217: 4209: 4198: 4195: 4187: 4181: 4171: 4168: 4164: 4159: 4153: 4147: 4144: 4138: 4130: 4118: 4105: 4099: 4090: 4086: 4083: 4079: 4072: 4069: 4065: 4062: 4056: 4049: 4046: 4036: 4033: 4029: 4026: 4022: 4015: 4012: 4003: 3993: 3990: 3980: 3977: 3974: 3971: 3965: 3957: 3948: 3939: 3926: 3909: 3906: 3903: 3897: 3886: 3877: 3874: 3871: 3868: 3854: 3845: 3837: 3834: 3832: 3816: 3813: 3795: 3792: 3789: 3783: 3772: 3763: 3760: 3757: 3754: 3740: 3731: 3723: 3720: 3718: 3702: 3699: 3675: 3669: 3658: 3654: 3641: 3634: 3631: 3627: 3624: 3618: 3615: 3606: 3603: 3594: 3589: 3573: 3569: 3547: 3544: 3541: 3535: 3532: 3526: 3523: 3520: 3512: 3492: 3476: 3467: 3463: 3460: 3457: 3453: 3449: 3443: 3435: 3421: 3408: 3404: 3401: 3395: 3392: 3389: 3381: 3370: 3367: 3364: 3356: 3351: 3332: 3328: 3324: 3316: 3308: 3303: 3285: 3281: 3269: 3265: 3254: 3240: 3217: 3214: 3211: 3197: 3187: 3183: 3169: 3165: 3162: 3154: 3145: 3137: 3134: 3131: 3124: 3096: 3092: 3089: 3081: 3072: 3064: 3060: 3056: 3053: 3050: 3036: 3027: 3019: 3016: 2981: 2978: 2968: 2965: 2962: 2956: 2947: 2944: 2934: 2925: 2922: 2907: 2904: 2877: 2874: 2870: 2867: 2860: 2857: 2851: 2842: 2839: 2827: 2823: 2816: 2801: 2797: 2793: 2788: 2785: 2782: 2778: 2777: 2775: 2771: 2770: 2769: 2755: 2752: 2746: 2735: 2732: 2729: 2720: 2717: 2708: 2689: 2683: 2679: 2671: 2667: 2663: 2653: 2651: 2647: 2643: 2638: 2636: 2620: 2616: 2613: 2610: 2607: 2602: 2589: 2586: 2583: 2558: 2554: 2545: 2525: 2522: 2517: 2504: 2501: 2498: 2481: 2476: 2472: 2468: 2462: 2454: 2451: 2448: 2445: 2442: 2438: 2428: 2424: 2420: 2414: 2403: 2401: 2397: 2393: 2388: 2383: 2367: 2361: 2355: 2352: 2346: 2343: 2340: 2328: 2325: 2322: 2314: 2306: 2301: 2296: 2289: 2286: 2283: 2273: 2263: 2260:; the square 2259: 2254: 2244: 2220: 2213: 2210: 2207: 2187: 2185: 2181: 2177: 2173: 2165: 2164: 2143: 2121: 2107: 2098: 2085: 2084: 2083: 2082:with itself, 2077: 2054: 2051: 2048: 2041: 2038: 2035: 2027: 2016: 2013: 2010: 2002: 1997: 1978: 1974: 1970: 1962: 1954: 1949: 1934: 1933: 1932: 1912: 1911:inner product 1908: 1907:Hilbert space 1905: 1902: 1886: 1878: 1874: 1869: 1867: 1856: 1847: 1833: 1825: 1817: 1809: 1803: 1790: 1779: 1775: 1771: 1765: 1750: 1747: 1739: 1729: 1725: 1721: 1715: 1714: 1710: 1705:This section 1703: 1699: 1694: 1693: 1685: 1683: 1682:string theory 1678: 1675: 1669: 1667: 1663: 1659: 1655: 1647: 1639: 1635: 1631: 1615: 1607: 1586: 1584: 1580: 1575: 1570: 1568: 1564: 1560: 1556: 1552: 1547: 1538: 1536: 1532: 1528: 1524: 1523: 1518: 1514: 1510: 1506: 1502: 1498: 1493: 1491: 1487: 1483: 1480: 1476: 1472: 1468: 1465:and negative 1464: 1463:probabilities 1460: 1456: 1451: 1449: 1445: 1441: 1437: 1433: 1429: 1426: 1422: 1418: 1416: 1411: 1407: 1403: 1399: 1395: 1390: 1386: 1383: 1379: 1374: 1372: 1368: 1364: 1360: 1356: 1352: 1348: 1344: 1339: 1337: 1333: 1329: 1325: 1305: 1302: 1297: 1294: 1285: 1269: 1245: 1242: 1237: 1234: 1212: 1204: 1188: 1181: 1163: 1160: 1157: 1154: 1132: 1111: 1103: 1091: 1086: 1084: 1079: 1077: 1072: 1071: 1069: 1068: 1060: 1057: 1055: 1052: 1050: 1047: 1045: 1042: 1040: 1037: 1035: 1032: 1030: 1027: 1025: 1022: 1020: 1017: 1015: 1012: 1010: 1007: 1005: 1002: 1000: 997: 995: 992: 990: 987: 985: 982: 980: 977: 975: 972: 970: 967: 965: 962: 960: 957: 955: 952: 950: 947: 945: 942: 940: 937: 935: 932: 930: 927: 925: 922: 920: 917: 915: 912: 910: 907: 905: 902: 900: 897: 895: 892: 890: 887: 885: 882: 880: 877: 875: 872: 870: 867: 865: 862: 860: 857: 855: 852: 850: 847: 845: 842: 840: 837: 835: 832: 831: 824: 823: 816: 813: 811: 808: 806: 803: 801: 798: 796: 793: 791: 790:Quantum chaos 788: 786: 783: 781: 778: 776: 773: 771: 768: 767: 761: 760: 752: 749: 747: 746:Transactional 744: 742: 739: 737: 736:Quantum logic 734: 732: 729: 727: 724: 718: 715: 714: 713: 710: 709: 708: 705: 703: 700: 698: 695: 693: 690: 688: 685: 683: 680: 679: 675: 670: 669: 661: 658: 656: 653: 651: 648: 646: 643: 641: 638: 637: 630: 629: 621: 618: 616: 613: 611: 608: 606: 603: 601: 598: 596: 593: 592: 588: 585: 584: 578: 577: 569: 566: 564: 561: 559: 556: 555: 549: 546: 545: 544: 541: 540: 536: 533: 531: 528: 526: 523: 521: 518: 516: 513: 511: 508: 506: 503: 501: 498: 496: 493: 491: 488: 487: 480: 479: 469: 466: 465: 464: 463:Wave function 461: 459: 456: 454: 451: 449: 446: 444: 443:Superposition 441: 439: 436: 434: 431: 429: 426: 424: 421: 419: 416: 414: 411: 409: 406: 404: 401: 400: 393: 392: 384: 381: 379: 376: 375: 371: 368: 366: 363: 361: 358: 357: 351: 350: 345: 342: 340: 337: 335: 332: 331: 330: 329: 325: 292: 286: 269: 266: 262: 254: 247: 246: 243: 240: 239: 235: 234: 226: 224: 220: 216: 212: 211:wave equation 208: 204: 200: 196: 192: 191:Hilbert space 188: 183: 165: 164:column matrix 161: 157: 153: 149: 145: 141: 137: 133: 128: 125: 120: 116: 115:normalization 112: 108: 104: 100: 96: 92: 88: 79: 74: 70: 69:quantum state 66: 62: 61:wave function 58: 49: 42: 37: 33: 28: 22: 21:Wave equation 18959:Quantum mind 18871:Franck–Hertz 18733:Klein–Gordon 18682:Formulations 18675:Formulations 18653: 18604:Interference 18594:Entanglement 18572:Ground state 18567:Energy level 18540:Fundamentals 18504:Introduction 18452: 18393: 18379:the original 18370: 18341: 18322: 18303: 18294: 18269: 18266:Weinberg, S. 18257:– via 18243: 18223: 18204: 18185: 18174:the original 18153: 18147: 18117: 18097: 18064: 18060: 18038: 18028:– via 18013: 17992: 17982:– via 17959: 17947:– via 17933: 17929:Lerner, R.G. 17902: 17869: 17862:Landau, L.D. 17839: 17824: 17795: 17791: 17781:– via 17760: 17755:ter Haar, D. 17735: 17715: 17695: 17681:– via 17667: 17643: 17622: 17616: 17607: 17603: 17576: 17570: 17563:Einstein, A. 17544: 17508: 17504: 17481: 17459: 17449:– via 17431: 17411: 17391: 17387: 17358: 17354: 17323: 17319: 17279: 17275: 17242: 17238: 17219: 17197: 17172: 17166: 17157: 17146: 17121: 17109: 17097: 17085: 17073: 17061: 17049: 17037: 17032:, Chapter 3. 17025: 17013: 17001: 16985: 16973: 16954: 16948: 16941:Zettili 2009 16936: 16924: 16917:Shankar 1994 16912: 16900: 16894:. p. 1. 16879: 16867: 16845:Shankar 1994 16840: 16828: 16816: 16804: 16792: 16780: 16772: 16764: 16752: 16740: 16711: 16682: 16670: 16658: 16646: 16626: 16614:. Retrieved 16601: 16595: 16583:. Retrieved 16575: 16566: 16554: 16549:, p. 6. 16542: 16530: 16523:Murdoch 1987 16518: 16498: 16486: 16474: 16462: 16256: 16247: 16233: 16224: 16219: 16211: 16206: 16182: 16174: 16158: 16141: 16116: 16109:wave packets 16102: 16094: 16090: 16086: 16080: 16059: 16052: 16046: 16037: 16014: 15999:Zorn's Lemma 15989: 15977: 15960: 15953: 15913:Faraday wave 15862:(e.g. Bohr, 15841: 15820: 15813: 15807: 15793: 15790: 15787: 15690: 15688: 15579: 15573: 15567: 15563: 15557: 15554: 15457: 15453: 15424: 15412: 15402: 15395: 15388: 15384: 15374: 15370: 15361: 15353: 15349: 15345: 15341: 15337: 15330: 15326: 15322: 15318: 15308: 15304: 15299: 15291: 15290: 15276: 15273:real numbers 15260: 15254: 15249: 15240: 15220: 15214: 15205: 15196: 15187: 15180: 15173: 15169: 15160: 15154: 15151: 15140: 15133: 15126: 15122: 15110: 15103: 15096: 15092: 15085: 15073: 15069: 15065: 15049: 15042: 15036: 15032: 15029: 15019: 15015: 14887: 14881: 14864: 14859: 14854: 14851: 14843: 14819: 14813: 14809: 14805: 14801: 14783: 14769: 14756: 14721: 14709: 14695: 14681: 14678: 14647: 14625: 14602: 14598: 14594: 14589: 14585: 14581: 14576: 14572: 14565: 14561: 14556: 14552: 14545: 14541: 14536: 14530: 14526: 14512: 14508: 14504: 14500: 14492: 14484: 14474: 14469: 14458: 14453: 14451: 14443:wave packets 14438: 14433: 14427: 14421: 14418: 14415: 14409: 14403: 14394: 14383:completeness 14379: 14361: 14357: 14209:, or in the 14200: 14188: 14184: 14174: 14170: 14091: 14084: 14059: 14031: 14021: 14012: 13994: 13987: 13983: 13979: 13970: 13966: 13962: 13955: 13953: 13942: 13937: 13931: 13927: 13908: 13904: 13900: 13893: 13885: 13881: 13877: 13870: 13839: 13828: 13826: 13818: 13813:vacuum state 13808: 13798: 13794: 13788: 13782: 13777: 13759: 13734:vector space 13719: 13695: 13691: 13687: 13677: 13671: 13661: 13653: 13646: 13642: 13638: 13633: 13630: 13616: 13612: 13608: 13604: 13600: 13589: 13585: 13574: 13567: 13563: 13549: 13545: 13538: 13524: 13520: 13514: 13507: 13162: 13156: 13145: 13141: 13134: 13127: 13121: 13016: 13001: 12969: 12759: 12749: 12722: 12716: 12712: 12696: 12687: 12678: 12669: 12660: 12650: 12643: 12636: 12341: 12337: 12220: 12210: 12206: 12197: 12182: 12165: 12161: 12149: 12141: 12133: 12124: 11932: 11910: 11862: 11743: 11570: 11563: 11502: 11283: 11002: 10999: 10621: 10612: 10606: 10602: 10593: 10588: 10581: 10575: 10571: 10562: 10559: 10352: 10344: 10327: 10320: 10316: 10309: 10305: 10302: 9819: 9816: 9813: 9402: 9399: 9270: 9268: 9261: 9257: 9254: 9043: 9039: 9029: 9022: 9012: 9005: 9002: 8993: 8989: 8986: 8974:all fermions 8973: 8965: 8823: 8817: 8815: 8808: 8792: 8788: 8708: 8695: 8691: 8689: 8667: 8662: 8646: 8190: 8180: 8168: 8025: 7769: 7762: 7758: 7754: 7750: 7746: 7742: 7738: 7734: 7730: 7723: 7716: 7713: 7592: 7135: 7125: 7118: 7114: 7110: 7106: 7097: 7083:can only be 7076: 7058: 7049: 7028: 6979: 6976: 6873: 6869: 6862: 6852: 6812: 6726: 6437: 6434: 6430:eigenvectors 6427: 6205: 6194: 6189: 5872: 5867: 5771:for a given 5766: 5751:dimensional 5701: 5657: 5653: 5563: 5230: 5088: 5000: 4887: 4733: 4730: 4612: 4565: 4562: 4549: 4529: 4520: 4509: 4492: 4485: 4385: 4283:one obtains 4119: 3927: 3673: 3667: 3664: 3655: 3571: 3567: 3422: 3283: 3279: 3267: 3263: 3255: 3193: 3184: 2825: 2821: 2814: 2805: 2681: 2677: 2673: 2642:vector space 2639: 2634: 2543: 2426: 2422: 2418: 2412: 2409: 2386: 2255: 2193: 2161: 2141: 2139: 2073: 1870: 1863: 1854: 1852:or momentum 1845: 1831: 1823: 1815: 1807: 1742: 1733: 1718:Please help 1706: 1679: 1673: 1670: 1633: 1587: 1582: 1578: 1571: 1548: 1544: 1537:were found. 1527:antiparticle 1521: 1494: 1458: 1455:relativistic 1452: 1420: 1414: 1375: 1340: 1327: 1326:, holds for 1099: 645:Klein–Gordon 581:Formulations 462: 418:Energy level 413:Entanglement 396:Fundamentals 383:Interference 334:Introduction 184: 159: 129: 123: 114: 77: 65:wavefunction 64: 60: 54: 19217:EPR paradox 18997:Quantum bus 18866:Double-slit 18844:Experiments 18810:Many-worlds 18748:Schrödinger 18712:Phase space 18702:Schrödinger 18692:Interaction 18649:Uncertainty 18619:Nonlocality 18614:Measurement 18609:Decoherence 18599:Hamiltonian 18291:Zurek, W.H. 17889:Online copy 17769:. pp. 17691:Greiner, W. 17625:: 121–128. 17114:Jaynes 2003 17090:Atkins 1974 17042:Conway 1990 16872:Treves 2006 16733:Atkins 1974 16616:12 February 16585:12 February 16559:Newton 2002 16444:Ludwig 1968 16163:Conway 1990 16007:Hamel basis 15938:Wave packet 15194:where each 14888:state space 14804:direction, 14622:unit sphere 14447:Hamel basis 14366:decay rates 13852:eigenvalues 13836:observables 13829:maximal set 13805:null vector 13709:orthonormal 13611:+ 1, ..., 13577:= 1, 2, ... 13534:Bohr radius 12766:, they are 12732:Bohr radius 12725:crystallite 12648:): setting 11871:, in which 8704:EPR paradox 7113:− 1, ..., − 7043:axis. (The 4888:If the set 2820:are called 2176:dimensional 1672:refer to a 1482:interaction 1469:. In 1927, 1404:. In 1927, 1034:von Neumann 1019:Schrödinger 795:EPR paradox 726:Many-worlds 660:Schrödinger 615:Schrödinger 610:Phase-space 600:Interaction 505:Double-slit 483:Experiments 458:Uncertainty 428:Nonlocality 423:Measurement 408:Decoherence 378:Hamiltonian 207:water waves 19255:Categories 19145:Extensions 18979:Technology 18825:Relational 18776:Copenhagen 18687:Heisenberg 18634:Tunnelling 18497:Background 18451:Einstein, 17175:(5): 367. 17102:Dirac 1982 16799:Chapter 4. 16757:Pauli 1927 16663:Hanle 1977 16440:Born 1926b 16421:Born 1926a 16189:, and its 15880:David Bohm 15856:Niels Bohr 15348:+ 1, ..., 15218:are in an 15158:are in an 14834:continuous 14766:Fock space 14642:[0, ∞) 14539:functions 14497:. The set 14092:orthogonal 14030:that does 13856:symmetries 13561:of degree 13160:and order 13154:of degree 10600:with spin 10569:with spin 8966:all bosons 8960:where the 7074:particle, 7039:along the 6428:since the 5767:Since the 4934:observable 4931:degenerate 4556:kets that 3491:plane wave 2660:See also: 1802:real parts 1563:Lamb shift 1203:wavelength 1102:Max Planck 1029:Sommerfeld 944:Heisenberg 939:Gutzwiller 879:de Broglie 827:Scientists 741:Relational 692:Copenhagen 595:Heisenberg 453:Tunnelling 354:Background 205:, such as 18851:Bell test 18721:Equations 18547:Born rule 18089:128228729 17812:121913205 17533:121466183 17429:(1992) . 17365:: 675–9. 17304:126244962 17267:119896026 16511:Bohr 1985 16479:Born 1927 16404:Citations 16376:α 16371:∑ 16367:⋯ 16356:α 16351:∑ 16339:α 16334:∑ 16330:≡ 16319:α 16312:… 16300:α 16287:α 16282:∑ 16278:≡ 16273:α 16269:∑ 16093:momentum 15772:ω 15745:ω 15739:α 15735:ρ 15729:Ω 15725:∫ 15716:∈ 15712:α 15707:∑ 15674:ω 15646:ω 15640:α 15636:ρ 15626:∫ 15617:∈ 15613:α 15608:∑ 15519:ω 15511:α 15504:Ψ 15482:ω 15476:α 15472:ρ 15449:at state 15245:and each 15238:× ... × Ω 14996:⟩ 14992:ω 14984:α 14964:ω 14956:α 14949:Ψ 14943:ω 14928:∫ 14923:α 14919:∑ 14912:⟩ 14909:Ψ 14770:tractable 14491:[0, 2 14270:Ψ 14264:Φ 14228:Born rule 14144:δ 14121:Ψ 14115:∗ 14106:Ψ 14102:∫ 13833:commuting 13490:ϕ 13484:θ 13471:ℓ 13463:⋅ 13417:ℓ 13406:− 13403:ℓ 13400:− 13387:ℓ 13324:− 13302:ℓ 13270:− 13267:ℓ 13264:− 13205:ϕ 13199:θ 13182:ℓ 13175:Ψ 13104:ϕ 13098:θ 13084:ℓ 13056:ϕ 13050:θ 13033:ℓ 13026:Ψ 12985:electron 12943:ℏ 12939:ω 12913:⋅ 12905:ℏ 12887:ω 12878:− 12870:⋅ 12846:ℏ 12843:π 12838:ω 12824:⋅ 12775:Ψ 12591:− 12530:≤ 12507:κ 12504:− 12476:κ 12446:− 12424:− 12353:Ψ 12313:≥ 12144:)⟩ 12076:… 12040:ψ 12034:ℏ 12017:− 11982:… 11946:Ψ 11894:∇ 11837:∂ 11829:∂ 11771:∇ 11714:∇ 11706:≪ 11689:∇ 11680:ℏ 11647:∇ 11615:∇ 11587:ρ 11541:∇ 11538:ρ 11422:ψ 11394:ρ 11372:ℏ 11342:⁡ 11317:ρ 11292:ψ 11262:ψ 11259:∇ 11254:∗ 11250:ψ 11233:ℏ 11220:∗ 11216:ψ 11212:∇ 11209:ψ 11206:− 11203:ψ 11200:∇ 11195:∗ 11191:ψ 11171:ℏ 11095:ψ 11067:ρ 11036:⋅ 11033:∇ 11021:∂ 11016:ρ 11013:∂ 10948:⋯ 10920:⋯ 10900:Ψ 10856:∫ 10852:⋯ 10814:∫ 10775:∫ 10717:∈ 10699:… 10653:∈ 10510:⋯ 10478:⋯ 10458:Ψ 10421:⋯ 10389:⋯ 10369:ρ 10251:⋯ 10219:⋯ 10193:Ψ 10164:⋯ 10132:⋯ 10110:∗ 10101:Ψ 10045:∫ 10041:⋯ 9986:∫ 9930:∫ 9909:∑ 9888:∑ 9884:⋯ 9864:∑ 9848:Ψ 9835:Ψ 9788:⏟ 9784:⟩ 9764:… 9726:… 9673:⏟ 9649:… 9611:… 9589:Ψ 9572:⏞ 9533:∫ 9529:⋯ 9491:∫ 9475:⏞ 9452:… 9431:∑ 9418:⟩ 9415:Ψ 9363:⋯ 9314:⋯ 9281:Ψ 9236:… 9200:… 9179:… 9161:… 9153:Ψ 9150:± 9139:… 9103:… 9082:… 9064:… 9056:Ψ 8998:identical 8943:… 8922:… 8904:… 8896:Ψ 8893:± 8882:… 8861:… 8843:… 8835:Ψ 8753:⋯ 8720:Ψ 8633:⟩ 8614:⊗ 8610:⟩ 8594:⟩ 8530:ξ 8509:ψ 8473:Ψ 8454:⟩ 8442:ξ 8433:⊗ 8429:⟩ 8417:ψ 8406:⟩ 8394:Ψ 8369:⟩ 8350:⊗ 8346:⟩ 8310:ξ 8289:ψ 8269:∫ 8253:∑ 8246:⟩ 8234:ξ 8225:⊗ 8221:⟩ 8209:ψ 8155:⟩ 8098:Ψ 8078:∫ 8062:∑ 8055:⟩ 8043:Ψ 7991:− 7977:Ψ 7955:− 7946:− 7932:Ψ 7925:⋮ 7906:− 7889:Ψ 7859:Ψ 7828:Ψ 7778:Ψ 7700:⟩ 7659:ξ 7646:− 7629:∑ 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4963:, by the 4945:λ 4914:⟩ 4905:ϕ 4865:⟩ 4862:ψ 4848:ϕ 4844:⟨ 4821:⟩ 4818:ψ 4804:ϕ 4800:⟨ 4797:⟩ 4788:ϕ 4773:∑ 4766:⟩ 4763:ψ 4749:⟩ 4746:ψ 4713:⟩ 4704:ϕ 4658:ϕ 4654:⟨ 4651:⟩ 4642:ϕ 4627:∑ 4595:⟩ 4586:ϕ 4452:ℏ 4430:Φ 4427:∫ 4421:ℏ 4418:π 4394:Ψ 4352:ℏ 4344:− 4327:Ψ 4324:∫ 4318:ℏ 4315:π 4291:Φ 4258:ℏ 4250:− 4239:ℏ 4236:π 4221:⟩ 4207:⟨ 4204:⇒ 4191:ℏ 4175:ℏ 4172:π 4142:⟩ 4128:⟨ 4094:Φ 4066:− 4057:δ 4040:Φ 4037:∫ 4020:⟩ 4001:⟨ 3984:Φ 3981:∫ 3969:⟩ 3955:⟨ 3943:Ψ 3940:∫ 3901:⟩ 3881:Φ 3878:∫ 3866:⟩ 3863:Ψ 3852:⟨ 3849:⟩ 3838:∫ 3828:⟩ 3825:Ψ 3811:⟩ 3808:Ψ 3787:⟩ 3767:Ψ 3764:∫ 3752:⟩ 3749:Ψ 3738:⟨ 3735:⟩ 3724:∫ 3714:⟩ 3711:Ψ 3697:⟩ 3694:Ψ 3628:− 3619:δ 3599:Ψ 3586:Ψ 3551:∞ 3548:≤ 3542:≤ 3539:∞ 3536:− 3509:Ψ 3472:ℏ 3432:Ψ 3378:Φ 3357:∗ 3348:Φ 3341:∞ 3336:∞ 3333:− 3329:∫ 3313:Φ 3300:Φ 3253:is time. 3206:Φ 3152:⟨ 3149:⟩ 3138:∫ 3109:⟩ 3106:Ψ 3079:⟨ 3076:⟩ 3065:∫ 3048:⟩ 3045:Ψ 3034:⟨ 3031:⟩ 3020:∫ 3014:⟩ 3011:Ψ 2972:Ψ 2960:⟩ 2941:⟨ 2929:Ψ 2926:∫ 2920:⟩ 2917:Ψ 2901:⟨ 2875:− 2861:δ 2855:⟩ 2836:⟨ 2750:⟩ 2724:Ψ 2721:∫ 2715:⟩ 2703:Ψ 2635:somewhere 2578:Ψ 2567:∞ 2562:∞ 2559:− 2555:∫ 2493:Ψ 2473:∫ 2452:≤ 2446:≤ 2356:ρ 2335:Ψ 2315:∗ 2311:Ψ 2278:Ψ 2202:Ψ 2118:‖ 2114:Ψ 2111:‖ 2102:Ψ 2096:Ψ 2058:∞ 2024:Ψ 2003:∗ 1994:Ψ 1987:∞ 1982:∞ 1979:− 1975:∫ 1959:Ψ 1946:Ψ 1901:separable 1707:does not 1666:causality 1567:Dirac sea 1495:In 1927, 1440:permanent 1428:algorithm 1425:iterative 1382:classical 1295:λ 1235:λ 1213:λ 1100:In 1900, 1059:Zeilinger 904:Ehrenfest 633:Equations 310:⟩ 307:Ψ 296:^ 284:⟩ 281:Ψ 258:ℏ 195:Born rule 166:(e.g., a 144:electrons 132:functions 111:measuring 95:Born rule 32:classical 19240:Category 19034:Timeline 18786:Ensemble 18766:Bayesian 18659:Collapse 18531:Glossary 18514:Timeline 18442:Archived 18392:(2002). 18293:(1983). 18268:(2013), 18139:(1926). 17980:66-30631 17868:(1977). 17821:(1958). 17779:66029628 17757:(1967). 17665:(1985). 17610:: 47–62. 17565:(1905). 17543:(1982). 17480:(1990). 17408:(1960). 17379:17798674 17361:(3172). 17314:(1927). 17312:Born, M. 17216:Bohr, N. 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17328:Bibcode 17284:Bibcode 17276:Z. Phys 17247:Bibcode 17239:Z. Phys 17177:Bibcode 16469:(1954). 15945:Remarks 15918:Fermion 15271:of the 15139:, ..., 15109:, ..., 14740:isospin 14338:, with 14193:is the 14078:. Also 13997:state. 13950:system. 13579:is the 13532:is the 12729:exciton 8672:isospin 8174:⊗ 7104:can be 7066:, is a 7035:is the 6856:is the 5646:state. 5564:Hence, 4563:If the 3237:is the 2262:modulus 1904:complex 1728:removed 1713:sources 1622:⁄ 1598:⁄ 1406:Hartree 1328:massive 1282:is the 1024:Simmons 1014:Rydberg 979:Moseley 959:Kramers 949:Hilbert 934:Glauber 929:Feynman 914:Everett 884:Compton 655:Rydberg 344:History 225:waves. 176:⁄ 156:isospin 148:photons 18886:Popper 18400: 18348: 18329: 18310: 18276: 18251: 18230: 18211: 18192: 18125: 18104: 18087: 18045: 18022: 17999: 17978: 17968: 17941: 17917: 17880: 17850: 17810: 17777: 17743: 17722: 17703: 17675: 17650: 17551: 17531: 17492: 17466: 17443: 17377: 17320:Nature 17302: 17265: 17226: 17204: 16993: 16961: 16260:Here: 16149:, the 16126:metric 16071:. 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15123:ω 15116:) 15113:n 15111:α 15107:2 15104:α 15100:1 15097:α 15093:α 15076:) 15074:t 15070:ω 15066:α 15052:m 15046:2 15040:1 15033:ω 15030:d 15020:ω 15016:α 14988:, 14979:| 14974:) 14971:t 14968:, 14960:, 14952:( 14936:m 14932:d 14915:= 14905:| 14865:L 14855:L 14816:. 14814:t 14810:z 14806:y 14802:x 14757:C 14718:. 14710:n 14704:. 14696:n 14687:n 14682:C 14644:. 14626:S 14606:. 14603:Z 14599:j 14595:i 14586:δ 14582:j 14580:( 14577:i 14573:e 14568:} 14566:Z 14562:i 14557:i 14553:e 14551:{ 14546:C 14542:Z 14533:) 14531:Z 14529:( 14527:l 14513:Z 14509:n 14505:π 14501:e 14499:{ 14493:π 14470:i 14468:- 14464:0 14459:L 14434:L 14428:e 14422:L 14410:L 14404:L 14347:Ψ 14340:Φ 14336:) 14321:Ψ 14314:Φ 14306:Ψ 14302:Φ 14288:, 14283:2 14278:| 14273:) 14267:, 14261:( 14257:| 14253:= 14250:p 14240:Ψ 14236:Φ 14232:p 14185:δ 14180:w 14175:n 14171:m 14156:, 14151:m 14148:n 14140:= 14137:V 14134:d 14130:w 14125:n 14110:m 14074:2 14070:1 14068:Φ 14064:Ψ 14056:Ψ 14050:2 14046:1 14044:Φ 14040:Φ 14036:Φ 14028:Ψ 14009:. 13991:) 13988:y 13984:S 13980:p 13974:) 13971:z 13967:S 13963:x 13938:z 13932:y 13928:x 13912:) 13909:y 13905:S 13901:p 13899:( 13894:y 13889:) 13886:z 13882:S 13878:x 13876:( 13871:z 13858:. 13801:Φ 13799:b 13795:a 13789:b 13783:a 13774:Φ 13770:Ψ 13711:. 13704:. 13698:) 13696:m 13692:ℓ 13688:n 13686:( 13681:. 13678:L 13672:L 13666:. 13662:L 13649:) 13647:m 13643:ℓ 13639:n 13637:( 13617:ℓ 13613:ℓ 13609:ℓ 13605:ℓ 13601:m 13590:n 13586:ℓ 13575:n 13568:ℓ 13564:n 13550:ℓ 13546:n 13539:L 13529:e 13527:e 13525:m 13523:/ 13521:ħ 13518:0 13511:0 13508:a 13493:) 13487:, 13481:( 13476:m 13467:Y 13459:) 13451:0 13447:a 13443:n 13438:r 13435:2 13429:( 13423:1 13420:+ 13414:2 13409:1 13397:n 13393:L 13382:) 13374:0 13370:a 13366:n 13361:r 13358:2 13352:( 13343:0 13339:a 13335:n 13331:/ 13327:r 13320:e 13311:] 13308:! 13305:) 13299:+ 13296:n 13293:( 13290:[ 13287:n 13284:2 13279:! 13276:) 13273:1 13261:n 13258:( 13250:3 13244:) 13236:0 13232:a 13228:n 13224:2 13219:( 13211:= 13208:) 13202:, 13196:, 13193:r 13190:( 13185:m 13179:n 13163:m 13157:ℓ 13148:) 13146:φ 13142:θ 13140:( 13135:ℓ 13128:Y 13122:R 13107:) 13101:, 13095:( 13089:m 13080:Y 13074:) 13071:r 13068:( 13065:R 13062:= 13059:) 13053:, 13047:, 13044:r 13041:( 13036:m 13030:n 12970:n 12953:) 12949:x 12936:m 12928:( 12921:n 12917:H 12902:2 12895:2 12891:x 12884:m 12874:e 12865:4 12861:/ 12857:1 12852:) 12835:m 12829:( 12817:! 12814:n 12808:n 12804:2 12799:1 12793:= 12790:) 12787:x 12784:( 12779:n 12762:n 12760:H 12700:l 12697:C 12691:l 12688:C 12682:l 12679:A 12673:r 12670:C 12664:r 12661:A 12654:r 12651:A 12646:x 12616:. 12613:a 12607:x 12600:x 12597:k 12594:i 12587:e 12580:l 12575:C 12571:+ 12566:x 12563:k 12560:i 12556:e 12549:r 12544:C 12536:, 12533:a 12526:| 12522:x 12518:| 12510:x 12500:e 12493:l 12488:B 12484:+ 12479:x 12472:e 12465:r 12460:B 12452:, 12449:a 12440:x 12433:x 12430:k 12427:i 12420:e 12413:l 12408:A 12404:+ 12399:x 12396:k 12393:i 12389:e 12382:r 12377:A 12370:{ 12365:= 12362:) 12359:x 12356:( 12342:κ 12338:k 12316:a 12309:| 12305:x 12301:| 12295:0 12288:a 12281:| 12277:x 12273:| 12265:0 12261:V 12254:{ 12249:= 12246:) 12243:x 12240:( 12237:V 12214:0 12211:V 12207:E 12201:0 12198:V 12168:) 12166:t 12164:( 12162:O 12150:O 12142:t 12134:O 12116:Ψ 12112:E 12098:, 12094:) 12089:N 12084:r 12079:, 12073:, 12068:2 12063:r 12058:, 12053:1 12048:r 12043:( 12030:/ 12026:t 12023:E 12020:i 12013:e 12009:= 12006:) 12003:t 12000:, 11995:N 11990:r 11985:, 11979:, 11974:2 11969:r 11964:, 11959:1 11954:r 11949:( 11936:N 11912:S 11897:S 11891:= 11881:P 11849:0 11846:= 11840:t 11832:S 11823:+ 11820:) 11816:x 11812:( 11809:V 11806:+ 11801:2 11796:| 11791:) 11788:t 11785:, 11781:x 11777:( 11774:S 11767:| 11760:m 11757:2 11753:1 11727:2 11722:| 11717:S 11710:| 11702:| 11698:S 11693:2 11684:| 11654:m 11650:S 11622:m 11618:S 11591:v 11584:= 11580:J 11548:m 11544:S 11532:= 11529:) 11526:t 11523:, 11519:x 11515:( 11511:J 11488:) 11485:t 11482:, 11478:x 11474:( 11471:S 11449:2 11444:| 11439:) 11436:t 11433:, 11429:x 11425:( 11418:| 11414:= 11411:) 11408:t 11405:, 11401:x 11397:( 11368:) 11365:t 11362:, 11358:x 11354:( 11351:S 11348:i 11334:) 11331:t 11328:, 11324:x 11320:( 11312:= 11309:) 11306:t 11303:, 11299:x 11295:( 11265:) 11246:( 11236:m 11228:= 11225:) 11187:( 11181:m 11178:i 11175:2 11166:= 11163:) 11160:t 11157:, 11153:x 11149:( 11145:J 11122:2 11117:| 11112:) 11109:t 11106:, 11102:x 11098:( 11091:| 11087:= 11084:) 11081:t 11078:, 11074:x 11070:( 11047:0 11044:= 11040:J 11030:+ 11024:t 10978:2 10973:| 10968:) 10964:t 10961:, 10956:N 10952:m 10943:1 10939:m 10935:, 10930:N 10925:r 10915:1 10910:r 10904:( 10896:| 10889:N 10884:r 10877:3 10873:d 10865:N 10861:R 10847:2 10842:r 10835:3 10831:d 10823:2 10819:R 10808:1 10803:r 10796:3 10792:d 10784:1 10780:R 10771:= 10768:) 10765:t 10762:( 10755:N 10751:m 10747:= 10742:N 10738:z 10734:s 10730:, 10725:N 10721:R 10712:N 10707:r 10702:, 10696:, 10691:1 10687:m 10683:= 10678:1 10674:z 10670:s 10666:, 10661:1 10657:R 10648:1 10643:r 10637:P 10622:t 10616:2 10613:m 10609:2 10607:z 10603:s 10597:2 10594:R 10585:1 10582:m 10578:1 10576:z 10572:s 10566:1 10563:R 10544:2 10539:| 10534:) 10530:t 10527:, 10522:N 10518:z 10514:s 10505:1 10501:z 10497:s 10493:, 10488:N 10483:r 10473:1 10468:r 10462:( 10454:| 10449:= 10445:) 10441:t 10438:, 10433:N 10429:z 10425:s 10416:1 10412:z 10408:s 10404:, 10399:N 10394:r 10384:1 10379:r 10373:( 10359:Ψ 10355:N 10338:i 10334:i 10330:i 10321:i 10310:i 10306:r 10303:d 10298:N 10290:N 10275:) 10271:t 10268:, 10263:N 10259:z 10255:s 10246:1 10242:z 10238:s 10234:, 10229:N 10224:r 10214:1 10209:r 10203:( 10197:2 10188:) 10184:t 10181:, 10176:N 10172:z 10168:s 10159:1 10155:z 10151:s 10147:, 10142:N 10137:r 10127:1 10122:r 10116:( 10105:1 10095:N 10090:r 10083:3 10079:d 10072:e 10069:c 10066:a 10063:p 10060:s 10056:l 10053:l 10050:a 10036:2 10031:r 10024:3 10020:d 10013:e 10010:c 10007:a 10004:p 10001:s 9997:l 9994:l 9991:a 9980:1 9975:r 9968:3 9964:d 9957:e 9954:c 9951:a 9948:p 9945:s 9941:l 9938:l 9935:a 9922:1 9918:z 9914:s 9901:2 9897:z 9893:s 9877:N 9873:z 9869:s 9860:= 9857:) 9852:2 9844:, 9839:1 9831:( 9820:N 9800:. 9779:N 9775:z 9771:s 9767:, 9761:, 9756:1 9752:z 9748:s 9744:, 9739:N 9734:r 9729:, 9723:, 9718:1 9713:r 9707:| 9669:) 9664:N 9660:z 9656:s 9652:, 9646:, 9641:1 9637:z 9633:s 9629:, 9624:N 9619:r 9614:, 9608:, 9603:1 9598:r 9593:( 9566:1 9561:r 9554:3 9550:d 9542:1 9538:R 9524:N 9519:r 9512:3 9508:d 9500:N 9496:R 9467:N 9463:z 9459:s 9455:, 9449:, 9444:1 9440:z 9436:s 9421:= 9411:| 9386:) 9383:t 9380:, 9375:N 9371:z 9367:s 9358:2 9354:z 9350:s 9346:, 9341:1 9337:z 9333:s 9329:, 9324:N 9319:r 9309:2 9304:r 9299:, 9294:1 9289:r 9284:( 9271:N 9262:i 9258:x 9240:) 9233:, 9228:2 9223:x 9218:, 9213:1 9208:x 9203:, 9197:, 9192:a 9187:r 9182:, 9176:, 9171:b 9166:r 9157:( 9147:= 9143:) 9136:, 9131:2 9126:x 9121:, 9116:1 9111:x 9106:, 9100:, 9095:b 9090:r 9085:, 9079:, 9074:a 9069:r 9060:( 9044:i 9040:r 9033:2 9030:x 9026:1 9023:x 9016:2 9013:r 9009:1 9006:r 8990:N 8970:− 8962:+ 8947:) 8940:, 8935:a 8930:r 8925:, 8919:, 8914:b 8909:r 8900:( 8890:= 8886:) 8879:, 8874:b 8869:r 8864:, 8858:, 8853:a 8848:r 8839:( 8809:N 8807:3 8803:t 8799:i 8793:i 8789:r 8774:) 8771:t 8768:, 8763:N 8758:r 8748:2 8743:r 8738:, 8733:1 8728:r 8723:( 8709:N 8668:J 8628:z 8624:s 8619:| 8606:r 8601:| 8597:= 8589:z 8585:s 8581:, 8577:r 8572:| 8552:) 8549:t 8546:, 8541:z 8537:s 8533:( 8526:) 8523:t 8520:, 8516:r 8512:( 8506:= 8503:) 8500:t 8497:, 8492:z 8488:s 8484:, 8480:r 8476:( 8451:) 8448:t 8445:( 8438:| 8426:) 8423:t 8420:( 8413:| 8409:= 8403:) 8400:t 8397:( 8390:| 8364:z 8360:s 8355:| 8342:r 8337:| 8332:) 8329:t 8326:, 8321:z 8317:s 8313:( 8306:) 8303:t 8300:, 8296:r 8292:( 8284:r 8277:3 8273:d 8262:z 8258:s 8249:= 8243:) 8240:t 8237:( 8230:| 8218:) 8215:t 8212:( 8205:| 8191:ξ 8181:ψ 8150:z 8146:s 8142:, 8138:r 8133:| 8128:) 8125:t 8122:, 8117:z 8113:s 8109:, 8105:r 8101:( 8093:r 8086:3 8082:d 8071:z 8067:s 8058:= 8052:) 8049:t 8046:( 8039:| 8009:] 8003:) 8000:t 7997:, 7994:s 7988:, 7984:r 7980:( 7970:) 7967:t 7964:, 7961:) 7958:1 7952:s 7949:( 7943:, 7939:r 7935:( 7918:) 7915:t 7912:, 7909:1 7903:s 7900:, 7896:r 7892:( 7882:) 7879:t 7876:, 7873:s 7870:, 7866:r 7862:( 7853:[ 7848:= 7845:) 7842:t 7839:, 7835:r 7831:( 7808:) 7805:t 7802:, 7797:z 7793:s 7789:, 7785:r 7781:( 7765:) 7763:t 7759:s 7755:ξ 7751:t 7747:s 7745:( 7743:ξ 7739:t 7735:s 7733:( 7731:ξ 7724:s 7722:2 7717:ξ 7695:z 7691:s 7686:| 7681:) 7678:t 7675:, 7670:z 7666:s 7662:( 7654:s 7649:s 7643:= 7638:z 7634:s 7625:= 7619:) 7616:t 7613:( 7606:| 7577:] 7571:1 7564:0 7550:0 7543:0 7537:[ 7532:) 7529:t 7526:, 7523:s 7517:( 7511:+ 7506:] 7500:0 7493:1 7479:0 7472:0 7466:[ 7461:) 7458:t 7455:, 7452:) 7449:1 7443:s 7440:( 7434:( 7428:+ 7422:+ 7417:] 7411:0 7404:0 7390:1 7383:0 7377:[ 7372:) 7369:t 7366:, 7363:1 7357:s 7354:( 7348:+ 7343:] 7337:0 7330:0 7316:0 7309:1 7303:[ 7298:) 7295:t 7292:, 7289:s 7286:( 7280:= 7275:] 7269:) 7266:t 7263:, 7260:s 7254:( 7244:) 7241:t 7238:, 7235:) 7232:1 7226:s 7223:( 7217:( 7200:) 7197:t 7194:, 7191:1 7185:s 7182:( 7172:) 7169:t 7166:, 7163:s 7160:( 7151:[ 7146:= 7126:s 7124:2 7119:s 7115:s 7111:s 7107:s 7100:z 7098:s 7093:s 7080:z 7077:s 7064:t 7059:r 7052:z 7050:s 7045:z 7041:z 7032:z 7029:s 7014:) 7011:t 7008:, 7003:z 6999:s 6995:( 6959:r 6954:| 6949:) 6946:t 6943:, 6939:r 6935:( 6927:r 6920:3 6916:d 6909:= 6903:) 6900:t 6897:( 6890:| 6876:) 6874:t 6870:r 6863:t 6853:r 6838:) 6835:t 6832:, 6828:r 6824:( 6788:z 6784:s 6779:| 6771:r 6766:| 6762:= 6754:z 6750:s 6746:, 6742:r 6737:| 6712:} 6707:i 6699:{ 6677:] 6669:s 6652:1 6649:+ 6646:s 6622:1 6616:s 6602:s 6591:[ 6586:= 6581:] 6568:| 6564:s 6544:| 6540:) 6537:1 6531:s 6528:( 6501:| 6497:1 6491:s 6474:| 6470:s 6461:[ 6456:= 6446:| 6412:] 6406:s 6398:0 6388:0 6383:0 6376:0 6371:) 6368:1 6362:s 6359:( 6346:0 6341:0 6307:0 6302:0 6292:1 6286:s 6281:0 6274:0 6269:0 6259:0 6254:s 6248:[ 6243:= 6238:z 6228:S 6216:1 6197:z 6195:s 6174:] 6168:1 6161:0 6147:0 6140:0 6134:[ 6123:s 6116:| 6111:, 6105:] 6099:0 6092:1 6078:0 6071:0 6065:[ 6054:) 6051:1 6045:s 6042:( 6035:| 6030:, 6023:, 6017:] 6011:0 6004:0 5990:1 5983:0 5977:[ 5966:1 5960:s 5956:| 5951:, 5945:] 5939:0 5932:0 5918:0 5911:1 5905:[ 5894:s 5890:| 5875:z 5873:s 5853:1 5850:+ 5847:s 5844:2 5819:2 5815:) 5811:1 5808:+ 5805:s 5802:2 5799:( 5779:s 5739:1 5736:+ 5733:s 5730:2 5710:s 5676:i 5667:| 5629:i 5620:| 5599:} 5589:| 5583:i 5572:{ 5549:} 5541:i 5532:| 5528:{ 5508:} 5500:) 5497:j 5494:( 5485:| 5481:{ 5460:| 5454:) 5451:j 5448:( 5432:) 5429:j 5426:( 5417:| 5411:j 5403:= 5388:P 5362:2 5357:| 5345:| 5329:P 5321:| 5317:= 5312:2 5307:| 5295:| 5289:) 5286:j 5283:( 5271:| 5265:j 5257:= 5254:) 5248:( 5239:P 5214:i 5185:} 5177:) 5174:j 5171:( 5162:| 5158:{ 5118:} 5110:i 5101:| 5097:{ 5073:2 5068:| 5056:| 5050:i 5038:| 5034:= 5031:) 5026:i 5018:( 5009:P 4980:i 4949:i 4917:} 4909:i 4900:| 4896:{ 4868:} 4858:| 4852:i 4841:{ 4814:| 4808:i 4792:i 4783:| 4777:i 4769:= 4759:| 4755:I 4752:= 4742:| 4716:} 4708:i 4699:| 4695:{ 4675:I 4672:= 4668:| 4662:i 4646:i 4637:| 4631:i 4623:= 4620:P 4598:} 4590:i 4581:| 4577:{ 4566:N 4510:L 4504:p 4500:x 4472:. 4468:p 4465:d 4460:x 4457:p 4449:i 4443:e 4439:) 4436:p 4433:( 4415:2 4411:1 4406:= 4403:) 4400:x 4397:( 4372:. 4368:x 4365:d 4360:x 4357:p 4349:i 4340:e 4336:) 4333:x 4330:( 4312:2 4308:1 4303:= 4300:) 4297:p 4294:( 4271:, 4266:x 4263:p 4255:i 4246:e 4233:2 4229:1 4224:= 4218:x 4214:| 4210:p 4199:x 4196:p 4188:i 4182:e 4169:2 4165:1 4160:= 4157:) 4154:x 4151:( 4148:p 4145:= 4139:p 4135:| 4131:x 4106:. 4103:) 4100:p 4097:( 4091:= 4084:p 4080:d 4077:) 4070:p 4063:p 4060:( 4054:) 4047:p 4043:( 4034:= 4027:p 4023:d 4013:p 4008:| 4004:p 3998:) 3991:p 3987:( 3978:= 3975:x 3972:d 3966:x 3962:| 3958:p 3952:) 3949:x 3946:( 3930:Ψ 3910:. 3907:p 3904:d 3898:p 3894:| 3890:) 3887:p 3884:( 3875:= 3872:p 3869:d 3859:| 3855:p 3846:p 3842:| 3835:= 3821:| 3817:I 3814:= 3804:| 3796:, 3793:x 3790:d 3784:x 3780:| 3776:) 3773:x 3770:( 3761:= 3758:x 3755:d 3745:| 3741:x 3732:x 3728:| 3721:= 3707:| 3703:I 3700:= 3690:| 3674:p 3668:x 3642:. 3639:) 3632:p 3625:p 3622:( 3616:= 3613:) 3604:p 3595:, 3590:p 3582:( 3554:} 3545:p 3533:, 3530:) 3527:t 3524:, 3521:x 3518:( 3513:p 3505:{ 3495:p 3477:, 3468:/ 3464:x 3461:p 3458:i 3454:e 3450:= 3447:) 3444:x 3441:( 3436:p 3409:. 3405:p 3402:d 3399:) 3396:t 3393:, 3390:p 3387:( 3382:2 3374:) 3371:t 3368:, 3365:p 3362:( 3352:1 3325:= 3322:) 3317:2 3309:, 3304:1 3296:( 3286:) 3284:t 3280:p 3278:( 3276:2 3274:Φ 3270:) 3268:t 3264:p 3262:( 3260:1 3258:Φ 3251:t 3235:p 3221:) 3218:t 3215:, 3212:p 3209:( 3170:. 3166:x 3163:d 3159:| 3155:x 3146:x 3142:| 3135:= 3132:I 3102:| 3097:) 3093:x 3090:d 3086:| 3082:x 3073:x 3069:| 3061:( 3057:= 3054:x 3051:d 3041:| 3037:x 3028:x 3024:| 3017:= 3007:| 2986:) 2979:x 2975:( 2969:= 2966:x 2963:d 2957:x 2953:| 2945:x 2938:) 2935:x 2932:( 2923:= 2913:| 2905:x 2881:) 2878:x 2868:x 2864:( 2858:= 2852:x 2848:| 2840:x 2815:x 2808:x 2756:x 2753:d 2747:x 2743:| 2739:) 2736:t 2733:, 2730:x 2727:( 2718:= 2712:) 2709:t 2706:( 2699:| 2684:) 2682:t 2678:x 2621:, 2617:1 2614:= 2611:x 2608:d 2603:2 2598:| 2593:) 2590:t 2587:, 2584:x 2581:( 2574:| 2540:t 2526:x 2523:d 2518:2 2513:| 2508:) 2505:t 2502:, 2499:x 2496:( 2489:| 2482:b 2477:a 2469:= 2466:) 2463:t 2460:( 2455:b 2449:x 2443:a 2439:P 2427:b 2423:x 2419:a 2413:x 2387:t 2368:, 2365:) 2362:x 2359:( 2353:= 2350:) 2347:t 2344:, 2341:x 2338:( 2332:) 2329:t 2326:, 2323:x 2320:( 2307:= 2302:2 2297:| 2293:) 2290:t 2287:, 2284:x 2281:( 2274:| 2251:t 2247:x 2239:t 2235:x 2221:, 2217:) 2214:t 2211:, 2208:x 2205:( 2168:Ψ 2156:Ψ 2148:Ψ 2136:, 2122:2 2108:= 2105:) 2099:, 2093:( 2080:Ψ 2070:. 2052:x 2049:d 2045:) 2042:t 2039:, 2036:x 2033:( 2028:2 2020:) 2017:t 2014:, 2011:x 2008:( 1998:1 1971:= 1968:) 1963:2 1955:, 1950:1 1942:( 1929:t 1924:2 1922:Ψ 1917:1 1915:Ψ 1887:t 1858:. 1855:p 1850:x 1842:p 1838:x 1832:p 1824:x 1818:) 1816:p 1810:) 1808:x 1780:. 1749:) 1743:( 1738:) 1734:( 1730:. 1716:. 1650:2 1642:1 1624:2 1620:3 1610:1 1600:2 1596:1 1590:0 1415:N 1320:, 1306:p 1303:h 1298:= 1270:h 1260:, 1246:p 1243:h 1238:= 1225:, 1189:p 1176:, 1164:f 1161:h 1158:= 1155:E 1145:, 1133:E 1112:f 1089:e 1082:t 1075:v 303:| 293:H 287:= 277:| 270:t 267:d 263:d 255:i 178:2 174:1 124:ψ 83:Ψ 78:ψ 23:.

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Index