Relativistic quantum mechanics

8988: 8324: 8983:{\displaystyle \psi (\mathbf {r} ,t)={\begin{bmatrix}\psi _{+,\,\sigma =s}(\mathbf {r} ,t)\\\psi _{+,\,\sigma =s-1}(\mathbf {r} ,t)\\\vdots \\\psi _{+,\,\sigma =-s+1}(\mathbf {r} ,t)\\\psi _{+,\,\sigma =-s}(\mathbf {r} ,t)\\\psi _{-,\,\sigma =s}(\mathbf {r} ,t)\\\psi _{-,\,\sigma =s-1}(\mathbf {r} ,t)\\\vdots \\\psi _{-,\,\sigma =-s+1}(\mathbf {r} ,t)\\\psi _{-,\,\sigma =-s}(\mathbf {r} ,t)\end{bmatrix}}\quad \rightleftharpoons \quad {\psi (\mathbf {r} ,t)}^{\dagger }{\begin{bmatrix}{\psi _{+,\,\sigma =s}(\mathbf {r} ,t)}^{\star }&{\psi _{+,\,\sigma =s-1}(\mathbf {r} ,t)}^{\star }&\cdots &{\psi _{-,\,\sigma =-s}(\mathbf {r} ,t)}^{\star }\end{bmatrix}}} 8257: 7787: 8252:{\displaystyle \psi (\mathbf {r} ,t)={\begin{bmatrix}\psi _{\sigma =s}(\mathbf {r} ,t)\\\psi _{\sigma =s-1}(\mathbf {r} ,t)\\\vdots \\\psi _{\sigma =-s+1}(\mathbf {r} ,t)\\\psi _{\sigma =-s}(\mathbf {r} ,t)\end{bmatrix}}\quad \rightleftharpoons \quad {\psi (\mathbf {r} ,t)}^{\dagger }={\begin{bmatrix}{\psi _{\sigma =s}(\mathbf {r} ,t)}^{\star }&{\psi _{\sigma =s-1}(\mathbf {r} ,t)}^{\star }&\cdots &{\psi _{\sigma =-s+1}(\mathbf {r} ,t)}^{\star }&{\psi _{\sigma =-s}(\mathbf {r} ,t)}^{\star }\end{bmatrix}}} 4495: 6871: 16640: 4200: 6614: 2667: 10187: 5575: 4490:{\displaystyle \rho ={\frac {i\hbar }{2mc^{2}}}\left(\psi ^{*}{\frac {\partial \psi }{\partial t}}-\psi {\frac {\partial \psi ^{*}}{\partial t}}\right)\,,\quad \mathbf {j} =-{\frac {i\hbar }{2m}}\left(\psi ^{*}\nabla \psi -\psi \nabla \psi ^{*}\right)\quad \rightleftharpoons \quad J^{\mu }={\frac {i\hbar }{2m}}(\psi ^{*}\partial ^{\mu }\psi -\psi \partial ^{\mu }\psi ^{*})} 4886: 5850: 3871: 4152: 3480: 6866:{\displaystyle \left({\frac {\hat {E}}{c}}+{\boldsymbol {\sigma }}\cdot {\hat {\mathbf {p} }}\right)\psi _{+}=0\,,\quad \left({\frac {\hat {E}}{c}}-{\boldsymbol {\sigma }}\cdot {\hat {\mathbf {p} }}\right)\psi _{-}=0\quad \rightleftharpoons \quad \sigma ^{\mu }{\hat {P}}_{\mu }\psi _{+}=0\,,\quad \sigma _{\mu }{\hat {P}}^{\mu }\psi _{-}=0\,,} 10892: 10709: 10314: 2441: 7497: 6456: 10919:(QED), are summarized below . More than half a century of experimental and theoretical research from the 1890s through to the 1950s in the new and mysterious quantum theory as it was up and coming revealed that a number of phenomena cannot be explained by QM alone. SR, found at the turn of the 20th century, was found to be a 10003: 2690:: infinite-order in space derivatives but only first order in the time derivative, which is inelegant and unwieldy. Again, there is the problem of the non-invariance of the energy operator, equated to the square root which is also not invariant. Another problem, less obvious and more severe, is that it can be shown to be 5372: 10732:. It was pointed out by L. Thomas that there is a second relativistic effect: An electric field component perpendicular to the electron velocity causes an additional acceleration of the electron perpendicular to its instantaneous velocity, so the electron moves in a curved path. The electron moves in a 9520: 2923: 5284: 9930: 11173:

and the apparent violation of causality upheld in SR: particles can appear to interact instantaneously at arbitrary distances. This was a misconception since information is not and cannot be transferred in the entangled states; rather the information transmission is in the process of measurement by

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its rest mass. Helicity indicates the orientations of the spin and translational momentum vectors. Helicity is frame-dependent because of the 3-momentum in the definition, and is quantized due to spin quantization, which has discrete positive values for parallel alignment, and negative values for

3311:, yet this equation alone isn't a sufficient foundation for RQM for a at least two reasons: one is that negative-energy states are solutions, another is the density (given below), and this equation as it stands is only applicable to spinless particles. This equation can be factored into the form: 1706:

should be equal, and ideally as low as possible, so that no initial values of the derivatives need to be specified. This is important for probability interpretations, exemplified below. The lowest possible order of any differential equation is the first (zeroth order derivatives would not form a

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This is not possible for all RWEs; and is one reason the Lorentz group theoretic approach is important and appealing: fundamental invariance and symmetries in space and time can be used to derive RWEs using appropriate group representations. The Lagrangian approach with field interpretation of

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and substituting this into the above Schrödinger equation gives a non-relativistic QM equation for the wavefunction: the procedure is a straightforward substitution of a simple expression. By contrast this is not as easy in RQM; the energy–momentum equation is quadratic in energy

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and was discovered by many people because of the straightforward way of obtaining it, notably by Schrödinger in 1925 before he found the non-relativistic equation named after him, and by Klein and Gordon in 1927, who included electromagnetic interactions in the equation. This

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suggest it isn't a scalar wavefunction as permitted in the KG equation, but must instead be a four-component entity. The Dirac equation still predicts negative energy solutions, so Dirac postulated that negative energy states are always occupied, because according to the

3317: 10746: 10202: 2662:{\displaystyle {\hat {H}}={\hat {E}}={\sqrt {c^{2}{\hat {\mathbf {p} }}\cdot {\hat {\mathbf {p} }}+(mc^{2})^{2}}}\quad \Rightarrow \quad i\hbar {\frac {\partial }{\partial t}}\psi ={\sqrt {c^{2}{\hat {\mathbf {p} }}\cdot {\hat {\mathbf {p} }}+(mc^{2})^{2}}}\,\psi } 10568: 9124: 7255: 10478: 7329: 2173: 3298: 3047:

and interaction terms with externally applied fields, similar to the classical potential energy term, as well as momentum terms like the classical kinetic energy term. A key difference is that relativistic Hamiltonians contain spin operators in the form of

1683: 11807: 10182:{\displaystyle J^{\alpha \beta }=2X^{}+{\frac {1}{m^{2}}}\varepsilon ^{\alpha \beta \gamma \delta }W_{\gamma }p_{\delta }\quad \rightleftharpoons \quad \mathbf {J} =\mathbf {X} \wedge \mathbf {P} +{\frac {1}{m^{2}}}\star (\mathbf {W} \wedge \mathbf {P} )} 6315: 3641: 2425: 5570:{\displaystyle \left(i\hbar {\frac {\partial }{\partial t}}-q\phi \right)\psi ={\frac {1}{2m}}{({\hat {\mathbf {p} }}-q\mathbf {A} )}^{2}\psi \quad \Leftrightarrow \quad {\hat {H}}={\frac {1}{2m}}{({\hat {\mathbf {p} }}-q\mathbf {A} )}^{2}+q\phi .} 10740:. It can be shown that the net result of this effect is that the spin–orbit interaction is reduced by half, as if the magnetic field experienced by the electron has only one-half the value, and the relativistic correction in the Hamiltonian is: 2011:

as one measures in a different frame boosted and/or rotated relative the original frame in consideration. The derivative operators, and hence the energy and 3-momentum operators, are also non-invariant and change under Lorentz transformations.

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There is also the problem of incorporating spin in the Hamiltonian, which isn't a prediction of the non-relativistic Schrödinger theory. Particles with spin have a corresponding spin magnetic moment quantized in units of

106: 4881:{\displaystyle {\hat {E}}\rightarrow {\hat {E}}-q\phi \,,\quad {\hat {\mathbf {p} }}\rightarrow {\hat {\mathbf {p} }}-q\mathbf {A} \quad \rightleftharpoons \quad {\hat {P}}_{\mu }\rightarrow {\hat {P}}_{\mu }-qA_{\mu }} 1197: 9276: 5845:{\displaystyle \left(i\hbar {\frac {\partial }{\partial t}}-q\phi \right)\psi =\left\psi \quad \Leftrightarrow \quad {\hat {H}}={\frac {1}{2m}}{({\boldsymbol {\sigma }}\cdot (\mathbf {p} -q\mathbf {A} ))}^{2}+q\phi } 1575: 5991: 3866:{\displaystyle \left({\hat {E}}-c{\boldsymbol {\alpha }}\cdot {\hat {\mathbf {p} }}-\beta mc^{2}\right)\psi =0\quad \Leftrightarrow \quad {\hat {H}}=c{\boldsymbol {\alpha }}\cdot {\hat {\mathbf {p} }}+\beta mc^{2}} 9129:

According to the relativistic energy-momentum relation, all massless particles travel at the speed of light, so particles traveling at the speed of light are also described by two-component spinors. Historically,

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For equations describing higher-spin particles, the inclusion of interactions is nowhere near as simple minimal coupling, they lead to incorrect predictions and self-inconsistencies. For spin greater than

5343:-mesons, experience the much stronger strong interaction in addition to the electromagnetic interaction. It does, however, correctly describe charged spinless bosons in the absence of other interactions. 5311:

components. Such solutions cannot in general describe particles with nonzero spin since spin components are not independent. Other constraint will have to be imposed for that, e.g. the Dirac equation for

4147:{\displaystyle \rho =\psi ^{\dagger }\psi ,\quad \mathbf {j} =\psi ^{\dagger }\gamma ^{0}{\boldsymbol {\gamma }}\psi \quad \rightleftharpoons \quad J^{\mu }=\psi ^{\dagger }\gamma ^{0}\gamma ^{\mu }\psi } 3475:{\displaystyle \left({\hat {E}}-c{\boldsymbol {\alpha }}\cdot {\hat {\mathbf {p} }}-\beta mc^{2}\right)\left({\hat {E}}+c{\boldsymbol {\alpha }}\cdot {\hat {\mathbf {p} }}+\beta mc^{2}\right)\psi =0\,,} 1486: 1881:

and non-relativistic QM, time is an absolute quantity all observers and particles can always agree on, "ticking away" in the background independent of space. Thus in non-relativistic QM one has for a

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formulations are both applicable in SR, but in QM the potentials enter the Hamiltonian (see above) and influence the motion of charged particles even in regions where the fields are zero. In 1964,

10887:{\displaystyle {\hat {H}}=-\mathbf {B} '\cdot {\hat {\boldsymbol {\mu }}}_{S}=-\left(\mathbf {B} +{\frac {\mathbf {E} \times \mathbf {v} }{2c^{2}}}\right)\cdot {\hat {\boldsymbol {\mu }}}_{S}\,.} 10309:{\displaystyle W_{\alpha }={\frac {1}{2}}\varepsilon _{\alpha \beta \gamma \delta }M^{\beta \gamma }p^{\delta }\quad \rightleftharpoons \quad \mathbf {W} =\star (\mathbf {M} \wedge \mathbf {P} )} 10704:{\displaystyle {\hat {H}}=-\mathbf {B} '\cdot {\hat {\boldsymbol {\mu }}}_{S}=-\left(\mathbf {B} +{\frac {\mathbf {E} \times \mathbf {v} }{c^{2}}}\right)\cdot {\hat {\boldsymbol {\mu }}}_{S}\,,} 7643: 7492:{\displaystyle {\hat {h}}={\hat {\mathbf {S} }}\cdot {\frac {\hat {\mathbf {p} }}{|\mathbf {p} |}}={\hat {\mathbf {S} }}\cdot {\frac {c{\hat {\mathbf {p} }}}{\sqrt {E^{2}-(m_{0}c^{2})^{2}}}}} 9015: 3706: 7134: 2073: 6451:{\displaystyle \psi ={\begin{pmatrix}\psi _{+}\\\psi _{-}\end{pmatrix}}={\begin{pmatrix}\psi _{+\uparrow }\\\psi _{+\downarrow }\\\psi _{-\uparrow }\\\psi _{-\downarrow }\end{pmatrix}}} 5339:

and the particle is described by a wavefunction, the solution to the KG equation. The equation is, as it stands, not always very useful, because massive spinless particles, such as the

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In the case where the charge is zero, the equation reduces trivially to the free KG equation so nonzero charge is assumed below. This is a scalar equation that is invariant under the

1067:'s work between 1927 to 1933 shaped the synthesis of special relativity and quantum mechanics. His work was instrumental, as he formulated the Dirac equation and also originated 3543: 2330: 7762:

were found in 1948 using Lorentz group theory, applicable for all free particles with any spin. Considering the factorization of the KG equation above, and more rigorously by

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pre-multiplying the energy operator (including the potential energy term), conventionally not written for simplicity and clarity (i.e. treated like the number 1). Here

3068: 9515:{\displaystyle \partial _{\mu }\left({\frac {\partial {\mathcal {L}}}{\partial (\partial _{\mu }\psi )}}\right)-{\frac {\partial {\mathcal {L}}}{\partial \psi }}=0\,} 6936: 4656:. Probability density and current also satisfy a continuity equation because probability is conserved, however this is only possible in the absence of interactions. 4997:

that is, the total energy of the particle is approximately the rest energy for small electric potentials, and the momentum is approximately the classical momentum.

2918:{\displaystyle {\hat {\boldsymbol {\mu }}}_{S}=-{\frac {g\mu _{B}}{\hbar }}{\hat {\mathbf {S} }}\,,\quad \left|{\boldsymbol {\mu }}_{S}\right|=-g\mu _{B}\sigma \,,} 15203: 5279:{\displaystyle {({\hat {E}}-q\phi )}^{2}\psi =c^{2}{({\hat {\mathbf {p} }}-q\mathbf {A} )}^{2}\psi +(mc^{2})^{2}\psi \quad \rightleftharpoons \quad \left\psi =0.} 15619: 12549: 9925:{\displaystyle M^{\alpha \beta }=X^{\alpha }P^{\beta }-X^{\beta }P^{\alpha }=2X^{}\quad \rightleftharpoons \quad \mathbf {M} =\mathbf {X} \wedge \mathbf {P} \,,} 5875: 11967: 7654: 1741: 15703: 15682: 15661: 15640: 15598: 15577: 15556: 15535: 11332: 15770: 9192:

case only allows a magnetic dipole, but for spin 1 particles magnetic quadrupoles and electric dipoles are also possible. For more on this topic, see

1035:, from which these predictions emerge automatically. By contrast, in non-relativistic quantum mechanics, terms have to be introduced artificially into the 14516: 14495: 989:

were originally formulated in a non-relativistic background, a few of them (e.g. the Dirac or path-integral formalism) also work with special relativity.

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particle. In the non-relativistic limit the equation reduces to the Schrödinger equation for a spinless charged particle in an electromagnetic field:

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Rȩbilas, K. (2013). "Comment on 'Elementary analysis of the special relativistic combination of velocities, Wigner rotation and Thomas precession'".

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uses invariant Lagrangians rather than Hamiltonian operators, since the latter can become extremely complicated, see (for example) Weinberg (1995).

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to predict the quantum dynamics of the particle. For space and time to be placed on equal footing, as in relativity, the orders of space and time

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in an external electromagnetic potential. As such, the equation cannot be applied to the description of atoms, since the electron is a spin

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which couples to the energy operator (time derivative), just as the other three matrices couple to the momentum operator (spatial derivatives).

6218:{\displaystyle \left(i\hbar {\frac {\partial }{\partial t}}-q\phi \right)\psi =\gamma ^{0}\left\psi \quad \rightleftharpoons \quad \left\psi =0} 13042:

Lorcé, Cédric (2009). "Electromagnetic Properties for Arbitrary Spin Particles: Part 1 − Electromagnetic Current and Multipole Decomposition".

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so that terms with mixed second-order derivatives cancel while the second-order derivatives purely in space and time remain. The first factor:

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Low, S.G. (1997). "Canonically Relativistic Quantum Mechanics: Representations of the Unitary Semidirect Heisenberg Group, U(1,3) *s H(1,3)".

16085: 15150: 9728:{\displaystyle {\mathcal {L}}=-{\frac {\hbar ^{2}}{m}}\eta ^{\mu \nu }\partial _{\mu }\psi ^{*}\partial _{\nu }\psi -mc^{2}\psi ^{*}\psi \,.} 1126: 958: 372: 4642:{\displaystyle {\frac {\partial \rho }{\partial t}}+\nabla \cdot \mathbf {J} =0\quad \rightleftharpoons \quad \partial _{\mu }J^{\mu }=0\,,} 3994:

is a wavefunction, the probability interpretation is not the same as in non-relativistic QM. Some RWEs do not predict a probability density

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which means the particle could arrive at a point before a pulse of light could. This would have to be remedied by the additional constraint

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Lorcé, Cédric (2009). "Electromagnetic Properties for Arbitrary Spin Particles: Part 2 − Natural Moments and Transverse Charge Densities".

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where the first term is already the non-relativistic magnetic moment interaction, and the second term the relativistic correction of order

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Substituting the energy and momentum operators directly into the energy–momentum relation may at first sight seem appealing, to obtain the

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of 1923 provided more evidence that special relativity does apply; in this case to a particle description of photon–electron scattering.

873: 1053:. A unique consequence of QFT that has been tested against other RQMs is the failure of conservation of particle number, for example in 5295: 2257: 2055: 16608: 13735: 11616: 11553: 15318: 9987:

of the rotating object. An additional relativistic-quantum term has to be added for particles with spin. For a particle of rest mass

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is not helpful for several reasons. The square root of the operators cannot be used as it stands; it would have to be expanded in a

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are common alternatives. All equations are in the position representation; for the momentum representation the equations have to be

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of particles, and the electrons and nuclei within atoms and molecules. Numerous results are attributed to the effects of spin.

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Masakatsu, K. (2012). "Superradiance Problem of Bosons and Fermions for Rotating Black Holes in Bargmann–Wigner Formulation".

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for further description of these spinors. In the non-relativistic limit the Dirac equation reduces to the Pauli equation (see

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A more modern approach to RWEs, first introduced during the time RWEs were developing for particles of any spin, is to apply

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pre-multiplying the energy operator conventionally not written. In RQM it is useful to take this as the zeroth Pauli matrix

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Relativistic Hamiltonians are analogous to those of non-relativistic QM in the following respect; there are terms including

16161: 1866: 459: 9407: 9379: 9119:{\displaystyle \psi (\mathbf {r} ,t)={\begin{pmatrix}\psi _{+}(\mathbf {r} ,t)\\\psi _{-}(\mathbf {r} ,t)\end{pmatrix}}} 14664: 14431: 7250:{\displaystyle \left_{+}=\gamma ^{\alpha }\gamma ^{\beta }+\gamma ^{\beta }\gamma ^{\alpha }=2\eta ^{\alpha \beta }\,,} 4528:

Instead, what appears look at first sight a "probability density" and "probability current" has to be reinterpreted as

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has to be added to the above non-relativistic Hamiltonian. On the contrary; a relativistic Hamiltonian introduces spin

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Bhaumik, Mani L. (2022). "How Dirac's Seminal Contributions Pave the Way for Comprehending Nature's Deeper Designs".

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letters in this definition, even though it is conventional to use Greek indices for vectors and tensors in spacetime.

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fermions to first-order relativistic corrections; one of the first attempts to describe such a relativistic quantum

2168:{\displaystyle \psi _{\sigma }(\mathbf {r} ,t)\rightarrow D(\Lambda )\psi _{\sigma }(\Lambda ^{-1}(\mathbf {r} ,t))} 16017: 14699: 9001:, there are only ever two-component spinor fields; one is for the particle in one helicity state corresponding to + 3293:{\displaystyle {\hat {E}}^{2}\psi =c^{2}{\hat {\mathbf {p} }}\cdot {\hat {\mathbf {p} }}\psi +(mc^{2})^{2}\psi \,,} 208: 10923:

component, leading to unification: RQM. Theoretical predictions and experiments mainly focused on the newly found

7295:. This is, however, still only an approximation, and the Hamiltonian includes numerous long and complicated sums. 1678:{\displaystyle {\hat {E}}=i\hbar {\frac {\partial }{\partial t}}\,,\quad {\hat {\mathbf {p} }}=-i\hbar \nabla \,,} 16575: 16284: 16279: 16002: 14749:

Henneaux, M.; Teitelboim, C. (1982). "Relativistic quantum mechanics of supersymmetric particles". Vol. 143.

11802:{\displaystyle \psi ={\begin{pmatrix}u\\v\end{pmatrix}}={\begin{pmatrix}u^{1}\\u^{2}\\v^{1}\\v^{2}\end{pmatrix}}} 11443: 10473:{\displaystyle \mathbf {B} '={\frac {\mathbf {E} \times \mathbf {v} }{c^{2}{\sqrt {1-\left(v/c\right)^{2}}}}}\,.} 10320: 2261: 2239:

as well as other labels, continuous or discrete, representing other quantum numbers are suppressed. One value of

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Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character

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The Hamiltonian operators in the Schrödinger picture are one approach to forming the differential equations for

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The Schrödinger/Pauli velocity operator can be defined for a massive particle using the classical definition

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is a four-component spinor field, which is conventionally split into two two-component spinors in the form:

3636:{\displaystyle \alpha _{i}\beta =-\beta \alpha _{i},\quad \alpha _{i}\alpha _{j}=-\alpha _{j}\alpha _{i}\,,} 1843:

This equation is also true in RQM, provided the Heisenberg operators are modified to be consistent with SR.

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Coester, F.; Polyzou, W.N. (1982). "Relativistic quantum mechanics of particles with direct interactions".

11207: 11195: 11088: 3962: 2420:{\displaystyle {\hat {H}}={\frac {{\hat {\mathbf {p} }}\cdot {\hat {\mathbf {p} }}}{2m}}+V(\mathbf {r} ,t)} 1407: 531: 526: 497: 348: 129: 3917:. The positive mass equation can continue to be used without loss of continuity. The matrices multiplying 1359:

were the first to propose the concept of spin. The inclusion of spin in the wavefunction incorporates the

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An automatic occurrence in the Dirac equation (and the Weyl equation) is the projection of the spin

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representation of the Lorentz group. This means that all of its solutions will belong to a direct sum of

1364: 1107: 565: 310: 300: 16664: 16643: 16405: 16213: 16184: 10336: 9160:, the RWE is not fixed by the particle's mass, spin, and electric charge; the electromagnetic moments ( 7265: 6590: 5865:

is not just a scalar wavefunction as in the non-relativistic Schrödinger equation, but a two-component

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absolute; any two observers moving relative to each other can measure different locations and times of

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Triebwasser, S.; Dayhoff, E.S. & Lamb, W.E. Jr. (1953). "Fine Structure of the Hydrogen Atom. V".

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Corben, H.C. (1993). "Factors of 2 in magnetic moments, spin–orbit coupling, and Thomas precession".

10991: 9779:. In RQM, the position and momentum operators are inserted directly where they appear in the orbital 9758: 9746: 9397: 4706: 4678: 4191: 3976: 3925: 3172: 1360: 516: 477: 430: 405: 328: 188: 14933:

Fronsdal, C.; Lundberg, L.E. (1997). "Relativistic Quantum Mechanics of Two Interacting Particles".

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is discovered: relativistic corrections to the spin of elementary particles with application in the

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Relativistic Quantum Physics: From Advanced Quantum Mechanics to Introductory Quantum Field Theory

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Jaroszewicz, T.; Kurzepa, P.S (1992). "Geometry of spacetime propagation of spinning particles".

12773: 12767: 12413: 12407: 11374: 11369: 11047: 10916: 3153:{\displaystyle {\hat {H}}={\hat {H}}(\mathbf {r} ,t,{\hat {\mathbf {p} }},{\hat {\mathbf {S} }})} 2249: 1238: 1068: 902: 769: 487: 395: 16603: 16151: 14791:

Ord, G.N. (1983). "Fractal space-time: a geometric analogue of relativistic quantum mechanics".

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measures the charge on the electron and finds experimental evidence of its quantization, in the

8993:

with the + subscript indicating the particle and − subscript for the antiparticle. However, for

7085:{\displaystyle \left=2i\varepsilon _{abc}\sigma _{c}\,,\quad \left_{+}=2\delta _{ab}\sigma _{0}} 1134: 445: 343: 109: 16514: 16289: 16269: 16194: 16062: 15837: 14727: 14271: 14249: 13564: 13531: 13368: 12138: 11137: 10365: 9935:

which are six components altogether: three are the non-relativistic 3-orbital angular momenta;

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tensor defined from the four-dimensional position and momentum of the particle, equivalently a

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Lamb, W.E. Jr. & Retherford, R.C. (1951). "Fine Structure of the Hydrogen Atom. Part II".

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measures of the anomalous magnetic electron moment, confirming one of QED's great predictions.

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For some RWEs, a Lagrangian can be found by inspection. For example, the Dirac Lagrangian is:

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Lamb, W.E. Jr. & Retherford, R.C. (1950). "Fine Structure of the Hydrogen Atom. Part I".

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Again this notation is not necessarily standard, the more advanced literature usually writes

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of atoms and rotation of macroscopic objects. In 1939 Wigner derived the Thomas precession.

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successfully diffract electrons, providing experimental evidence of wave-particle duality.

11011: 10999: 10959: 7292: 3049: 2272: 1882: 1203: 1082:(not necessarily in the literature), and where space and time components can be collected, 1045: 1036: 997: 560: 472: 198: 155: 9005:

and the other for the antiparticle in the opposite helicity state corresponding to −

7754:

of various spins. In 1936, Dirac extended his equation to all fermions, three years later

5928:{\displaystyle \psi ={\begin{pmatrix}\psi _{\uparrow }\\\psi _{\downarrow }\end{pmatrix}}} 2728:

is finite and zero elsewhere, then at any later time the equation predicts delocalization

1847: 804: 8: 16555: 16524: 16469: 16449: 16357: 16314: 16169: 16095: 16022: 16012: 15924: 14917: 14870: 13909:

Lamb, W.E. Jr. & Retherford, R.C. (1952). "Fine Structure of the Hydrogen Atom. IV".

13658: 13432: 13207: 12707: 12602: 12358: 12304: 12250: 11170: 11122: 11051: 9193: 9169: 8263: 7719:{\displaystyle {\hat {h}}={\hat {\mathbf {S} }}\cdot {\frac {c{\hat {\mathbf {p} }}}{E}}} 7308: 6602: 4905: 4551: 4001: 2697: 2691: 1878: 1833:{\displaystyle {\frac {d}{dt}}A={\frac {1}{i\hbar }}+{\frac {\partial }{\partial t}}A\,,} 1259: 1001: 966: 962: 938: 930: 659: 467: 385: 213: 193: 145: 15181: 15076: 15039: 15002: 14956: 14909: 14862: 14833: 14804: 14767: 14723: 14677: 14388: 13958: 13922: 13886: 13859: 13831: 13794: 13757: 13710: 13608: 13397: 13271: 13119:

Relativistic Quantum Mechanics: With Applications in Condensed Matter and Atomic Physics

13086: 13020: 12983: 12955: 12893: 12846: 12346: 12292: 12238: 11942:

Relativistic Quantum Mechanics: With Applications in Condensed Matter and Atomic Physics

11129: 5985:

In RQM, the Dirac equation can also incorporate minimal coupling, rewritten from above;

4668:

is a simple way to include the electromagnetic interaction. For one charged particle of

16615: 16484: 16382: 16090: 16037: 15929: 15051: 15014: 14988: 14942: 14921: 14895: 14874: 14578: 14538: 14195: 13409: 13383: 13307: 13098: 13072: 13043: 12975: 12967: 12912: 12877: 12858: 12832: 12383: 12111: 12093: 12056: 11448: 11317: 9402: 7108: 6900:

The Pauli and gamma matrices were introduced here, in theoretical physics, rather than

6594: 4653: 3308: 2935: 2004: 1961: 1957: 1711: 1703: 1442: 1420: 1372: 1079: 986: 978: 970: 380: 305: 238: 150: 14973: 11972:. Springer Series in Solid-State Sciences Series. Vol. 134. Springer. p. 6. 6924: 6589:

to the appropriate electrostatic potential, additional relativistic terms include the

16625: 16534: 16504: 16432: 16395: 16390: 16372: 16337: 16327: 16042: 16007: 15990: 15893: 15832: 15730: 15709: 15688: 15667: 15646: 15625: 15604: 15583: 15562: 15541: 15520: 15499: 15478: 15457: 15429: 15408: 15387: 15366: 15345: 15324: 15303: 15296: 15277: 15270: 15251: 15232: 15213: 15185: 15156: 15135: 15114: 15055: 15047: 15018: 14925: 14878: 14812: 14779: 14741: 14634: 14613: 14589: 14549: 14522: 14501: 14480: 14459: 14437: 14413: 14392: 14363: 14342: 14321: 14300: 14279: 14257: 14235: 14214: 14179: 14158: 14139: 14120: 14101: 14077: 14056: 14035: 14003: 13934: 13898: 13664: 13637: 13568: 13557: 13535: 13524: 13502: 13474: 13443: 13413: 13344: 13317: 13295: 13275: 13240: 13213: 13186: 13180: 13156: 13145: 13123: 13028: 12979: 12917: 12862: 12803: 12777: 12743: 12718: 12685: 12660: 12633: 12608: 12578: 12553: 12528: 12503: 12475: 12448: 12417: 12194: 12169: 12142: 12131: 12115: 12064: 12036: 12009: 11973: 11946: 11919: 11892: 11852: 11527:

etc., but this would clutter expressions with unnecessary subscripts. The subscripts

11406: 11386: 11259: 11203: 11066: 10967: 10332: 7314: 7105: 4158: 1945: 1851: 1585: 1103: 1087: 926: 906: 814: 789: 729: 724: 624: 590: 570: 168: 27: 14231:

Perspectives on Quantum Reality: Non-Relativistic, Relativistic, and Field-Theoretic

13102: 11998: 11881: 11812:

but here we show informally the correspondence of energy, helicity, and spin states.

11206:

was published in a paper on the EPR paradox, showing that QM cannot be derived from

7275:

was found to describe two or more electromagnetically interacting massive spin

899: 16352: 16347: 16204: 16100: 15846: 15080: 15043: 15006: 14960: 14913: 14866: 14837: 14808: 14771: 14737: 14681: 14173: 13993: 13962: 13926: 13890: 13863: 13835: 13798: 13752: 13714: 13612: 13401: 13090: 13024: 12959: 12907: 12897: 12850: 12350: 12296: 12242: 12103: 11453: 11411: 11328: 11174:

two observers (one observer has to send a signal to the other, which cannot exceed

11003: 10987: 10932: 9788: 7120: 7116: 6901: 4987:{\displaystyle E-e\phi \approx mc^{2}\,,\quad \mathbf {p} \approx m\mathbf {v} \,,} 4665: 3520: 2303: 2282: 1855: 1380: 1352: 1024: 934: 910: 819: 809: 799: 699: 679: 664: 634: 502: 390: 15407:. Springer series on atomic, optical, and plasma physics. Vol. 63. Springer. 12823:

Esposito, S. (2011). "Searching for an equation: Dirac, Majorana and the others".

9596:{\displaystyle {\mathcal {L}}={\overline {\psi }}(\gamma ^{\mu }P_{\mu }-mc)\psi } 1978:

corresponding to events, and the energy and 3-momentum combine naturally into the

1371:, rederived by Pauli a year later. This is the explanation for a diverse range of 16582: 16509: 16489: 16459: 16422: 16417: 16322: 16146: 15423: 15291: 15265: 14974:"Spin–orbital motion and Thomas precession in the classical and quantum theories" 13781: 13697: 13303: 13176: 11416: 11391: 11321: 11133: 10955: 10928: 10552:{\displaystyle \mathbf {B} '={\frac {\mathbf {E} \times \mathbf {v} }{c^{2}}}\,,} 9173: 6885: 6300: 4669: 4537: 4533: 3647: 3026:{\displaystyle {\hat {H}}_{B}=-\mathbf {B} \cdot {\hat {\boldsymbol {\mu }}}_{S}} 2042: 1949: 1581: 1356: 1241: 1075: 1054: 844: 714: 694: 440: 280: 15687:. Theoretical and Computational Chemistry. Vol. 11. Elsevier. p. 208. 13590:"Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?" 11550:

This spinor notation is not necessarily standard; the literature usually writes

10323:. For more on relativistic spin, see (for example) Troshin & Tyurin (1994). 16560: 16529: 16519: 16141: 16131: 15965: 15724: 14695: 14091: 13094: 11538: 11468: 11458: 11433: 11364: 11336: 11187: 11162: 11114: 11103: 11081: 11070: 11058: 11002:, which are consistent with special relativity and quantum mechanics. By 1927, 10983: 10971: 10924: 9984: 7272: 6233: 5856: 5617: 5613: 4918: 4529: 4021: 3949: 3877: 2953: 2286: 2225: 1384: 1344: 1212: 1130: 1095: 1032: 1020: 942: 918: 779: 739: 719: 689: 669: 619: 585: 435: 425: 218: 15855: 15131:

Gauge Theories in Particle Physics: From Relativistic Quantum Mechanics to QED

12854: 12246: 11313:

establishes the field of QED, also coining the term "quantum electrodynamics".

10915:

The events which led to and established RQM, and the continuation beyond into

9131: 16658: 16479: 16332: 16223: 16057: 16027: 15980: 15104: 14841: 14775: 14206: 14007: 13299: 12354: 12300: 11396: 11141: 11040: 11036: 11024: 11007: 10975: 10950: 7774: 7763: 7751: 6877: 4921:. In the following, the non-relativistic limit refers to the limiting cases: 4901: 4505: 4177: 3881: 2945: 2790: 2207: 2197: 2064: 2016: 1979: 1392: 1099: 839: 834: 764: 734: 704: 575: 521: 248: 223: 101:{\displaystyle i\hbar {\frac {d}{dt}}|\Psi \rangle ={\hat {H}}|\Psi \rangle } 16:

Quantum mechanics taking into account particles near or at the speed of light

14964: 13930: 13719: 13692: 12107: 12004:. Manchester Physics Series (3rd ed.). John Wiley & Sons. pp. 11254:

levels of hydrogen, due to the interaction between the electron and vacuum.

9138:

in 1913, prior to the spinors revealed in the RWEs following the year 1927.

16362: 15975: 15970: 14686: 14659: 14573: 13998: 13981: 13938: 13902: 13894: 13867: 13839: 13803: 13776: 13617: 13494: 12921: 11887:. Manchester Physics Series (3rd ed.). John Wiley & Sons. p. 11473: 11158: 11099: 11077: 10936: 9762: 8293: 7770: 7755: 7261: 6562: 6557: 5866: 3953: 2676:

before the momentum operator, raised to a power in each term, could act on

2673: 2253: 1368: 1348: 1216: 1192:{\displaystyle i\hbar {\frac {\partial }{\partial t}}\psi ={\hat {H}}\psi } 1058: 829: 824: 759: 744: 709: 203: 14783: 13966: 13944: 12902: 6519:

corresponds to a similar particle with the same mass and spin states, but

3166: 16400: 15752: 13772: 11255: 10940: 9271:{\displaystyle {\hat {\mathbf {v} }}={\frac {1}{m}}{\hat {\mathbf {p} }}} 6601:. In ordinary QM these terms have to be put in by hand and treated using 6598: 3524: 3040:

as a requirement of enforcing the relativistic energy-momentum relation.

1570:{\displaystyle E^{2}=c^{2}\mathbf {p} \cdot \mathbf {p} +(mc^{2})^{2}\,.} 1446: 794: 749: 684: 639: 14993: 14947: 14900: 13388: 15173:

Relativistic quantum mechanics and introduction to quantum field theory

14655: 14630:

Group Theory in Quantum Mechanics: An Introduction to Its Present Usage

14451: 14025: 12971: 11310: 11297: 11219: 11183: 11136:

by atomic nuclei bound in a solid, useful for accurate measurements of

10944: 10193: 7766:

theory, it becomes apparent to introduce spin in the form of matrices.

6920: 6905: 6551: 6547: 1064: 993: 784: 754: 674: 649: 644: 629: 15010: 13405: 6605:. The positive energies do account accurately for the fine structure. 16254: 15950: 15199: 14754:

Fanchi, J.R. (1986). "Parametrizing relativistic quantum mechanics".

12497: 11293: 10722:, but this disagrees with experimental atomic spectra by a factor of 6916: 6554:. This was the first interpretation and prediction of a particle and 4659: 3937: 3880:. The other factor is also the Dirac equation, but for a particle of 3044: 1411: 1388: 1321:. This is an additional discrete variable the wavefunction requires; 1113: 946: 275: 15862: 15084: 14849:

Mann, R.B.; Ralph, T.C. (2012). "Relativistic quantum information".

14171: 12963: 12061:

QED and the Men Who Made It: Dyson, Feynman, Schwinger, and Tomonaga

3905:, and comparison with the KG equation determines the constraints on 14320:. Texts and Monographs in Physics Series (2nd ed.). Springer. 13873:

Lamb, W.E. Jr. (1952). "Fine Structure of the Hydrogen Atom. III".

12098: 11118: 11043:

into several components in the presence of a static magnetic field.

11028: 9784: 6881: 6566: 5601: 3163:

is a function of space, time, and the momentum and spin operators.

1433: 1406:

A fundamental prediction of special relativity is the relativistic

1091: 1086:

is shown also (frequently used in the literature), in addition the

1004: 914: 654: 15097: 14359:

Theoretical Chemistry and Physics of Heavy and Superheavy Elements

13466:

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

13077: 13048: 12837: 12388: 11262:

experimentally measure stimulated radio-frequency transitions the

11224:

In 1947, the Lamb shift was discovered: a small difference in the

11194:, questioning the status of electromagnetic potentials in QM. The 6608:

Within RQM, for massless particles the Dirac equation reduces to:

5005:

In RQM, the KG equation admits the minimal coupling prescription;

13587: 12440:

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

11478: 11152: 11110: 10963: 5303:

representations. Solutions that do not belong to the irreducible

2266: 1735:

contain the time dependence, governed by the equation of motion:

1275: 1016: 887: 13982:"The quantum theory of the emission and absorption of radiation" 11655:{\displaystyle \psi ={\begin{pmatrix}\chi \\\eta \end{pmatrix}}} 11606:{\displaystyle \psi ={\begin{pmatrix}u^{1}\\u^{2}\end{pmatrix}}} 10326: 7119:, and have a connection to the components of the flat spacetime 1696:

consistent with the energy–momentum relation, and is solved for

15404:

Relativistic Many-body Theory: A New Field-theoretical Approach

12937:"On Unitary Representations Of The Inhomogeneous Lorentz Group" 10995: 10736:, and this additional precession of the electron is called the 9135: 1424: 1396: 1071:, both of which were successful in combining the two theories. 14971: 13908: 13845: 13817: 13693:"Significance of electromagnetic potentials in quantum theory" 12035:. Vol. 2. North-Holland Publishing Company. p. 875. 10562:

so the non-relativistic spin interaction Hamiltonian becomes:

9364:{\displaystyle {\hat {\mathbf {v} }}={\frac {i}{\hbar }}\left} 5335:

The electromagnetic field is treated classically according to

4550:. The density and current of electric charge always satisfy a 15443: 12878:"Group theoretical discussion of relativistic wave equations" 12502:. Shaum's outlines (2nd ed.). McGraw–Hill. p. 181. 12137:. Manchester Physics Series. John Wiley & Sons. pp. 10979: 6913: 6909: 5347: 1960:. The position and time coordinates combine naturally into a 1714:

is another formulation of QM, in which case the wavefunction

1400: 1279: 15198: 15127: 13501:. Oxford University Press. pp. 168–169, 176, 263, 228. 13336: 12737: 12627: 11427: 11296:

radiation. An explanation of the Lamb shift is presented by

11144:, and in the analysis of nuclear electromagnetic moments in 7734:

The Dirac equation can only describe particles of spin

14155:

Relativistic Quantum Mechanics (Pure & Applied Physics)

13777:"Fine Structure of the Hydrogen Atom by a Microwave Method" 13006: 12063:. Princeton, N.J: Princeton University Press. p. 573. 11080:

is discovered: splitting of spectral lines due to a static

6573:

for how). When applied a one-electron atom or ion, setting

3933: 992:

Key features common to all RQMs include: the prediction of

14932: 14748: 12596: 12594: 12467: 11300:. Papers on the effect were published in the early 1950s. 10951:

Relativistic description of particles in quantum phenomena

2245:

may occur more than once depending on the representation.

1481:{\displaystyle p={\sqrt {\mathbf {p} \cdot \mathbf {p} }}} 1074:

In this article, the equations are written in familiar 3D

15449: 13259: 11303: 5332:, it can only be interpreted as a system with zero spin. 2185:

is a finite-dimensional representation, in other words a

15575: 15248:

Relativistic Quantum Fields (Pure & Applied Physics)

15245: 14819: 14710:

Antoine, J.P. (2004). "Relativistic Quantum Mechanics".

13369:"The Thomas precession factor in spin–orbit interaction" 13239:(2nd ed.). Cambridge University Press. p. 62. 12875: 12128: 9752: 6880:, a considerable simplification applicable for massless 5328:, see below. Thus if a system satisfies the KG equation 1049:(QFT), in which elementary particles are interpreted as 15684:

Relativistic Electronic Structure Theory - Fundamentals

12591: 11991: 11989: 11069:, showed that atoms possess an internal structure: the 9217:, and substituting quantum operators in the usual way: 7750:. Beyond the Dirac equation, RWEs have been applied to 4664:

Including interactions in RWEs is generally difficult.

3167:

The Klein–Gordon and Dirac equations for free particles

14560:

hyperfine structure in relativistic quantum mechanics.

13462: 12436: 11736: 11707: 11631: 11568: 11132:: resonant and recoil-free emission and absorption of 9047: 8804: 8356: 8041: 7819: 6373: 6330: 5890: 2682:. As a result of the power series, the space and time 14848: 14544:(2nd ed.). University of Chicago Press. p. 11695: 11619: 11556: 11125:, confirming the theoretical prediction of positrons. 11091:: experimental evidence of spin and its quantization. 10749: 10571: 10502: 10383: 10205: 10006: 9800: 9615: 9534: 9419: 9294: 9226: 9018: 8327: 7790: 7657: 7584: 7332: 7137: 6939: 6617: 6318: 5994: 5938:

where the subscripts ↑ and ↓ refer to the "spin up" (

5878: 5633: 5375: 5014: 4930: 4732: 4563: 4546:

is not a wavefunction at all, but reinterpreted as a

4203: 4033: 3720: 3659: 3546: 3320: 3184: 3071: 2970: 2802: 2444: 2333: 2076: 1744: 1600: 1497: 1454: 1146: 1039:

to achieve agreement with experimental observations.

39: 15603:. Graduate texts in physics. Springer. p. 149. 14493: 14299:. Vol. 2 (5th ed.). Taylor & Francis. 13690: 12656:

Relativistic quantum mechanics of leptons and fields

12498:

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

11986: 2701:: if the particle is initially localized at a point 14600:

magnetic moments in relativistic quantum mechanics.

13294: 11874: 11872: 11870: 11868: 9385: 9176:

would contribute also). For example, the spin

7648:For massless particles the helicity simplifies to: 4525:may be freely chosen, the density can be negative. 2435:momentum leading to difficulties. Naively setting: 15540:(3 ed.). Butterworth-Heinemann. p. 308. 15425:Relativistic Quantum theory of atoms and molecules 15295: 15269: 14577: 14566: 14537: 14194: 11911: 11801: 11654: 11605: 10886: 10703: 10551: 10472: 10364:, will in its own frame of reference experience a 10308: 10181: 9924: 9727: 9595: 9514: 9406:), then generate the differential equation by the 9363: 9270: 9118: 8982: 8251: 7718: 7637: 7491: 7249: 7084: 6865: 6450: 6217: 5927: 5844: 5569: 5346:The KG equation is applicable to spinless charged 5278: 4986: 4880: 4660:Spin and electromagnetically interacting particles 4641: 4489: 4146: 3865: 3700: 3635: 3474: 3292: 3152: 3025: 2917: 2661: 2419: 2167: 1832: 1677: 1569: 1480: 1191: 1114:Combining special relativity and quantum mechanics 1042:The most successful (and most widely used) RQM is 100: 15680: 15621:Handbook of Nanophysics: Functional Nanomaterials 15428:. Atomic, optical, and plasma physics. Springer. 15379: 15229:Elementary particle physics, Quantum Field Theory 15209:Relativistic quantum mechanics and quantum fields 14405: 14334: 14270: 14248: 14172:Feynman, R.P.; Leighton, R.B.; Sands, M. (1965). 12742:(1st ed.). Prentice Hall. pp. 632–635. 10982:due to first order relativistic corrections. The 10344:classical electromagnetism and special relativity 9745:is the subject of QFT rather than RQM: Feynman's 7268:, but is not the subject of special relativity). 2320:, with the corresponding quantum operator in the 16656: 15833:"Quantum Mechanics Foundations and Applications" 15799: 15764:"Relativistic Quantum Mechanics (Lecture Notes)" 14409:Relativistic Quantum Measurement and Decoherence 13469:(2nd ed.). John Wiley & Sons. pp. 11865: 11678:, this informal identification is commonly made. 11533:labeling spin values are not to be confused for 7638:{\displaystyle {\sqrt {E^{2}-(m_{0}c^{2})^{2}}}} 965:, more specifically quantizing the equations of 15761: 15754:Relativistic Quantum Mechanics, an Introduction 15498:(4th ed.). Cengage Learning. p. 105. 15358: 15320:Relativistic Quantum Mechanics and Field Theory 15169: 15098:Relativistic quantum mechanics and field theory 12493: 12491: 12443:(2nd ed.). John Wiley & Sons. p. 11402:Dirac equation in the algebra of physical space 11353: 3701:{\displaystyle \alpha _{i}^{2}=\beta ^{2}=I\,,} 1858:are equivalent, later furthered by Dirac using 1211:corresponding to the system. The solution is a 15836:. Alabama, USA: Taylor & Francis. p. 14429: 13872: 13771: 13588:Einstein, A.; Podolsky, B.; Rosen, N. (1935). 12401: 12399: 12165:Relativistic Quantum Mechanics. Wave Equations 11153:Quantum non-locality and relativistic locality 9396:. An equivalent alternative is to determine a 2267:Non-relativistic and relativistic Hamiltonians 981:. Although the earlier formulations, like the 15878: 15847:"The Particle Electron and Thomas Precession" 15829: 15666:(2nd ed.). Manchester University Press. 15617: 15554: 15533: 15516:Introduction to the Electron Theory of Metals 15341:Advanced Quantum Mechanics: A Practical Guide 15226: 15148: 14709: 14227: 14048: 13426: 13340:General relativity and John Archibald Wheeler 12381: 11969:Magnetism in the Solid State: An Introduction 11844: 10327:Thomas precession and spin–orbit interactions 3519:are not simply numbers or vectors, but 4 × 4 959:mathematical formulation of quantum mechanics 867: 15512: 15470: 15400: 15337: 15062: 14753: 14479:(3rd ed.). Courier Dover Publications. 14355: 14192: 14152: 12822: 12765: 12761: 12759: 12652: 12632:(1st ed.). Prentice Hall. p. 634. 12488: 12405: 12321: 12267: 12213: 3062:, so in general a relativistic Hamiltonian: 95: 69: 15750: 15638: 15519:. Cambridge University Press. p. 387. 15491: 15421: 15302:. Vol. 3. Cambridge University Press. 15276:. Vol. 2. Cambridge University Press. 15103: 15025: 14700:"Exclusion principle and quantum mechanics" 14535: 14514: 14472: 14450: 14376: 14313: 14292: 14069: 13656: 13554: 13521: 13366: 13232: 13205: 13185:. Vol. 1. Cambridge University Press. 13142: 13122:. Cambridge University Press. p. 206. 13115: 12797: 12704: 12600: 12547: 12396: 12188: 12161: 12028: 11938: 8310:values in each case), altogether forming a 7758:and Pauli rederived the same equation. The 7562:, which is the helicity (for the spin 3948:In non-relativistic quantum mechanics, the 3932:from positive to negative energy levels in 1580:These equations are used together with the 15885: 15871: 15659: 15444:Quantum theory and applications in general 15362:General Principles of Quantum Field Theory 15113:. Cambridge University Press. p. 10. 14605: 14580:The theory of groups and quantum mechanics 14114: 13362: 13360: 12934: 11995: 11878: 6470:corresponds to a particle with 4-momentum 4677:in an electromagnetic field, given by the 2952:. For a particle in an externally applied 2258:Representation theory of the Lorentz group 1122:to be consistent with special relativity. 874: 860: 15722: 15701: 15316: 15134:. Vol. 1 (3rd ed.). CRC Press. 14992: 14972:Bordovitsyn, V.A.; Myagkii, A.N. (2004). 14946: 14899: 14885: 14790: 14731: 14685: 14626: 13997: 13802: 13756: 13718: 13631: 13616: 13387: 13076: 13047: 12911: 12901: 12836: 12756: 12679: 12572: 12522: 12471:Quantum Mechanics Non-Relativistic Theory 12387: 12097: 11428:Particle physics and quantum field theory 11333:anomalous magnetic moment of the electron 10880: 10697: 10545: 10466: 9918: 9721: 9511: 8933: 8871: 8820: 8725: 8672: 8615: 8571: 8524: 8471: 8414: 8370: 8284:components for the particle, and another 8262:where the expression on the right is the 7518:is the total energy of the particle, and 7510:the spin operator for a particle of spin 7298: 7243: 7008: 6859: 6809: 6686: 6228:and was the first equation to accurately 4980: 4959: 4769: 4635: 4313: 3943: 3694: 3629: 3468: 3286: 2911: 2864: 2655: 1826: 1671: 1637: 1563: 1255:, describing the behavior of the system. 15596: 15290: 15264: 13736:"On the Einstein Podolsky Rosen Paradox" 13212:. Vintage Books. pp. 437, 566–569. 13175: 12207: 12055: 11965: 9285:value. In RQM, the Dirac theory, it is: 977:(RQM) is quantum mechanics applied with 15844: 15386:(2nd ed.). John Wiley & Sons. 15365:(2nd ed.). Springer. p. 272. 15128:Aitchison, I.J.R.; Hey, A.J.G. (2002). 14500:(2nd ed.). Springer. p. 245. 14133: 14119:(3rd ed.). John Wiley & Sons. 14097:General Principles of Quantum Mechanics 13357: 13337:Ciufolini, I.; Matzner, R.R.A. (2010). 13263:Spin phenomena in particle interactions 12738:Bransden, B.H.; Joachain, C.J. (1983). 12628:Bransden, B.H.; Joachain, C.J. (1983). 12083: 11380: 10865: 10785: 10682: 10607: 9408:field-theoretic Euler–Lagrange equation 9196:and (for example) Cédric Lorcé (2009). 6717: 6644: 6060: 5855:by means of the 2 × 2 5794: 5706: 4086: 3823: 3745: 3414: 3345: 3008: 2875: 2807: 16657: 15456:. PHI Learning Pvt. Ltd. p. 395. 14584:. Courier Dover Publications. p. 14406:Breuer, H.P.; Petruccione, F. (2000). 14138:. Vol. 1. John Wiley & Sons. 13493: 12474:. Vol. 3. Elsevier. p. 455. 12412:(2nd ed.). McGraw Hill. pp. 12168:(3rd ed.). Springer. p. 70. 11662:etc., but in the context of spin 11304:Development of quantum electrodynamics 6232:spin, a consequence of the 4 × 4 2000:of a dynamic particle, as measured in 1133:of any quantum system is given by the 15892: 15866: 14694: 14654: 14205: 14090: 14024: 13979: 13062: 13041: 12772:(2nd ed.). McGraw Hill. p. 12468:Landau, L.D.; Lifshitz, E.M. (1981). 11996:Martin, B.R.; Shaw, G. (2008-12-03). 11879:Martin, B.R.; Shaw, G. (2008-12-03). 11821:Some authors, including Penrose, use 11484:Modern searches for Lorentz violation 10962:; a particle description of light as 10346:, an electron moving with a velocity 9753:Relativistic quantum angular momentum 9374:which must have eigenvalues between ± 7769:The wavefunctions are multicomponent 6919:, because they satisfy the important 5307:representation will have two or more 15537:Radiochemistry and nuclear chemistry 15450:Aruldhas, G.; Rajagopal, P. (2005). 15152:Introduction to Elementary Particles 14572: 13733: 13260:Troshin, S.M.; Tyurin, N.E. (1994). 13151:. Demystified. McGraw-Hill. p. 12769:McGraw Hill Encyclopaedia of Physics 12713:. Demystified. McGraw Hill. p. 12409:McGraw Hill Encyclopaedia of Physics 11348: 10907:is predicted by the Dirac equation. 9199: 1867:representations of the Lorentz group 1379:of atoms, nuclei (and therefore all 969:by replacing dynamical variables by 15576:Nolting, W.; Ramakanth, A. (2008). 15246:Bjorken, J.D.; Drell, S.D. (1965). 14153:Bjorken, J.D.; Drell, S.D. (1964). 13758:10.1103/PhysicsPhysiqueFizika.1.195 12876:Bargmann, V.; Wigner, E.P. (1948). 12607:. Vintage Books. pp. 620–621. 12129:Forshaw, J.R.; Smith, A.G. (2009). 11848:Introduction to High Energy Physics 11109:1932 Experimental discovery of the 6546:, that is, negative energy states, 4508:. Since the initial values of both 1962:four-dimensional spacetime position 1391:), to the quark configurations and 1312: − 1, ... , − 909:(QM). This theory is applicable to 13: 15853: 15474:Electronic properties of materials 15092: 14665:Proceedings of the Royal Society A 14648: 14213:(2nd ed.). World Scientific. 13563:. John Wiley & Sons. pp. 13530:. John Wiley & Sons. pp. 10897:In the case of RQM, the factor of 9680: 9660: 9618: 9537: 9496: 9489: 9484: 9456: 9449: 9442: 9437: 9421: 7545:operator on the 3-momentum (times 6904:itself. They have applications to 6012: 6008: 5651: 5647: 5393: 5389: 4610: 4587: 4575: 4567: 4465: 4446: 4376: 4364: 4299: 4284: 4266: 4258: 2566: 2562: 2133: 2113: 2045:, all one-particle quantum states 1814: 1810: 1668: 1628: 1624: 1258:Every particle has a non-negative 1159: 1155: 955:Non-relativistic quantum mechanics 406:Sum-over-histories (path integral) 92: 66: 22:Part of a series of articles about 14: 16701: 15744: 15561:. World Scientific. p. 443. 15338:Nazarov, Y.V.; Danon, J. (2013). 14380:Advanced visual quantum mechanics 14034:(4th ed.). Clarendon Press. 14018: 13463:Resnick, R.; Eisberg, R. (1985). 12437:Resnick, R.; Eisberg, R. (1985). 12077: 11213: 11210:if locality is to be maintained. 11084:(compare with the Zeeman effect). 9382:for more theoretical background. 9317: 7128:in the anticommutation relation: 6510:, as before). The other 2-spinor 6003: 5642: 5384: 4418: 4335: 4216: 2845: 2557: 1872: 1775: 1665: 1619: 1375:behavior and phenomena: from the 1343:Historically, in the early 1920s 1150: 43: 16639: 16638: 15856:"Spin and the Thomas precession" 15802:"Relativistic Quantum Mechanics" 15782:"Relativistic Quantum Mechanics" 15600:Quantum Physics in the Nanoworld 14660:"Relativistic Quantum Mechanics" 13775:; Retherford, Robert C. (1947). 12550:"Relativistic quantum mechanics" 12191:"Relativistic quantum mechanics" 10831: 10823: 10812: 10770: 10653: 10645: 10634: 10592: 10528: 10520: 10505: 10409: 10401: 10386: 10299: 10291: 10277: 10172: 10164: 10133: 10125: 10117: 9914: 9906: 9898: 9606:and Klein–Gordon Lagrangian is: 9386:Relativistic quantum Lagrangians 9346: 9299: 9281:which has eigenvalues that take 9258: 9231: 9095: 9064: 9026: 8952: 8893: 8836: 8779: 8744: 8697: 8637: 8587: 8543: 8496: 8436: 8386: 8335: 8221: 8174: 8116: 8066: 8013: 7978: 7938: 7885: 7842: 7798: 7700: 7677: 7428: 7405: 7386: 7370: 7352: 6728: 6655: 6092: 6075: 5816: 5805: 5728: 5717: 5541: 5524: 5463: 5446: 5093: 5076: 4976: 4965: 4812: 4795: 4778: 4594: 4319: 4058: 3834: 3756: 3425: 3356: 3241: 3224: 3137: 3120: 3103: 2997: 2854: 2614: 2597: 2508: 2491: 2404: 2373: 2356: 2149: 2091: 1646: 1530: 1522: 1472: 1464: 929:. The theory has application in 14567:Group theory in quantum physics 14497:Quantum Mechanics: Fundamentals 14494:Gottfried, K.; Yan, T. (2003). 14476:The Quantum Theory of Radiation 14433:A Course in Theoretical Physics 14424:Relativistic quantum mechanics. 14356:Kaldor, U.; Wilson, S. (2003). 14341:. Alpha Science International. 14178:. Vol. 3. Addison-Wesley. 14031:Principles of Quantum Mechanics 13973: 13811: 13765: 13727: 13691:Aharonov, Y.; Bohm, D. (1959). 13684: 13650: 13636:. Addison Wesley. p. 192. 13625: 13581: 13548: 13515: 13487: 13456: 13438:(3rd ed.). Wiley. p. 13420: 13330: 13288: 13253: 13226: 13199: 13169: 13136: 13109: 13056: 13035: 13000: 12928: 12869: 12816: 12791: 12731: 12698: 12684:. Addison Wesley. p. 423. 12673: 12646: 12621: 12577:. Addison Wesley. p. 415. 12566: 12541: 12527:. Addison Wesley. p. 425. 12516: 12461: 12430: 12375: 12182: 12155: 12122: 11815: 11681: 11544: 11509:Other common notations include 11503: 11444:Relativistic Heavy Ion Collider 10275: 10271: 10115: 10111: 9896: 9892: 9380:Foldy–Wouthuysen transformation 9172:are arbitrary. (Theoretically, 9134:found the most general form of 8769: 8765: 8003: 7999: 7729: 7012: 6813: 6763: 6759: 6690: 6128: 6124: 5757: 5753: 5606:Non relativistically, spin was 5484: 5480: 5146: 5142: 4963: 4820: 4816: 4773: 4608: 4604: 4398: 4394: 4317: 4097: 4093: 4056: 3803: 3799: 3582: 2868: 2553: 2549: 2262:symmetries in quantum mechanics 2210:containing components with the 1641: 16588:Relativistic quantum mechanics 15344:. Cambridge University Press. 14918:10.1088/0264-9381/29/22/220301 14871:10.1088/0264-9381/29/22/220301 14633:. Courier Dover Publications. 14456:Intermediate Quantum Mechanics 14338:Relativistic quantum mechanics 14317:Relativistic Quantum Mechanics 14055:. Courier Dover Publications. 13499:Quanta: A handbook of concepts 12740:Physics of Atoms and Molecules 12630:Physics of Atoms and Molecules 12049: 12022: 11959: 11945:. Cambridge University Press. 11932: 11915:Relativistic Quantum Chemistry 11905: 11851:. Cambridge University Press. 11838: 11360:Relativistic quantum chemistry 11208:local hidden-variable theories 11017: 10868: 10788: 10756: 10685: 10610: 10578: 10483:In the non-relativistic limit 10303: 10287: 10272: 10176: 10160: 10112: 10047: 10031: 9893: 9887: 9871: 9587: 9555: 9468: 9452: 9350: 9333: 9303: 9262: 9235: 9105: 9091: 9074: 9060: 9036: 9022: 8962: 8948: 8903: 8889: 8846: 8832: 8789: 8775: 8766: 8754: 8740: 8707: 8693: 8647: 8633: 8597: 8583: 8553: 8539: 8506: 8492: 8446: 8432: 8396: 8382: 8345: 8331: 8231: 8217: 8184: 8170: 8126: 8112: 8076: 8062: 8023: 8009: 8000: 7988: 7974: 7948: 7934: 7895: 7881: 7852: 7838: 7808: 7794: 7773:, which can be represented as 7704: 7681: 7664: 7624: 7600: 7477: 7453: 7432: 7409: 7391: 7381: 7374: 7356: 7339: 6831: 6781: 6760: 6732: 6703: 6659: 6630: 6435: 6418: 6401: 6384: 6182: 6154: 6144: 6125: 6096: 6079: 6068: 5912: 5898: 5823: 5820: 5801: 5790: 5764: 5754: 5735: 5732: 5713: 5702: 5545: 5528: 5517: 5491: 5481: 5467: 5450: 5439: 5252: 5243: 5234: 5206: 5196: 5191: 5163: 5153: 5143: 5130: 5113: 5097: 5080: 5069: 5041: 5026: 5017: 4850: 4840: 4828: 4817: 4799: 4788: 4782: 4754: 4745: 4739: 4695:defined by the magnetic field 4605: 4484: 4432: 4395: 4094: 3838: 3810: 3800: 3760: 3732: 3429: 3401: 3360: 3332: 3274: 3257: 3245: 3228: 3192: 3147: 3141: 3124: 3099: 3093: 3078: 3011: 2978: 2858: 2810: 2644: 2627: 2618: 2601: 2550: 2538: 2521: 2512: 2495: 2466: 2451: 2414: 2400: 2377: 2360: 2340: 2162: 2159: 2145: 2129: 2116: 2110: 2104: 2101: 2087: 1802: 1796: 1781: 1650: 1607: 1554: 1537: 1180: 1127:postulate of quantum mechanics 1118:One approach is to modify the 975:Relativistic quantum mechanics 917:up to those comparable to the 892:relativistic quantum mechanics 556:Relativistic quantum mechanics 88: 81: 62: 1: 16566:Quantum statistical mechanics 16343:Quantum differential calculus 16265:Delayed-choice quantum eraser 16048:Symmetry in quantum mechanics 15624:. CRC Press. pp. 40–43. 14274:; Diu, B.; Laloë, F. (1977). 14252:; Diu, B.; Laloë, F. (1977). 14052:Lectures on Quantum Mechanics 13680:: The entangled quantum world 12274:spin. II. Massless Particles" 11912:Reiher, M.; Wolf, A. (2009). 11831: 11165:published a paper concerning 9781:relativistic angular momentum 9761:is formed from the classical 6884:. This time there is a 2 × 2 5580: 4157:where the dagger denotes the 2054:locally transform under some 1854:show that wave mechanics and 1694:partial differential equation 1299:-projection quantum numbers; 1088:Einstein summation convention 596:Quantum statistical mechanics 15495:Introduction to Spectroscopy 15380:Mandl, F.; Shaw, G. (2010). 15298:The Quantum Theory of Fields 15272:The Quantum Theory of Fields 14540:The Physics of Time Reversal 14201:(3rd ed.). McGraw-Hill. 13559:Introductory Nuclear Physics 13526:Introductory Nuclear Physics 13182:The Quantum Theory of Fields 13029:10.1016/0003-4916(92)90176-M 12882:Proc. Natl. Acad. Sci. U.S.A 12802:. Demystified. McGraw-Hill. 11497: 11354:Atomic physics and chemistry 9997:angular momentum tensor is: 9757:In non-relativistic QM, the 9550: 5620:in 1927 for particles in an 3979:, circa 1927. In RQM, while 3963:probability density function 1363:(1925) and the more general 7: 16368:Quantum stochastic calculus 16358:Quantum measurement problem 16280:Mach–Zehnder interferometer 15579:Quantum theory of magnetism 15558:Theory of nuclear reactions 14981:American Journal of Physics 14175:Feynman Lectures on Physics 13376:American Journal of Physics 11343: 10734:rotating frame of reference 10321:Pauli–Lubanski pseudovector 10192:where the star denotes the 7305:Helicity (particle physics) 4018:positive-definite functions 4011:probability current density 1846:Historically, around 1926, 1108:position and momentum space 566:Quantum information science 10: 16706: 15705:Ideas of Quantum Chemistry 15681:Schwerdtfeger, P. (2002). 15048:10.1088/0143-0807/34/3/L55 14813:10.1088/0305-4470/16/9/012 14742:10.1088/0305-4470/37/4/B01 14335:Parthasarathy, R. (2010). 14278:. Vol. 2. Wiley VCH. 14256:. Vol. 1. Wiley VCH. 14211:Advanced Quantum Mechanics 13980:Dirac, Paul (1927-03-01). 13095:10.1103/PhysRevD.79.113011 11217: 11169:of particles, questioning 10910: 10352:through an electric field 7506:is the momentum operator, 7312: 7302: 7111:. The gamma matrices form 6876:the first of which is the 5599: 3309:relativistically invariant 2270: 2247: 1690:relativistic wave equation 1591:, which are respectively: 1078:notation and use hats for 1023:, and quantum dynamics of 961:applied in the context of 16634: 16596: 16548: 16428:Quantum complexity theory 16406:Quantum cellular automata 16381: 16313: 16247: 16160: 16124: 16111:Path integral formulation 16078: 15943: 15900: 15800:Miller, David J. (2008). 15708:. Elsevier. p. 676. 15645:. Springer. p. 256. 15477:. Springer. p. 395. 15323:. John Wiley & Sons. 15155:. John Wiley & Sons. 14521:. Springer. p. 220. 14436:. John Wiley & Sons. 13471:57, 114–116, 125–126, 272 13434:Classical Electrodynamics 13343:. Springer. p. 329. 12855:10.1016/j.aop.2012.02.016 12247:10.1103/PhysRev.133.B1318 11918:. John Wiley & Sons. 11063:Geiger–Marsden experiment 10958:in 1905 explained of the 10358:but not a magnetic field 9759:angular momentum operator 9747:path integral formulation 8318:-component spinor field: 7760:Bargmann–Wigner equations 7260:(This can be extended to 5000: 4904:that has a corresponding 4707:electric scalar potential 4679:magnetic vector potential 4540:. Then, the wavefunction 3977:Copenhagen interpretation 3056:runs over the spin index 1395:(hence the properties of 1377:electronic configurations 1361:Pauli exclusion principle 16495:Quantum machine learning 16475:Quantum key distribution 16465:Quantum image processing 16455:Quantum error correction 16305:Wheeler's delayed choice 15762:Lukačević, Igor (2013). 15642:Solid-State Spectroscopy 15359:Bogolubov, N.N. (1989). 15170:Capri, Anton Z. (2002). 14842:10.1103/PhysRevD.26.1348 14776:10.1103/PhysRevA.34.1677 12659:. Springer. p. 54. 12355:10.1103/PhysRev.181.1893 12301:10.1103/PhysRev.134.B882 12193:. Springer. p. 34. 11439:Two-body Dirac equations 11089:Stern–Gerlach experiment 7526:antiparallel alignment. 6930:relations respectively: 4188:probability four-current 3936:would be forbidden. See 2948:, so they interact with 2007:, change according to a 1707:differential equation). 1408:energy–momentum relation 1249:of the particle at time 1031:. The key result is the 951:condensed matter physics 601:Quantum machine learning 354:Wheeler's delayed-choice 16411:Quantum finite automata 15793:University of Cambridge 14965:10.1103/PhysRevD.1.3247 14609:Group Theory in Physics 14454:; Jackiw, R.W. (1997). 14430:Shepherd, P.J. (2013). 14115:Merzbacher, E. (1998). 13931:10.1103/PhysRev.86.1014 13720:10.1103/PhysRev.115.485 12552:. Springer. p. 5. 12133:Dynamics and Relativity 12108:10.12743/quanta.v8i1.96 11375:Fine-structure constant 11370:Electron spin resonance 11190:publish a paper on the 10974:; the splitting of the 10917:quantum electrodynamics 9166:magnetic dipole moments 9162:electric dipole moments 5293:one-dimensional scalar 4161:(authors usually write 4020:of space and time. The 2961:, the interaction term 2250:Generator (mathematics) 1365:spin–statistics theorem 1316: + 1, − 1274:is an integer, odd for 1069:quantum electrodynamics 311:Leggett–Garg inequality 16515:Quantum neural network 15845:Calvert, J.B. (2003). 15830:Swanson, D.G. (2007). 15618:Sattler, K.D. (2010). 15555:Sitenko, A.G. (1990). 15534:Choppin, G.R. (2002). 15227:Nagashima, Y. (2010). 15149:Griffiths, D. (2008). 14687:10.1098/rspa.1932.0094 14228:Clifton, R.K. (2011). 14049:Dirac, P.A.M. (1964). 13999:10.1098/rspa.1927.0039 13895:10.1103/PhysRev.85.259 13868:10.1103/PhysRev.81.222 13840:10.1103/PhysRev.79.549 13804:10.1103/PhysRev.72.241 13618:10.1103/PhysRev.47.777 11845:Perkins, D.H. (2000). 11803: 11656: 11607: 11146:hyperfine interactions 11138:gravitational redshift 11128:1958 Discovery of the 10888: 10705: 10553: 10474: 10337:spin–orbit interaction 10310: 10183: 9965:, and the other three 9926: 9729: 9597: 9516: 9365: 9272: 9120: 8984: 8292:for the corresponding 8253: 7720: 7639: 7493: 7299:Helicity and chirality 7251: 7086: 6867: 6591:spin–orbit interaction 6452: 6219: 5929: 5846: 5571: 5280: 4988: 4882: 4643: 4491: 4148: 3944:Densities and currents 3930:electronic transitions 3867: 3702: 3637: 3476: 3294: 3154: 3027: 2950:electromagnetic fields 2938:for the particle, and 2919: 2663: 2421: 2169: 2020:Lorentz transformation 2009:Lorentz transformation 1942:relativistic mechanics 1834: 1679: 1571: 1482: 1419:, and in a particular 1193: 1029:electromagnetic fields 925:, and can accommodate 102: 16540:Quantum teleportation 16068:Wave–particle duality 15809:University of Glasgow 15751:Pfeifer, W. (2008) . 15513:Mizutani, U. (2001). 15471:Hummel, R.E. (2011). 15401:Lindgren, I. (2011). 14193:Schiff, L.I. (1968). 13967:10.1103/PhysRev.89.98 13312:. Macmillan. p. 13306:(15 September 1973). 12944:Annals of Mathematics 12903:10.1073/pnas.34.5.211 12766:Parker, C.B. (1994). 12653:Grandy, W.T. (1991). 12406:Parker, C.B. (1994). 12322:Weinberg, S. (1969). 12268:Weinberg, S. (1964). 12214:Weinberg, S. (1964). 11804: 11657: 11608: 11324:, influential in QED. 11182:violate SR. In 1959, 11098:studies splitting of 10992:wave–particle duality 10889: 10706: 10554: 10475: 10311: 10184: 9927: 9730: 9598: 9517: 9366: 9273: 9121: 8985: 8254: 7721: 7640: 7494: 7252: 7087: 6868: 6488:and two spin states ( 6453: 6220: 5930: 5847: 5622:electromagnetic field 5572: 5281: 4989: 4883: 4644: 4492: 4192:Klein–Gordon equation 4149: 3868: 3703: 3638: 3523:that are required to 3477: 3295: 3173:Klein–Gordon equation 3155: 3054:matrix multiplication 3028: 2920: 2743:everywhere, even for 2688:completely asymmetric 2664: 2422: 2279:classical Hamiltonian 2248:Further information: 2170: 1860:transformation theory 1835: 1680: 1572: 1483: 1194: 1084:tensor index notation 998:spin magnetic moments 296:Elitzur–Vaidman 286:Davisson–Germer 103: 16690:Theory of relativity 16670:Mathematical physics 16571:Quantum field theory 16500:Quantum metamaterial 16445:Quantum cryptography 16175:Consistent histories 15789:Cavendish Laboratory 15639:Kuzmany, H. (2009). 15492:Pavia, D.L. (2005). 15422:Grant, I.P. (2007). 15383:Quantum Field Theory 15212:. World Scientific. 15176:. World Scientific. 14612:. World Scientific. 14536:Sachs, R.G. (1987). 14515:Schwabl, F. (2010). 14473:Heitler, W. (1954). 14377:Thaller, B. (2005). 14314:Pilkuhn, H. (2005). 14293:Rae, A.I.M. (2008). 14134:Messiah, A. (1961). 14070:Thaller, B. (2010). 13657:Penrose, R. (2005). 13555:Krane, K.S. (1988). 13522:Krane, K.S. (1988). 13367:Kroemer, H. (2003). 13266:. World Scientific. 13236:Quantum Field Theory 13233:Ryder, L.H. (1996). 13206:Penrose, R. (2005). 13143:Labelle, P. (2010). 13116:Strange, P. (1998). 12798:Labelle, P. (2010). 12709:Quantum Field Theory 12705:McMahon, D. (2008). 12601:Penrose, R. (2005). 12548:Wachter, A. (2011). 12189:Wachter, A. (2011). 12162:Greiner, W. (2000). 12029:Messiah, A. (1981). 11939:Strange, P. (1998). 11693: 11617: 11554: 11381:Mathematical physics 11192:Aharonov–Bohm effect 11167:quantum entanglement 11033:mass-to-charge ratio 11000:de Broglie relations 10960:photoelectric effect 10747: 10569: 10500: 10381: 10203: 10004: 9798: 9613: 9532: 9417: 9292: 9224: 9016: 8325: 7788: 7655: 7582: 7330: 7293:many-particle system 7135: 6937: 6615: 6316: 5992: 5876: 5631: 5373: 5012: 4928: 4730: 4561: 4201: 4031: 3718: 3657: 3544: 3318: 3182: 3069: 2968: 2800: 2442: 2331: 2281:for a particle in a 2273:Hamiltonian operator 2074: 1883:many particle system 1742: 1724:, and the operators 1598: 1495: 1452: 1410:; for a particle of 1204:Hamiltonian operator 1144: 1135:Schrödinger equation 1046:quantum field theory 1037:Hamiltonian operator 561:Quantum field theory 473:Consistent histories 110:Schrödinger equation 37: 16556:Quantum fluctuation 16525:Quantum programming 16485:Quantum logic gates 16470:Quantum information 16450:Quantum electronics 15925:Classical mechanics 15660:Reid, J.M. (1984). 15182:2002rqmi.book.....C 15077:1993AmJPh..61..551C 15040:2013EJPh...34L..55R 15003:2004AmJPh..72...51K 14957:1970PhRvD...1.3247F 14910:2012CQGra..29v0301M 14863:2012CQGra..29v0301M 14851:Class. Quantum Grav 14834:1982PhRvD..26.1348C 14805:1983JPhA...16.1869O 14768:1986PhRvA..34.1677F 14724:2004JPhA...37.1463P 14678:1932RSPSA.136..453D 14606:Tung, W.K. (1985). 14389:2005avqm.book.....T 13959:1953PhRv...89...98T 13923:1952PhRv...86.1014L 13887:1952PhRv...85..259L 13860:1951PhRv...81..222L 13832:1950PhRv...79..549L 13795:1947PhRv...72..241L 13734:Bell, John (1964). 13711:1959PhRv..115..485A 13660:The Road to Reality 13609:1935PhRv...47..777E 13398:2004AmJPh..72...51K 13272:1994sppi.book.....T 13209:The Road to Reality 13087:2009PhRvD..79k3011L 13021:1992AnPhy.216..226J 12956:1939AnMat..40..149W 12935:Wigner, E. (1937). 12894:1948PNAS...34..211B 12847:2012AnPhy.327.1617E 12604:The Road to Reality 12347:1969PhRv..181.1893W 12293:1964PhRv..134..882W 12239:1964PhRv..133.1318W 12233:(5B): B1318–B1332. 12057:Schweber, Silvan S. 11464:Chirality (physics) 11292:hydrogen levels by 11171:quantum nonlocality 11157:In 1935, Einstein, 11052:oil drop experiment 11035:. Discovery of the 10366:Lorentz-transformed 9194:multipole expansion 9170:spin quantum number 8264:Hermitian conjugate 7781:of space and time: 7309:Chirality (physics) 6603:perturbation theory 6299:. There is a 4 × 4 5960:) and "spin down" ( 5337:Maxwell's equations 4906:4-momentum operator 4552:continuity equation 4536:when multiplied by 4013:) because they are 4002:probability current 3674: 2322:Schrödinger picture 2206:is thought of as a 1946:spatial coordinates 1879:classical mechanics 1704:partial derivatives 1260:spin quantum number 1120:Schrödinger picture 1104:Fourier transformed 983:Schrödinger picture 967:classical mechanics 963:Galilean relativity 939:accelerator physics 931:high energy physics 913:propagating at all 349:Stern–Gerlach 146:Classical mechanics 16609:in popular culture 16391:Quantum algorithms 16239:Von Neumann–Wigner 16219:Objective collapse 15930:Old quantum theory 15723:Kumar, M. (2009). 15702:Piela, L. (2006). 15663:The Atomic Nucleus 15317:Gross, F. (2008). 15204:Hwang, W.Y. Pauchy 14627:Heine, V. (1993). 14458:. Addison-Wesley. 14073:The Dirac Equation 13632:Abers, E. (2004). 12680:Abers, E. (2004). 12573:Abers, E. (2004). 12523:Abers, E. (2004). 11799: 11793: 11722: 11652: 11646: 11603: 11597: 11449:Symmetry (physics) 11061:scattering in the 11039:: the splitting a 10884: 10701: 10549: 10470: 10306: 10179: 9983:are boosts of the 9922: 9725: 9593: 9512: 9403:Lagrangian density 9361: 9268: 9116: 9110: 8997:particles of spin 8980: 8974: 8759: 8249: 8243: 7993: 7716: 7635: 7489: 7247: 7109:Levi-Civita symbol 7082: 6863: 6595:gyromagnetic ratio 6448: 6442: 6359: 6215: 5925: 5919: 5842: 5612:introduced in the 5609:phenomenologically 5567: 5276: 4984: 4878: 4654:conserved quantity 4639: 4487: 4144: 3863: 3698: 3660: 3646:and square to the 3633: 3521:Hermitian matrices 3472: 3290: 3150: 3023: 2915: 2659: 2417: 2218:allowed values of 2165: 1830: 1712:Heisenberg picture 1675: 1567: 1478: 1421:frame of reference 1373:subatomic particle 1189: 987:Heisenberg picture 979:special relativity 927:massless particles 537:Von Neumann–Wigner 517:Objective-collapse 316:Mach–Zehnder 306:Leggett inequality 301:Franck–Hertz 151:Old quantum theory 98: 16665:Quantum mechanics 16652: 16651: 16626:Quantum mysticism 16604:Schrödinger's cat 16535:Quantum simulator 16505:Quantum metrology 16433:Quantum computing 16396:Quantum amplifier 16373:Quantum spacetime 16338:Quantum cosmology 16328:Quantum chemistry 16043:Scattering theory 15991:Zero-point energy 15986:Degenerate levels 15894:Quantum mechanics 15736:978-1-84831-035-3 15715:978-0-08-046676-7 15694:978-0-08-054046-7 15673:978-0-7190-0978-5 15652:978-3-642-01480-2 15631:978-1-4200-7553-3 15610:978-3-642-31238-0 15597:Luth, H. (2013). 15589:978-3-540-85416-6 15568:978-9971-5-0482-3 15547:978-0-7506-7463-8 15526:978-0-521-58709-9 15505:978-0-495-11478-9 15484:978-1-4419-8164-6 15463:978-81-203-2597-5 15435:978-0-387-34671-7 15414:978-1-4419-8309-1 15393:978-0-471-49683-0 15372:978-0-7923-0540-8 15351:978-0-521-76150-5 15330:978-3-527-61734-0 15309:978-0-521-66000-6 15283:978-0-521-55002-4 15257:978-0-07-005494-3 15238:978-3-527-40962-4 15219:978-981-02-0608-6 15191:978-981-238-137-8 15162:978-3-527-61847-7 15141:978-0-8493-8775-3 15120:978-1-139-50432-4 15011:10.1119/1.1615526 14941:(12): 3247–3258. 14894:(22): 2197–2209. 14640:978-0-486-67585-5 14619:978-9971-966-56-0 14555:978-0-226-73331-9 14528:978-3-540-71933-5 14518:Quantum Mechanics 14507:978-0-387-95576-6 14486:978-0-486-64558-2 14465:978-0-201-32831-8 14443:978-1-118-51692-8 14419:978-3-540-41061-4 14398:978-0-387-27127-9 14369:978-1-4020-1371-3 14348:978-1-84265-573-3 14327:978-3-540-28522-9 14306:978-1-58488-970-0 14296:Quantum Mechanics 14285:978-0-471-16435-7 14276:Quantum Mechanics 14263:978-0-471-16433-3 14254:Quantum Mechanics 14241:978-90-481-4643-7 14220:978-981-4383-40-0 14197:Quantum Mechanics 14185:978-0-201-02118-9 14164:978-0-07-005493-6 14145:978-0-471-59766-7 14136:Quantum Mechanics 14126:978-0-471-88702-7 14117:Quantum Mechanics 14107:978-3-540-09842-3 14083:978-3-642-08134-7 14062:978-0-486-41713-4 14041:978-0-19-852011-5 13670:978-0-09-944068-0 13663:. Vintage Books. 13643:978-0-13-146100-0 13634:Quantum Mechanics 13574:978-0-471-80553-3 13541:978-0-471-80553-3 13508:978-0-19-855493-6 13480:978-0-471-87373-0 13449:978-0-471-30932-1 13406:10.1119/1.1615526 13350:978-90-481-3735-0 13323:978-0-7167-0344-0 13281:978-981-02-1692-4 13246:978-0-521-47814-4 13219:978-0-09-944068-0 13192:978-0-521-55001-7 13162:978-0-07-163641-4 13129:978-0-521-56583-7 13065:Physical Review D 13009:Annals of Physics 12825:Annals of Physics 12809:978-0-07-163641-4 12783:978-0-07-051400-3 12749:978-0-582-44401-0 12724:978-0-07-154382-8 12691:978-0-13-146100-0 12682:Quantum Mechanics 12666:978-0-7923-1049-5 12639:978-0-582-44401-0 12614:978-0-09-944068-0 12584:978-0-13-146100-0 12575:Quantum Mechanics 12559:978-90-481-3645-2 12534:978-0-13-146100-0 12525:Quantum Mechanics 12509:978-0-07-162358-2 12500:Quantum Mechanics 12481:978-0-08-050348-6 12454:978-0-471-87373-0 12423:978-0-07-051400-3 12287:(4B): B882–B896. 12200:978-90-481-3645-2 12175:978-3-540-67457-3 12148:978-0-470-01460-8 12070:978-0-691-21328-6 12042:978-0-7204-0045-8 12032:Quantum Mechanics 12015:978-0-470-03294-7 11979:978-3-540-43183-1 11966:Mohn, P. (2003). 11952:978-0-521-56583-7 11925:978-3-527-62749-3 11898:978-0-470-03294-7 11858:978-0-521-62196-0 11494: 11493: 11407:Casimir invariant 11387:Quantum spacetime 11031:and measures its 10935:; by considering 10871: 10851: 10791: 10759: 10738:Thomas precession 10688: 10668: 10613: 10581: 10543: 10464: 10461: 10333:Thomas precession 10227: 10155: 10070: 9644: 9553: 9503: 9472: 9353: 9336: 9320: 9306: 9265: 9252: 9238: 9200:Velocity operator 9168:) allowed by the 8270:particle of spin 7714: 7707: 7684: 7667: 7633: 7487: 7486: 7435: 7412: 7396: 7377: 7359: 7342: 7321:helicity operator 7315:spin polarization 7106:three-dimensional 6834: 6784: 6735: 6711: 6706: 6662: 6638: 6633: 6157: 6082: 6019: 5786: 5767: 5698: 5658: 5531: 5513: 5494: 5453: 5435: 5400: 5209: 5166: 5083: 5029: 4853: 4831: 4802: 4785: 4757: 4742: 4700:= ∇ × 4582: 4430: 4347: 4306: 4273: 4238: 4159:Hermitian adjoint 3841: 3813: 3763: 3735: 3432: 3404: 3363: 3335: 3248: 3231: 3195: 3144: 3127: 3096: 3081: 3014: 2981: 2861: 2848: 2813: 2653: 2621: 2604: 2573: 2547: 2515: 2498: 2469: 2454: 2392: 2380: 2363: 2343: 1821: 1799: 1779: 1758: 1653: 1635: 1610: 1476: 1202:using a suitable 1183: 1166: 1025:charged particles 911:massive particles 907:quantum mechanics 884: 883: 591:Scattering theory 571:Quantum computing 344:Schrödinger's cat 276:Bell's inequality 84: 59: 28:Quantum mechanics 16697: 16680:Particle physics 16675:Electromagnetism 16642: 16641: 16353:Quantum geometry 16348:Quantum dynamics 16205:Superdeterminism 16101:Matrix mechanics 15956:Bra–ket notation 15887: 15880: 15873: 15864: 15863: 15859: 15850: 15841: 15835: 15826: 15824: 15823: 15817: 15811:. Archived from 15806: 15796: 15786: 15777: 15775: 15769:. Archived from 15768: 15758: 15740: 15719: 15698: 15677: 15656: 15635: 15614: 15593: 15572: 15551: 15530: 15509: 15488: 15467: 15439: 15418: 15397: 15376: 15355: 15334: 15313: 15301: 15287: 15275: 15261: 15242: 15223: 15195: 15166: 15145: 15124: 15088: 15059: 15022: 14996: 14978: 14968: 14950: 14929: 14903: 14882: 14845: 14828:(6): 1348–1367. 14816: 14799:(9): 1869–1884. 14787: 14762:(3): 1677–1681. 14750: 14745: 14735: 14706: 14704: 14691: 14689: 14672:(829): 453–464. 14644: 14623: 14602: 14583: 14562: 14543: 14532: 14511: 14490: 14469: 14447: 14426: 14402: 14373: 14352: 14331: 14310: 14289: 14267: 14245: 14224: 14202: 14200: 14189: 14168: 14149: 14130: 14111: 14087: 14066: 14045: 14012: 14011: 14001: 13992:(767): 243–265. 13977: 13971: 13970: 13942: 13917:(6): 1014–1022. 13906: 13871: 13843: 13815: 13809: 13808: 13806: 13769: 13763: 13762: 13760: 13740: 13731: 13725: 13724: 13722: 13688: 13682: 13674: 13654: 13648: 13647: 13629: 13623: 13622: 13620: 13594: 13585: 13579: 13578: 13562: 13552: 13546: 13545: 13529: 13519: 13513: 13512: 13491: 13485: 13484: 13460: 13454: 13453: 13437: 13424: 13418: 13417: 13391: 13373: 13364: 13355: 13354: 13334: 13328: 13327: 13292: 13286: 13285: 13257: 13251: 13250: 13230: 13224: 13223: 13203: 13197: 13196: 13173: 13167: 13166: 13150: 13140: 13134: 13133: 13113: 13107: 13106: 13080: 13060: 13054: 13053: 13051: 13039: 13033: 13032: 13004: 12998: 12997: 12995: 12994: 12988: 12982:. Archived from 12941: 12932: 12926: 12925: 12915: 12905: 12873: 12867: 12866: 12840: 12831:(6): 1617–1644. 12820: 12814: 12813: 12795: 12789: 12787: 12763: 12754: 12753: 12735: 12729: 12728: 12712: 12702: 12696: 12695: 12677: 12671: 12670: 12650: 12644: 12643: 12625: 12619: 12618: 12598: 12589: 12588: 12570: 12564: 12563: 12545: 12539: 12538: 12520: 12514: 12513: 12495: 12486: 12485: 12465: 12459: 12458: 12434: 12428: 12427: 12403: 12394: 12393: 12391: 12379: 12373: 12372: 12370: 12369: 12363: 12357:. Archived from 12341:(5): 1893–1899. 12332: 12318: 12316: 12315: 12309: 12303:. Archived from 12278: 12264: 12262: 12261: 12255: 12249:. Archived from 12224: 12211: 12205: 12204: 12186: 12180: 12179: 12159: 12153: 12152: 12136: 12126: 12120: 12119: 12101: 12081: 12075: 12074: 12053: 12047: 12046: 12026: 12020: 12019: 12003: 12000:Particle Physics 11993: 11984: 11983: 11963: 11957: 11956: 11936: 11930: 11929: 11909: 11903: 11902: 11886: 11883:Particle Physics 11876: 11863: 11862: 11842: 11826: 11819: 11813: 11808: 11806: 11805: 11800: 11798: 11797: 11790: 11789: 11776: 11775: 11762: 11761: 11748: 11747: 11727: 11726: 11685: 11679: 11677: 11675: 11674: 11671: 11668: 11661: 11659: 11658: 11653: 11651: 11650: 11612: 11610: 11609: 11604: 11602: 11601: 11594: 11593: 11580: 11579: 11548: 11542: 11532: 11526: 11517: 11507: 11412:Casimir operator 11349: 11290: 11289: 11285: 11275: 11274: 11270: 11252: 11251: 11247: 11237: 11236: 11232: 11130:Mössbauer effect 10933:particle physics 10906: 10905: 10901: 10893: 10891: 10890: 10885: 10879: 10878: 10873: 10872: 10864: 10857: 10853: 10852: 10850: 10849: 10848: 10835: 10834: 10826: 10820: 10815: 10799: 10798: 10793: 10792: 10784: 10777: 10773: 10761: 10760: 10752: 10731: 10730: 10726: 10721: 10710: 10708: 10707: 10702: 10696: 10695: 10690: 10689: 10681: 10674: 10670: 10669: 10667: 10666: 10657: 10656: 10648: 10642: 10637: 10621: 10620: 10615: 10614: 10606: 10599: 10595: 10583: 10582: 10574: 10558: 10556: 10555: 10550: 10544: 10542: 10541: 10532: 10531: 10523: 10517: 10512: 10508: 10492: 10479: 10477: 10476: 10471: 10465: 10463: 10462: 10460: 10459: 10454: 10450: 10446: 10426: 10424: 10423: 10413: 10412: 10404: 10398: 10393: 10389: 10373: 10363: 10357: 10351: 10315: 10313: 10312: 10307: 10302: 10294: 10280: 10270: 10269: 10260: 10259: 10247: 10246: 10228: 10220: 10215: 10214: 10188: 10186: 10185: 10180: 10175: 10167: 10156: 10154: 10153: 10141: 10136: 10128: 10120: 10110: 10109: 10100: 10099: 10090: 10089: 10071: 10069: 10068: 10056: 10051: 10050: 10038: 10037: 10019: 10018: 9992: 9982: 9976: 9970: 9964: 9954: 9944: 9931: 9929: 9928: 9923: 9917: 9909: 9901: 9891: 9890: 9878: 9877: 9859: 9858: 9849: 9848: 9836: 9835: 9826: 9825: 9813: 9812: 9789:exterior algebra 9778: 9744: 9734: 9732: 9731: 9726: 9717: 9716: 9707: 9706: 9688: 9687: 9678: 9677: 9668: 9667: 9658: 9657: 9645: 9640: 9639: 9630: 9622: 9621: 9602: 9600: 9599: 9594: 9577: 9576: 9567: 9566: 9554: 9546: 9541: 9540: 9521: 9519: 9518: 9513: 9504: 9502: 9494: 9493: 9492: 9482: 9477: 9473: 9471: 9464: 9463: 9447: 9446: 9445: 9435: 9429: 9428: 9400:(really meaning 9395: 9370: 9368: 9367: 9362: 9360: 9356: 9355: 9354: 9349: 9344: 9338: 9337: 9329: 9321: 9313: 9308: 9307: 9302: 9297: 9277: 9275: 9274: 9269: 9267: 9266: 9261: 9256: 9253: 9245: 9240: 9239: 9234: 9229: 9216: 9191: 9189: 9188: 9185: 9182: 9159: 9157: 9156: 9153: 9150: 9125: 9123: 9122: 9117: 9115: 9114: 9098: 9090: 9089: 9067: 9059: 9058: 9029: 8989: 8987: 8986: 8981: 8979: 8978: 8971: 8970: 8965: 8955: 8947: 8946: 8912: 8911: 8906: 8896: 8888: 8887: 8855: 8854: 8849: 8839: 8831: 8830: 8798: 8797: 8792: 8782: 8764: 8763: 8747: 8739: 8738: 8700: 8692: 8691: 8640: 8632: 8631: 8590: 8582: 8581: 8546: 8538: 8537: 8499: 8491: 8490: 8439: 8431: 8430: 8389: 8381: 8380: 8338: 8317: 8309: 8303: 8291: 8283: 8275: 8258: 8256: 8255: 8250: 8248: 8247: 8240: 8239: 8234: 8224: 8216: 8215: 8193: 8192: 8187: 8177: 8169: 8168: 8135: 8134: 8129: 8119: 8111: 8110: 8085: 8084: 8079: 8069: 8061: 8060: 8032: 8031: 8026: 8016: 7998: 7997: 7981: 7973: 7972: 7941: 7933: 7932: 7888: 7880: 7879: 7845: 7837: 7836: 7801: 7749: 7747: 7746: 7743: 7740: 7725: 7723: 7722: 7717: 7715: 7710: 7709: 7708: 7703: 7698: 7691: 7686: 7685: 7680: 7675: 7669: 7668: 7660: 7644: 7642: 7641: 7636: 7634: 7632: 7631: 7622: 7621: 7612: 7611: 7596: 7595: 7586: 7577: 7575: 7574: 7571: 7568: 7561: 7544: 7542: 7541: 7538: 7535: 7498: 7496: 7495: 7490: 7488: 7485: 7484: 7475: 7474: 7465: 7464: 7449: 7448: 7439: 7438: 7437: 7436: 7431: 7426: 7419: 7414: 7413: 7408: 7403: 7397: 7395: 7394: 7389: 7384: 7378: 7373: 7368: 7366: 7361: 7360: 7355: 7350: 7344: 7343: 7335: 7290: 7288: 7287: 7284: 7281: 7262:curved spacetime 7256: 7254: 7253: 7248: 7242: 7241: 7223: 7222: 7213: 7212: 7200: 7199: 7190: 7189: 7177: 7176: 7171: 7167: 7166: 7165: 7153: 7152: 7127: 7121:Minkowski metric 7117:Clifford algebra 7103: 7091: 7089: 7088: 7083: 7081: 7080: 7071: 7070: 7052: 7051: 7046: 7042: 7041: 7040: 7028: 7027: 7007: 7006: 6997: 6996: 6972: 6968: 6967: 6966: 6954: 6953: 6902:pure mathematics 6896: 6872: 6870: 6869: 6864: 6852: 6851: 6842: 6841: 6836: 6835: 6827: 6823: 6822: 6802: 6801: 6792: 6791: 6786: 6785: 6777: 6773: 6772: 6752: 6751: 6742: 6738: 6737: 6736: 6731: 6726: 6720: 6712: 6707: 6699: 6697: 6679: 6678: 6669: 6665: 6664: 6663: 6658: 6653: 6647: 6639: 6634: 6626: 6624: 6588: 6582: 6545: 6534: 6518: 6509: 6508: 6506: 6505: 6502: 6499: 6487: 6481: 6469: 6457: 6455: 6454: 6449: 6447: 6446: 6439: 6438: 6422: 6421: 6405: 6404: 6388: 6387: 6364: 6363: 6356: 6355: 6342: 6341: 6308: 6298: 6224: 6222: 6221: 6216: 6205: 6201: 6200: 6199: 6181: 6180: 6165: 6164: 6159: 6158: 6150: 6143: 6142: 6120: 6116: 6115: 6114: 6099: 6095: 6084: 6083: 6078: 6073: 6063: 6050: 6049: 6034: 6030: 6020: 6018: 6007: 5981: 5980: 5978: 5977: 5974: 5971: 5959: 5958: 5956: 5955: 5952: 5949: 5934: 5932: 5931: 5926: 5924: 5923: 5916: 5915: 5902: 5901: 5864: 5851: 5849: 5848: 5843: 5832: 5831: 5826: 5819: 5808: 5797: 5787: 5785: 5774: 5769: 5768: 5760: 5749: 5745: 5744: 5743: 5738: 5731: 5720: 5709: 5699: 5697: 5686: 5673: 5669: 5659: 5657: 5646: 5596: 5594: 5593: 5590: 5587: 5576: 5574: 5573: 5568: 5554: 5553: 5548: 5544: 5533: 5532: 5527: 5522: 5514: 5512: 5501: 5496: 5495: 5487: 5476: 5475: 5470: 5466: 5455: 5454: 5449: 5444: 5436: 5434: 5423: 5415: 5411: 5401: 5399: 5388: 5365: 5363: 5362: 5359: 5356: 5327: 5325: 5324: 5321: 5318: 5306: 5302: 5298: 5285: 5283: 5282: 5277: 5266: 5262: 5261: 5260: 5255: 5237: 5233: 5232: 5217: 5216: 5211: 5210: 5202: 5194: 5190: 5189: 5174: 5173: 5168: 5167: 5159: 5138: 5137: 5128: 5127: 5106: 5105: 5100: 5096: 5085: 5084: 5079: 5074: 5066: 5065: 5050: 5049: 5044: 5031: 5030: 5022: 4993: 4991: 4990: 4985: 4979: 4968: 4958: 4957: 4916: 4899: 4887: 4885: 4884: 4879: 4877: 4876: 4861: 4860: 4855: 4854: 4846: 4839: 4838: 4833: 4832: 4824: 4815: 4804: 4803: 4798: 4793: 4787: 4786: 4781: 4776: 4759: 4758: 4750: 4744: 4743: 4735: 4722: 4704: 4694: 4676: 4666:Minimal coupling 4648: 4646: 4645: 4640: 4628: 4627: 4618: 4617: 4597: 4583: 4581: 4573: 4565: 4545: 4524: 4513: 4503: 4496: 4494: 4493: 4488: 4483: 4482: 4473: 4472: 4454: 4453: 4444: 4443: 4431: 4429: 4421: 4413: 4408: 4407: 4393: 4389: 4388: 4387: 4363: 4362: 4348: 4346: 4338: 4330: 4322: 4312: 4308: 4307: 4305: 4297: 4296: 4295: 4282: 4274: 4272: 4264: 4256: 4254: 4253: 4239: 4237: 4236: 4235: 4219: 4211: 4185: 4175: 4167: 4153: 4151: 4150: 4145: 4140: 4139: 4130: 4129: 4120: 4119: 4107: 4106: 4089: 4084: 4083: 4074: 4073: 4061: 4049: 4048: 4009:(really meaning 4008: 3999: 3993: 3974: 3960: 3922: 3916: 3910: 3904: 3872: 3870: 3869: 3864: 3862: 3861: 3843: 3842: 3837: 3832: 3826: 3815: 3814: 3806: 3789: 3785: 3784: 3783: 3765: 3764: 3759: 3754: 3748: 3737: 3736: 3728: 3707: 3705: 3704: 3699: 3687: 3686: 3673: 3668: 3642: 3640: 3639: 3634: 3628: 3627: 3618: 3617: 3602: 3601: 3592: 3591: 3578: 3577: 3556: 3555: 3536: 3518: 3512: 3481: 3479: 3478: 3473: 3458: 3454: 3453: 3452: 3434: 3433: 3428: 3423: 3417: 3406: 3405: 3397: 3389: 3385: 3384: 3383: 3365: 3364: 3359: 3354: 3348: 3337: 3336: 3328: 3299: 3297: 3296: 3291: 3282: 3281: 3272: 3271: 3250: 3249: 3244: 3239: 3233: 3232: 3227: 3222: 3219: 3218: 3203: 3202: 3197: 3196: 3188: 3159: 3157: 3156: 3151: 3146: 3145: 3140: 3135: 3129: 3128: 3123: 3118: 3106: 3098: 3097: 3089: 3083: 3082: 3074: 3061: 3032: 3030: 3029: 3024: 3022: 3021: 3016: 3015: 3007: 3000: 2989: 2988: 2983: 2982: 2974: 2960: 2943: 2933: 2924: 2922: 2921: 2916: 2907: 2906: 2888: 2884: 2883: 2878: 2863: 2862: 2857: 2852: 2849: 2844: 2843: 2842: 2829: 2821: 2820: 2815: 2814: 2806: 2788: 2775: 2769: 2753: 2742: 2727: 2709: 2681: 2668: 2666: 2665: 2660: 2654: 2652: 2651: 2642: 2641: 2623: 2622: 2617: 2612: 2606: 2605: 2600: 2595: 2592: 2591: 2582: 2574: 2572: 2561: 2548: 2546: 2545: 2536: 2535: 2517: 2516: 2511: 2506: 2500: 2499: 2494: 2489: 2486: 2485: 2476: 2471: 2470: 2462: 2456: 2455: 2447: 2426: 2424: 2423: 2418: 2407: 2393: 2391: 2383: 2382: 2381: 2376: 2371: 2365: 2364: 2359: 2354: 2350: 2345: 2344: 2336: 2319: 2304:potential energy 2301: 2244: 2238: 2232: 2223: 2217: 2205: 2196: 2184: 2174: 2172: 2171: 2166: 2152: 2144: 2143: 2128: 2127: 2094: 2086: 2085: 2062: 2053: 2040: 1999: 1977: 1936: 1856:matrix mechanics 1839: 1837: 1836: 1831: 1822: 1820: 1809: 1801: 1800: 1792: 1780: 1778: 1767: 1759: 1757: 1746: 1734: 1722:time-independent 1719: 1701: 1684: 1682: 1681: 1676: 1655: 1654: 1649: 1644: 1636: 1634: 1623: 1612: 1611: 1603: 1576: 1574: 1573: 1568: 1562: 1561: 1552: 1551: 1533: 1525: 1520: 1519: 1507: 1506: 1487: 1485: 1484: 1479: 1477: 1475: 1467: 1462: 1445:in terms of the 1440: 1431: 1418: 1339: 1320: 1295: 1287: 1273: 1266: 1254: 1248: 1232: 1210: 1198: 1196: 1195: 1190: 1185: 1184: 1176: 1167: 1165: 1154: 1014: 1013: 1009: 935:particle physics 876: 869: 862: 503:Superdeterminism 156:Bra–ket notation 107: 105: 104: 99: 91: 86: 85: 77: 65: 60: 58: 47: 19: 18: 16705: 16704: 16700: 16699: 16698: 16696: 16695: 16694: 16655: 16654: 16653: 16648: 16630: 16616:Wigner's friend 16592: 16583:Quantum gravity 16544: 16530:Quantum sensing 16510:Quantum network 16490:Quantum machine 16460:Quantum imaging 16423:Quantum circuit 16418:Quantum channel 16377: 16323:Quantum biology 16309: 16285:Elitzur–Vaidman 16260:Davisson–Germer 16243: 16195:Hidden-variable 16185:de Broglie–Bohm 16162:Interpretations 16156: 16120: 16074: 15961:Complementarity 15939: 15896: 15891: 15821: 15819: 15815: 15804: 15784: 15780: 15773: 15766: 15747: 15737: 15716: 15695: 15674: 15653: 15632: 15611: 15590: 15569: 15548: 15527: 15506: 15485: 15464: 15446: 15436: 15415: 15394: 15373: 15352: 15331: 15310: 15284: 15258: 15250:. McGraw-Hill. 15239: 15231:. Vol. 1. 15220: 15192: 15163: 15142: 15121: 15100: 15095: 15093:Further reading 15085:10.1119/1.17207 14994:physics/0310016 14976: 14948:physics/9703008 14901:physics/9703008 14733:10.1.1.499.2793 14702: 14651: 14649:Selected papers 14641: 14620: 14596: 14569: 14556: 14529: 14508: 14487: 14466: 14444: 14420: 14399: 14370: 14349: 14328: 14307: 14286: 14264: 14242: 14221: 14186: 14165: 14157:. McGraw-Hill. 14146: 14127: 14108: 14084: 14063: 14042: 14021: 14016: 14015: 13978: 13974: 13943: 13907: 13844: 13816: 13812: 13782:Physical Review 13773:Lamb, Willis E. 13770: 13766: 13738: 13732: 13728: 13698:Physical Review 13689: 13685: 13671: 13655: 13651: 13644: 13630: 13626: 13603:(10): 777–780. 13592: 13586: 13582: 13575: 13553: 13549: 13542: 13520: 13516: 13509: 13492: 13488: 13481: 13461: 13457: 13450: 13425: 13421: 13389:physics/0310016 13371: 13365: 13358: 13351: 13335: 13331: 13324: 13293: 13289: 13282: 13258: 13254: 13247: 13231: 13227: 13220: 13204: 13200: 13193: 13174: 13170: 13163: 13141: 13137: 13130: 13114: 13110: 13061: 13057: 13040: 13036: 13005: 13001: 12992: 12990: 12986: 12964:10.2307/1968551 12939: 12933: 12929: 12874: 12870: 12821: 12817: 12810: 12796: 12792: 12784: 12764: 12757: 12750: 12736: 12732: 12725: 12703: 12699: 12692: 12678: 12674: 12667: 12651: 12647: 12640: 12626: 12622: 12615: 12599: 12592: 12585: 12571: 12567: 12560: 12546: 12542: 12535: 12521: 12517: 12510: 12496: 12489: 12482: 12466: 12462: 12455: 12435: 12431: 12424: 12404: 12397: 12380: 12376: 12367: 12365: 12361: 12330: 12324:"Feynman Rules 12320: 12313: 12311: 12307: 12276: 12270:"Feynman Rules 12266: 12259: 12257: 12253: 12222: 12216:"Feynman Rules 12212: 12208: 12201: 12187: 12183: 12176: 12160: 12156: 12149: 12127: 12123: 12082: 12078: 12071: 12054: 12050: 12043: 12027: 12023: 12016: 11994: 11987: 11980: 11964: 11960: 11953: 11937: 11933: 11926: 11910: 11906: 11899: 11877: 11866: 11859: 11843: 11839: 11834: 11829: 11820: 11816: 11792: 11791: 11785: 11781: 11778: 11777: 11771: 11767: 11764: 11763: 11757: 11753: 11750: 11749: 11743: 11739: 11732: 11731: 11721: 11720: 11714: 11713: 11703: 11702: 11694: 11691: 11690: 11686: 11682: 11672: 11669: 11666: 11665: 11663: 11645: 11644: 11638: 11637: 11627: 11626: 11618: 11615: 11614: 11596: 11595: 11589: 11585: 11582: 11581: 11575: 11571: 11564: 11563: 11555: 11552: 11551: 11549: 11545: 11528: 11524: 11519: 11515: 11510: 11508: 11504: 11500: 11495: 11430: 11422: 11417:Wigner D-matrix 11392:Spin connection 11383: 11356: 11346: 11331:calculates the 11322:renormalization 11320:begins work on 11306: 11291: 11287: 11283: 11282: 11276: 11272: 11268: 11267: 11253: 11249: 11245: 11244: 11238: 11234: 11230: 11229: 11222: 11216: 11196:EM field tensor 11155: 11134:gamma radiation 11104:magnetic fields 11020: 11010:and separately 10956:Albert Einstein 10953: 10929:nuclear physics 10913: 10903: 10899: 10898: 10874: 10863: 10862: 10861: 10844: 10840: 10836: 10830: 10822: 10821: 10819: 10811: 10810: 10806: 10794: 10783: 10782: 10781: 10769: 10768: 10751: 10750: 10748: 10745: 10744: 10728: 10724: 10723: 10715: 10691: 10680: 10679: 10678: 10662: 10658: 10652: 10644: 10643: 10641: 10633: 10632: 10628: 10616: 10605: 10604: 10603: 10591: 10590: 10573: 10572: 10570: 10567: 10566: 10537: 10533: 10527: 10519: 10518: 10516: 10504: 10503: 10501: 10498: 10497: 10484: 10455: 10442: 10438: 10434: 10433: 10425: 10419: 10415: 10414: 10408: 10400: 10399: 10397: 10385: 10384: 10382: 10379: 10378: 10369: 10368:magnetic field 10359: 10353: 10347: 10329: 10298: 10290: 10276: 10265: 10261: 10252: 10248: 10233: 10229: 10219: 10210: 10206: 10204: 10201: 10200: 10171: 10163: 10149: 10145: 10140: 10132: 10124: 10116: 10105: 10101: 10095: 10091: 10076: 10072: 10064: 10060: 10055: 10043: 10039: 10030: 10026: 10011: 10007: 10005: 10002: 10001: 9988: 9978: 9972: 9966: 9956: 9946: 9936: 9913: 9905: 9897: 9883: 9879: 9870: 9866: 9854: 9850: 9844: 9840: 9831: 9827: 9821: 9817: 9805: 9801: 9799: 9796: 9795: 9766: 9755: 9740: 9712: 9708: 9702: 9698: 9683: 9679: 9673: 9669: 9663: 9659: 9650: 9646: 9635: 9631: 9629: 9617: 9616: 9614: 9611: 9610: 9572: 9568: 9562: 9558: 9545: 9536: 9535: 9533: 9530: 9529: 9495: 9488: 9487: 9483: 9481: 9459: 9455: 9448: 9441: 9440: 9436: 9434: 9430: 9424: 9420: 9418: 9415: 9414: 9391: 9388: 9345: 9343: 9342: 9328: 9327: 9326: 9322: 9312: 9298: 9296: 9295: 9293: 9290: 9289: 9257: 9255: 9254: 9244: 9230: 9228: 9227: 9225: 9222: 9221: 9205: 9202: 9186: 9183: 9180: 9179: 9177: 9174:magnetic charge 9154: 9151: 9146: 9145: 9143: 9109: 9108: 9094: 9085: 9081: 9078: 9077: 9063: 9054: 9050: 9043: 9042: 9025: 9017: 9014: 9013: 8973: 8972: 8966: 8951: 8926: 8922: 8921: 8920: 8918: 8913: 8907: 8892: 8864: 8860: 8859: 8858: 8856: 8850: 8835: 8813: 8809: 8808: 8807: 8800: 8799: 8793: 8778: 8771: 8770: 8758: 8757: 8743: 8718: 8714: 8711: 8710: 8696: 8665: 8661: 8658: 8657: 8651: 8650: 8636: 8608: 8604: 8601: 8600: 8586: 8564: 8560: 8557: 8556: 8542: 8517: 8513: 8510: 8509: 8495: 8464: 8460: 8457: 8456: 8450: 8449: 8435: 8407: 8403: 8400: 8399: 8385: 8363: 8359: 8352: 8351: 8334: 8326: 8323: 8322: 8311: 8305: 8297: 8285: 8277: 8271: 8242: 8241: 8235: 8220: 8202: 8198: 8197: 8196: 8194: 8188: 8173: 8149: 8145: 8144: 8143: 8141: 8136: 8130: 8115: 8094: 8090: 8089: 8088: 8086: 8080: 8065: 8050: 8046: 8045: 8044: 8037: 8036: 8027: 8012: 8005: 8004: 7992: 7991: 7977: 7959: 7955: 7952: 7951: 7937: 7913: 7909: 7906: 7905: 7899: 7898: 7884: 7863: 7859: 7856: 7855: 7841: 7826: 7822: 7815: 7814: 7797: 7789: 7786: 7785: 7744: 7741: 7738: 7737: 7735: 7732: 7699: 7697: 7696: 7692: 7690: 7676: 7674: 7673: 7659: 7658: 7656: 7653: 7652: 7627: 7623: 7617: 7613: 7607: 7603: 7591: 7587: 7585: 7583: 7580: 7579: 7572: 7569: 7566: 7565: 7563: 7550: 7539: 7536: 7533: 7532: 7530: 7524: 7480: 7476: 7470: 7466: 7460: 7456: 7444: 7440: 7427: 7425: 7424: 7420: 7418: 7404: 7402: 7401: 7390: 7385: 7380: 7379: 7369: 7367: 7365: 7351: 7349: 7348: 7334: 7333: 7331: 7328: 7327: 7323:is defined by; 7317: 7311: 7303:Main articles: 7301: 7285: 7282: 7279: 7278: 7276: 7264:by introducing 7234: 7230: 7218: 7214: 7208: 7204: 7195: 7191: 7185: 7181: 7172: 7161: 7157: 7148: 7144: 7143: 7139: 7138: 7136: 7133: 7132: 7123: 7101: 7096: 7076: 7072: 7063: 7059: 7047: 7036: 7032: 7023: 7019: 7018: 7014: 7013: 7002: 6998: 6986: 6982: 6962: 6958: 6949: 6945: 6944: 6940: 6938: 6935: 6934: 6929: 6895: 6889: 6886:identity matrix 6847: 6843: 6837: 6826: 6825: 6824: 6818: 6814: 6797: 6793: 6787: 6776: 6775: 6774: 6768: 6764: 6747: 6743: 6727: 6725: 6724: 6716: 6698: 6696: 6695: 6691: 6674: 6670: 6654: 6652: 6651: 6643: 6625: 6623: 6622: 6618: 6616: 6613: 6612: 6584: 6574: 6540: 6524: 6517: 6511: 6503: 6500: 6497: 6496: 6494: 6489: 6483: 6471: 6468: 6462: 6441: 6440: 6431: 6427: 6424: 6423: 6414: 6410: 6407: 6406: 6397: 6393: 6390: 6389: 6380: 6376: 6369: 6368: 6358: 6357: 6351: 6347: 6344: 6343: 6337: 6333: 6326: 6325: 6317: 6314: 6313: 6304: 6301:identity matrix 6296: 6289: 6282: 6268: 6261: 6254: 6236: 6195: 6191: 6176: 6172: 6160: 6149: 6148: 6147: 6138: 6134: 6133: 6129: 6110: 6106: 6091: 6074: 6072: 6071: 6067: 6059: 6055: 6051: 6045: 6041: 6011: 6006: 5999: 5995: 5993: 5990: 5989: 5975: 5972: 5969: 5968: 5966: 5961: 5953: 5950: 5947: 5946: 5944: 5939: 5918: 5917: 5911: 5907: 5904: 5903: 5897: 5893: 5886: 5885: 5877: 5874: 5873: 5860: 5827: 5815: 5804: 5793: 5789: 5788: 5778: 5773: 5759: 5758: 5739: 5727: 5716: 5705: 5701: 5700: 5690: 5685: 5684: 5680: 5650: 5645: 5638: 5634: 5632: 5629: 5628: 5604: 5598: 5591: 5588: 5585: 5584: 5582: 5549: 5540: 5523: 5521: 5520: 5516: 5515: 5505: 5500: 5486: 5485: 5471: 5462: 5445: 5443: 5442: 5438: 5437: 5427: 5422: 5392: 5387: 5380: 5376: 5374: 5371: 5370: 5360: 5357: 5354: 5353: 5351: 5322: 5319: 5316: 5315: 5313: 5304: 5300: 5294: 5256: 5242: 5241: 5228: 5224: 5212: 5201: 5200: 5199: 5195: 5185: 5181: 5169: 5158: 5157: 5156: 5152: 5151: 5147: 5133: 5129: 5123: 5119: 5101: 5092: 5075: 5073: 5072: 5068: 5067: 5061: 5057: 5045: 5021: 5020: 5016: 5015: 5013: 5010: 5009: 5003: 4975: 4964: 4953: 4949: 4929: 4926: 4925: 4914: 4909: 4897: 4892: 4872: 4868: 4856: 4845: 4844: 4843: 4834: 4823: 4822: 4821: 4811: 4794: 4792: 4791: 4777: 4775: 4774: 4749: 4748: 4734: 4733: 4731: 4728: 4727: 4709: 4696: 4681: 4672: 4670:electric charge 4662: 4652:as charge is a 4623: 4619: 4613: 4609: 4593: 4574: 4566: 4564: 4562: 4559: 4558: 4541: 4538:electric charge 4534:current density 4515: 4509: 4501: 4478: 4474: 4468: 4464: 4449: 4445: 4439: 4435: 4422: 4414: 4412: 4403: 4399: 4383: 4379: 4358: 4354: 4353: 4349: 4339: 4331: 4329: 4318: 4298: 4291: 4287: 4283: 4281: 4265: 4257: 4255: 4249: 4245: 4244: 4240: 4231: 4227: 4220: 4212: 4210: 4202: 4199: 4198: 4181: 4163: 4162: 4135: 4131: 4125: 4121: 4115: 4111: 4102: 4098: 4085: 4079: 4075: 4069: 4065: 4057: 4044: 4040: 4032: 4029: 4028: 4004: 3995: 3980: 3965: 3956: 3946: 3926:Pauli principle 3918: 3912: 3906: 3885: 3857: 3853: 3833: 3831: 3830: 3822: 3805: 3804: 3779: 3775: 3755: 3753: 3752: 3744: 3727: 3726: 3725: 3721: 3719: 3716: 3715: 3682: 3678: 3669: 3664: 3658: 3655: 3654: 3648:identity matrix 3623: 3619: 3613: 3609: 3597: 3593: 3587: 3583: 3573: 3569: 3551: 3547: 3545: 3542: 3541: 3528: 3514: 3510: 3503: 3496: 3486: 3448: 3444: 3424: 3422: 3421: 3413: 3396: 3395: 3394: 3390: 3379: 3375: 3355: 3353: 3352: 3344: 3327: 3326: 3325: 3321: 3319: 3316: 3315: 3277: 3273: 3267: 3263: 3240: 3238: 3237: 3223: 3221: 3220: 3214: 3210: 3198: 3187: 3186: 3185: 3183: 3180: 3179: 3169: 3136: 3134: 3133: 3119: 3117: 3116: 3102: 3088: 3087: 3073: 3072: 3070: 3067: 3066: 3057: 3052:, in which the 3017: 3006: 3005: 3004: 2996: 2984: 2973: 2972: 2971: 2969: 2966: 2965: 2956: 2939: 2929: 2902: 2898: 2879: 2874: 2873: 2869: 2853: 2851: 2850: 2838: 2834: 2830: 2828: 2816: 2805: 2804: 2803: 2801: 2798: 2797: 2786: 2781: 2760: 2755: 2744: 2729: 2721: 2711: 2708: 2702: 2677: 2647: 2643: 2637: 2633: 2613: 2611: 2610: 2596: 2594: 2593: 2587: 2583: 2581: 2565: 2560: 2541: 2537: 2531: 2527: 2507: 2505: 2504: 2490: 2488: 2487: 2481: 2477: 2475: 2461: 2460: 2446: 2445: 2443: 2440: 2439: 2403: 2384: 2372: 2370: 2369: 2355: 2353: 2352: 2351: 2349: 2335: 2334: 2332: 2329: 2328: 2306: 2289: 2275: 2269: 2264: 2240: 2234: 2228: 2226:quantum numbers 2219: 2211: 2201: 2186: 2179: 2148: 2136: 2132: 2123: 2119: 2090: 2081: 2077: 2075: 2072: 2071: 2058: 2051: 2046: 2043:Minkowski space 2022: 2015:Under a proper 2005:reference frame 1982: 1964: 1950:coordinate time 1934: 1927: 1920: 1909: 1902: 1895: 1885: 1875: 1813: 1808: 1791: 1790: 1771: 1766: 1750: 1745: 1743: 1740: 1739: 1725: 1715: 1697: 1688:to construct a 1645: 1643: 1642: 1627: 1622: 1602: 1601: 1599: 1596: 1595: 1557: 1553: 1547: 1543: 1529: 1521: 1515: 1511: 1502: 1498: 1496: 1493: 1492: 1471: 1463: 1461: 1453: 1450: 1449: 1436: 1427: 1414: 1322: 1300: 1289: 1283: 1268: 1262: 1250: 1244: 1242:position vector 1219: 1206: 1175: 1174: 1158: 1153: 1145: 1142: 1141: 1116: 1094:are used here; 1076:vector calculus 1055:matter creation 1011: 1007: 1006: 905:formulation of 880: 851: 850: 849: 614: 606: 605: 551: 550:Advanced topics 543: 542: 541: 493:Hidden-variable 483:de Broglie–Bohm 462: 460:Interpretations 452: 451: 450: 420: 412: 411: 410: 368: 360: 359: 358: 325: 281:CHSH inequality 270: 262: 261: 260: 189:Complementarity 183: 175: 174: 173: 141: 112: 87: 76: 75: 61: 51: 46: 38: 35: 34: 17: 12: 11: 5: 16703: 16693: 16692: 16687: 16685:Atomic physics 16682: 16677: 16672: 16667: 16650: 16649: 16647: 16646: 16635: 16632: 16631: 16629: 16628: 16623: 16618: 16613: 16612: 16611: 16600: 16598: 16594: 16593: 16591: 16590: 16585: 16580: 16579: 16578: 16568: 16563: 16561:Casimir effect 16558: 16552: 16550: 16546: 16545: 16543: 16542: 16537: 16532: 16527: 16522: 16520:Quantum optics 16517: 16512: 16507: 16502: 16497: 16492: 16487: 16482: 16477: 16472: 16467: 16462: 16457: 16452: 16447: 16442: 16441: 16440: 16430: 16425: 16420: 16415: 16414: 16413: 16403: 16398: 16393: 16387: 16385: 16379: 16378: 16376: 16375: 16370: 16365: 16360: 16355: 16350: 16345: 16340: 16335: 16330: 16325: 16319: 16317: 16311: 16310: 16308: 16307: 16302: 16297: 16295:Quantum eraser 16292: 16287: 16282: 16277: 16272: 16267: 16262: 16257: 16251: 16249: 16245: 16244: 16242: 16241: 16236: 16231: 16226: 16221: 16216: 16211: 16210: 16209: 16208: 16207: 16192: 16187: 16182: 16177: 16172: 16166: 16164: 16158: 16157: 16155: 16154: 16149: 16144: 16139: 16134: 16128: 16126: 16122: 16121: 16119: 16118: 16113: 16108: 16103: 16098: 16093: 16088: 16082: 16080: 16076: 16075: 16073: 16072: 16071: 16070: 16065: 16055: 16050: 16045: 16040: 16035: 16030: 16025: 16020: 16015: 16010: 16005: 16000: 15995: 15994: 15993: 15988: 15983: 15978: 15968: 15966:Density matrix 15963: 15958: 15953: 15947: 15945: 15941: 15940: 15938: 15937: 15932: 15927: 15922: 15921: 15920: 15910: 15904: 15902: 15898: 15897: 15890: 15889: 15882: 15875: 15867: 15861: 15860: 15851: 15842: 15827: 15797: 15778: 15776:on 2014-08-26. 15759: 15746: 15745:External links 15743: 15742: 15741: 15735: 15726:Quantum (book) 15720: 15714: 15699: 15693: 15678: 15672: 15657: 15651: 15636: 15630: 15615: 15609: 15594: 15588: 15573: 15567: 15552: 15546: 15531: 15525: 15510: 15504: 15489: 15483: 15468: 15462: 15453:Modern Physics 15445: 15442: 15441: 15440: 15434: 15419: 15413: 15398: 15392: 15377: 15371: 15356: 15350: 15335: 15329: 15314: 15308: 15288: 15282: 15262: 15256: 15243: 15237: 15224: 15218: 15196: 15190: 15167: 15161: 15146: 15140: 15125: 15119: 15099: 15096: 15094: 15091: 15090: 15089: 15060: 15034:(3): L55–L61. 15023: 14969: 14930: 14883: 14857:(22): 220301. 14846: 14817: 14788: 14751: 14746: 14707: 14692: 14650: 14647: 14646: 14645: 14639: 14624: 14618: 14603: 14594: 14568: 14565: 14564: 14563: 14554: 14533: 14527: 14512: 14506: 14491: 14485: 14470: 14464: 14448: 14442: 14427: 14418: 14403: 14397: 14374: 14368: 14353: 14347: 14332: 14326: 14311: 14305: 14290: 14284: 14268: 14262: 14246: 14240: 14225: 14219: 14203: 14190: 14184: 14169: 14163: 14150: 14144: 14131: 14125: 14112: 14106: 14088: 14082: 14067: 14061: 14046: 14040: 14020: 14019:Selected books 14017: 14014: 14013: 13972: 13881:(2): 259–276. 13854:(2): 222–232. 13826:(4): 549–572. 13810: 13789:(3): 241–243. 13764: 13751:(3): 195–200. 13726: 13705:(3): 485–491. 13683: 13669: 13649: 13642: 13624: 13580: 13573: 13547: 13540: 13514: 13507: 13486: 13479: 13455: 13448: 13419: 13356: 13349: 13329: 13322: 13287: 13280: 13252: 13245: 13225: 13218: 13198: 13191: 13168: 13161: 13135: 13128: 13108: 13071:(11): 113011. 13055: 13034: 13015:(2): 226–267. 12999: 12950:(1): 149–204. 12927: 12868: 12815: 12808: 12790: 12782: 12755: 12748: 12730: 12723: 12697: 12690: 12672: 12665: 12645: 12638: 12620: 12613: 12590: 12583: 12565: 12558: 12540: 12533: 12515: 12508: 12487: 12480: 12460: 12453: 12429: 12422: 12395: 12374: 12206: 12199: 12181: 12174: 12154: 12147: 12121: 12076: 12069: 12048: 12041: 12021: 12014: 11985: 11978: 11958: 11951: 11931: 11924: 11904: 11897: 11864: 11857: 11836: 11835: 11833: 11830: 11828: 11827: 11814: 11811: 11810: 11796: 11788: 11784: 11780: 11779: 11774: 11770: 11766: 11765: 11760: 11756: 11752: 11751: 11746: 11742: 11738: 11737: 11735: 11730: 11725: 11719: 11716: 11715: 11712: 11709: 11708: 11706: 11701: 11698: 11680: 11649: 11643: 11640: 11639: 11636: 11633: 11632: 11630: 11625: 11622: 11600: 11592: 11588: 11584: 11583: 11578: 11574: 11570: 11569: 11567: 11562: 11559: 11543: 11539:Pauli matrices 11535:tensor indices 11522: 11513: 11501: 11499: 11496: 11492: 11491: 11487: 11486: 11481: 11476: 11471: 11469:Standard model 11466: 11461: 11459:CPT invariance 11456: 11451: 11446: 11441: 11436: 11434:Zitterbewegung 11429: 11426: 11424: 11420: 11419: 11414: 11409: 11404: 11399: 11394: 11389: 11382: 11379: 11378: 11377: 11372: 11367: 11365:Breit equation 11362: 11355: 11352: 11347: 11345: 11342: 11341: 11340: 11325: 11314: 11305: 11302: 11281: 11266: 11243: 11228: 11218:Main article: 11215: 11214:The Lamb shift 11212: 11204:Bell's theorem 11200:EM 4-potential 11154: 11151: 11150: 11149: 11126: 11107: 11092: 11085: 11082:electric field 11074: 11071:atomic nucleus 11059:Alpha particle 11055: 11044: 11027:discovers the 11019: 11016: 10984:Compton effect 10976:spectral lines 10972:fine structure 10952: 10949: 10925:atomic physics 10912: 10909: 10895: 10894: 10883: 10877: 10870: 10867: 10860: 10856: 10847: 10843: 10839: 10833: 10829: 10825: 10818: 10814: 10809: 10805: 10802: 10797: 10790: 10787: 10780: 10776: 10772: 10767: 10764: 10758: 10755: 10712: 10711: 10700: 10694: 10687: 10684: 10677: 10673: 10665: 10661: 10655: 10651: 10647: 10640: 10636: 10631: 10627: 10624: 10619: 10612: 10609: 10602: 10598: 10594: 10589: 10586: 10580: 10577: 10560: 10559: 10548: 10540: 10536: 10530: 10526: 10522: 10515: 10511: 10507: 10481: 10480: 10469: 10458: 10453: 10449: 10445: 10441: 10437: 10432: 10429: 10422: 10418: 10411: 10407: 10403: 10396: 10392: 10388: 10328: 10325: 10317: 10316: 10305: 10301: 10297: 10293: 10289: 10286: 10283: 10279: 10274: 10268: 10264: 10258: 10255: 10251: 10245: 10242: 10239: 10236: 10232: 10226: 10223: 10218: 10213: 10209: 10190: 10189: 10178: 10174: 10170: 10166: 10162: 10159: 10152: 10148: 10144: 10139: 10135: 10131: 10127: 10123: 10119: 10114: 10108: 10104: 10098: 10094: 10088: 10085: 10082: 10079: 10075: 10067: 10063: 10059: 10054: 10049: 10046: 10042: 10036: 10033: 10029: 10025: 10022: 10017: 10014: 10010: 9985:centre of mass 9933: 9932: 9921: 9916: 9912: 9908: 9904: 9900: 9895: 9889: 9886: 9882: 9876: 9873: 9869: 9865: 9862: 9857: 9853: 9847: 9843: 9839: 9834: 9830: 9824: 9820: 9816: 9811: 9808: 9804: 9754: 9751: 9736: 9735: 9724: 9720: 9715: 9711: 9705: 9701: 9697: 9694: 9691: 9686: 9682: 9676: 9672: 9666: 9662: 9656: 9653: 9649: 9643: 9638: 9634: 9628: 9625: 9620: 9604: 9603: 9592: 9589: 9586: 9583: 9580: 9575: 9571: 9565: 9561: 9557: 9552: 9549: 9544: 9539: 9523: 9522: 9510: 9507: 9501: 9498: 9491: 9486: 9480: 9476: 9470: 9467: 9462: 9458: 9454: 9451: 9444: 9439: 9433: 9427: 9423: 9387: 9384: 9372: 9371: 9359: 9352: 9348: 9341: 9335: 9332: 9325: 9319: 9316: 9311: 9305: 9301: 9279: 9278: 9264: 9260: 9251: 9248: 9243: 9237: 9233: 9201: 9198: 9127: 9126: 9113: 9107: 9104: 9101: 9097: 9093: 9088: 9084: 9080: 9079: 9076: 9073: 9070: 9066: 9062: 9057: 9053: 9049: 9048: 9046: 9041: 9038: 9035: 9032: 9028: 9024: 9021: 8991: 8990: 8977: 8969: 8964: 8961: 8958: 8954: 8950: 8945: 8942: 8939: 8936: 8932: 8929: 8925: 8919: 8917: 8914: 8910: 8905: 8902: 8899: 8895: 8891: 8886: 8883: 8880: 8877: 8874: 8870: 8867: 8863: 8857: 8853: 8848: 8845: 8842: 8838: 8834: 8829: 8826: 8823: 8819: 8816: 8812: 8806: 8805: 8803: 8796: 8791: 8788: 8785: 8781: 8777: 8774: 8768: 8762: 8756: 8753: 8750: 8746: 8742: 8737: 8734: 8731: 8728: 8724: 8721: 8717: 8713: 8712: 8709: 8706: 8703: 8699: 8695: 8690: 8687: 8684: 8681: 8678: 8675: 8671: 8668: 8664: 8660: 8659: 8656: 8653: 8652: 8649: 8646: 8643: 8639: 8635: 8630: 8627: 8624: 8621: 8618: 8614: 8611: 8607: 8603: 8602: 8599: 8596: 8593: 8589: 8585: 8580: 8577: 8574: 8570: 8567: 8563: 8559: 8558: 8555: 8552: 8549: 8545: 8541: 8536: 8533: 8530: 8527: 8523: 8520: 8516: 8512: 8511: 8508: 8505: 8502: 8498: 8494: 8489: 8486: 8483: 8480: 8477: 8474: 8470: 8467: 8463: 8459: 8458: 8455: 8452: 8451: 8448: 8445: 8442: 8438: 8434: 8429: 8426: 8423: 8420: 8417: 8413: 8410: 8406: 8402: 8401: 8398: 8395: 8392: 8388: 8384: 8379: 8376: 8373: 8369: 8366: 8362: 8358: 8357: 8355: 8350: 8347: 8344: 8341: 8337: 8333: 8330: 8260: 8259: 8246: 8238: 8233: 8230: 8227: 8223: 8219: 8214: 8211: 8208: 8205: 8201: 8195: 8191: 8186: 8183: 8180: 8176: 8172: 8167: 8164: 8161: 8158: 8155: 8152: 8148: 8142: 8140: 8137: 8133: 8128: 8125: 8122: 8118: 8114: 8109: 8106: 8103: 8100: 8097: 8093: 8087: 8083: 8078: 8075: 8072: 8068: 8064: 8059: 8056: 8053: 8049: 8043: 8042: 8040: 8035: 8030: 8025: 8022: 8019: 8015: 8011: 8008: 8002: 7996: 7990: 7987: 7984: 7980: 7976: 7971: 7968: 7965: 7962: 7958: 7954: 7953: 7950: 7947: 7944: 7940: 7936: 7931: 7928: 7925: 7922: 7919: 7916: 7912: 7908: 7907: 7904: 7901: 7900: 7897: 7894: 7891: 7887: 7883: 7878: 7875: 7872: 7869: 7866: 7862: 7858: 7857: 7854: 7851: 7848: 7844: 7840: 7835: 7832: 7829: 7825: 7821: 7820: 7818: 7813: 7810: 7807: 7804: 7800: 7796: 7793: 7775:column vectors 7752:free particles 7731: 7728: 7727: 7726: 7713: 7706: 7702: 7695: 7689: 7683: 7679: 7672: 7666: 7663: 7630: 7626: 7620: 7616: 7610: 7606: 7602: 7599: 7594: 7590: 7522: 7500: 7499: 7483: 7479: 7473: 7469: 7463: 7459: 7455: 7452: 7447: 7443: 7434: 7430: 7423: 7417: 7411: 7407: 7400: 7393: 7388: 7383: 7376: 7372: 7364: 7358: 7354: 7347: 7341: 7338: 7300: 7297: 7273:Breit equation 7258: 7257: 7246: 7240: 7237: 7233: 7229: 7226: 7221: 7217: 7211: 7207: 7203: 7198: 7194: 7188: 7184: 7180: 7175: 7170: 7164: 7160: 7156: 7151: 7147: 7142: 7099: 7093: 7092: 7079: 7075: 7069: 7066: 7062: 7058: 7055: 7050: 7045: 7039: 7035: 7031: 7026: 7022: 7017: 7011: 7005: 7001: 6995: 6992: 6989: 6985: 6981: 6978: 6975: 6971: 6965: 6961: 6957: 6952: 6948: 6943: 6927: 6925:anticommutator 6893: 6874: 6873: 6862: 6858: 6855: 6850: 6846: 6840: 6833: 6830: 6821: 6817: 6812: 6808: 6805: 6800: 6796: 6790: 6783: 6780: 6771: 6767: 6762: 6758: 6755: 6750: 6746: 6741: 6734: 6730: 6723: 6719: 6715: 6710: 6705: 6702: 6694: 6689: 6685: 6682: 6677: 6673: 6668: 6661: 6657: 6650: 6646: 6642: 6637: 6632: 6629: 6621: 6571:Dirac equation 6556:corresponding 6552:negated charge 6550:momentum, and 6515: 6466: 6459: 6458: 6445: 6437: 6434: 6430: 6426: 6425: 6420: 6417: 6413: 6409: 6408: 6403: 6400: 6396: 6392: 6391: 6386: 6383: 6379: 6375: 6374: 6372: 6367: 6362: 6354: 6350: 6346: 6345: 6340: 6336: 6332: 6331: 6329: 6324: 6321: 6294: 6287: 6280: 6266: 6259: 6252: 6234:gamma matrices 6226: 6225: 6214: 6211: 6208: 6204: 6198: 6194: 6190: 6187: 6184: 6179: 6175: 6171: 6168: 6163: 6156: 6153: 6146: 6141: 6137: 6132: 6127: 6123: 6119: 6113: 6109: 6105: 6102: 6098: 6094: 6090: 6087: 6081: 6077: 6070: 6066: 6062: 6058: 6054: 6048: 6044: 6040: 6037: 6033: 6029: 6026: 6023: 6017: 6014: 6010: 6005: 6002: 5998: 5936: 5935: 5922: 5914: 5910: 5906: 5905: 5900: 5896: 5892: 5891: 5889: 5884: 5881: 5857:Pauli matrices 5853: 5852: 5841: 5838: 5835: 5830: 5825: 5822: 5818: 5814: 5811: 5807: 5803: 5800: 5796: 5792: 5784: 5781: 5777: 5772: 5766: 5763: 5756: 5752: 5748: 5742: 5737: 5734: 5730: 5726: 5723: 5719: 5715: 5712: 5708: 5704: 5696: 5693: 5689: 5683: 5679: 5676: 5672: 5668: 5665: 5662: 5656: 5653: 5649: 5644: 5641: 5637: 5614:Pauli equation 5600:Main article: 5597: 5579: 5578: 5577: 5566: 5563: 5560: 5557: 5552: 5547: 5543: 5539: 5536: 5530: 5526: 5519: 5511: 5508: 5504: 5499: 5493: 5490: 5483: 5479: 5474: 5469: 5465: 5461: 5458: 5452: 5448: 5441: 5433: 5430: 5426: 5421: 5418: 5414: 5410: 5407: 5404: 5398: 5395: 5391: 5386: 5383: 5379: 5287: 5286: 5275: 5272: 5269: 5265: 5259: 5254: 5251: 5248: 5245: 5240: 5236: 5231: 5227: 5223: 5220: 5215: 5208: 5205: 5198: 5193: 5188: 5184: 5180: 5177: 5172: 5165: 5162: 5155: 5150: 5145: 5141: 5136: 5132: 5126: 5122: 5118: 5115: 5112: 5109: 5104: 5099: 5095: 5091: 5088: 5082: 5078: 5071: 5064: 5060: 5056: 5053: 5048: 5043: 5040: 5037: 5034: 5028: 5025: 5019: 5002: 4999: 4995: 4994: 4983: 4978: 4974: 4971: 4967: 4962: 4956: 4952: 4948: 4945: 4942: 4939: 4936: 4933: 4919:four-potential 4912: 4895: 4889: 4888: 4875: 4871: 4867: 4864: 4859: 4852: 4849: 4842: 4837: 4830: 4827: 4819: 4814: 4810: 4807: 4801: 4797: 4790: 4784: 4780: 4772: 4768: 4765: 4762: 4756: 4753: 4747: 4741: 4738: 4661: 4658: 4650: 4649: 4638: 4634: 4631: 4626: 4622: 4616: 4612: 4607: 4603: 4600: 4596: 4592: 4589: 4586: 4580: 4577: 4572: 4569: 4530:charge density 4498: 4497: 4486: 4481: 4477: 4471: 4467: 4463: 4460: 4457: 4452: 4448: 4442: 4438: 4434: 4428: 4425: 4420: 4417: 4411: 4406: 4402: 4397: 4392: 4386: 4382: 4378: 4375: 4372: 4369: 4366: 4361: 4357: 4352: 4345: 4342: 4337: 4334: 4328: 4325: 4321: 4316: 4311: 4304: 4301: 4294: 4290: 4286: 4280: 4277: 4271: 4268: 4263: 4260: 4252: 4248: 4243: 4234: 4230: 4226: 4223: 4218: 4215: 4209: 4206: 4155: 4154: 4143: 4138: 4134: 4128: 4124: 4118: 4114: 4110: 4105: 4101: 4096: 4092: 4088: 4082: 4078: 4072: 4068: 4064: 4060: 4055: 4052: 4047: 4043: 4039: 4036: 4022:Dirac equation 3975:. This is the 3950:square modulus 3945: 3942: 3878:Dirac equation 3874: 3873: 3860: 3856: 3852: 3849: 3846: 3840: 3836: 3829: 3825: 3821: 3818: 3812: 3809: 3802: 3798: 3795: 3792: 3788: 3782: 3778: 3774: 3771: 3768: 3762: 3758: 3751: 3747: 3743: 3740: 3734: 3731: 3724: 3709: 3708: 3697: 3693: 3690: 3685: 3681: 3677: 3672: 3667: 3663: 3644: 3643: 3632: 3626: 3622: 3616: 3612: 3608: 3605: 3600: 3596: 3590: 3586: 3581: 3576: 3572: 3568: 3565: 3562: 3559: 3554: 3550: 3508: 3501: 3494: 3483: 3482: 3471: 3467: 3464: 3461: 3457: 3451: 3447: 3443: 3440: 3437: 3431: 3427: 3420: 3416: 3412: 3409: 3403: 3400: 3393: 3388: 3382: 3378: 3374: 3371: 3368: 3362: 3358: 3351: 3347: 3343: 3340: 3334: 3331: 3324: 3301: 3300: 3289: 3285: 3280: 3276: 3270: 3266: 3262: 3259: 3256: 3253: 3247: 3243: 3236: 3230: 3226: 3217: 3213: 3209: 3206: 3201: 3194: 3191: 3168: 3165: 3161: 3160: 3149: 3143: 3139: 3132: 3126: 3122: 3115: 3112: 3109: 3105: 3101: 3095: 3092: 3086: 3080: 3077: 3034: 3033: 3020: 3013: 3010: 3003: 2999: 2995: 2992: 2987: 2980: 2977: 2954:magnetic field 2934:is the (spin) 2926: 2925: 2914: 2910: 2905: 2901: 2897: 2894: 2891: 2887: 2882: 2877: 2872: 2867: 2860: 2856: 2847: 2841: 2837: 2833: 2827: 2824: 2819: 2812: 2809: 2784: 2719: 2706: 2670: 2669: 2658: 2650: 2646: 2640: 2636: 2632: 2629: 2626: 2620: 2616: 2609: 2603: 2599: 2590: 2586: 2580: 2577: 2571: 2568: 2564: 2559: 2556: 2552: 2544: 2540: 2534: 2530: 2526: 2523: 2520: 2514: 2510: 2503: 2497: 2493: 2484: 2480: 2474: 2468: 2465: 2459: 2453: 2450: 2428: 2427: 2416: 2413: 2410: 2406: 2402: 2399: 2396: 2390: 2387: 2379: 2375: 2368: 2362: 2358: 2348: 2342: 2339: 2287:kinetic energy 2271:Main article: 2268: 2265: 2176: 2175: 2164: 2161: 2158: 2155: 2151: 2147: 2142: 2139: 2135: 2131: 2126: 2122: 2118: 2115: 2112: 2109: 2106: 2103: 2100: 2097: 2093: 2089: 2084: 2080: 2056:representation 2049: 1932: 1925: 1918: 1907: 1900: 1893: 1874: 1873:Space and time 1871: 1841: 1840: 1829: 1825: 1819: 1816: 1812: 1807: 1804: 1798: 1795: 1789: 1786: 1783: 1777: 1774: 1770: 1765: 1762: 1756: 1753: 1749: 1686: 1685: 1674: 1670: 1667: 1664: 1661: 1658: 1652: 1648: 1640: 1633: 1630: 1626: 1621: 1618: 1615: 1609: 1606: 1578: 1577: 1566: 1560: 1556: 1550: 1546: 1542: 1539: 1536: 1532: 1528: 1524: 1518: 1514: 1510: 1505: 1501: 1474: 1470: 1466: 1460: 1457: 1385:periodic table 1367:(1939) due to 1200: 1199: 1188: 1182: 1179: 1173: 1170: 1164: 1161: 1157: 1152: 1149: 1131:time evolution 1115: 1112: 1096:Gaussian units 1033:Dirac equation 1021:fine structure 957:refers to the 943:atomic physics 919:speed of light 882: 881: 879: 878: 871: 864: 856: 853: 852: 848: 847: 842: 837: 832: 827: 822: 817: 812: 807: 802: 797: 792: 787: 782: 777: 772: 767: 762: 757: 752: 747: 742: 737: 732: 727: 722: 717: 712: 707: 702: 697: 692: 687: 682: 677: 672: 667: 662: 657: 652: 647: 642: 637: 632: 627: 622: 616: 615: 612: 611: 608: 607: 604: 603: 598: 593: 588: 586:Density matrix 583: 578: 573: 568: 563: 558: 552: 549: 548: 545: 544: 540: 539: 534: 529: 524: 519: 514: 509: 508: 507: 506: 505: 490: 485: 480: 475: 470: 464: 463: 458: 457: 454: 453: 449: 448: 443: 438: 433: 428: 422: 421: 418: 417: 414: 413: 409: 408: 403: 398: 393: 388: 383: 377: 376: 375: 369: 366: 365: 362: 361: 357: 356: 351: 346: 340: 339: 338: 337: 336: 334:Delayed-choice 329:Quantum eraser 324: 323: 318: 313: 308: 303: 298: 293: 288: 283: 278: 272: 271: 268: 267: 264: 263: 259: 258: 257: 256: 246: 241: 236: 231: 226: 221: 219:Quantum number 216: 211: 206: 201: 196: 191: 185: 184: 181: 180: 177: 176: 172: 171: 166: 160: 159: 158: 153: 148: 142: 139: 138: 135: 134: 133: 132: 127: 122: 114: 113: 108: 97: 94: 90: 83: 80: 74: 71: 68: 64: 57: 54: 50: 45: 42: 31: 30: 24: 23: 15: 9: 6: 4: 3: 2: 16702: 16691: 16688: 16686: 16683: 16681: 16678: 16676: 16673: 16671: 16668: 16666: 16663: 16662: 16660: 16645: 16637: 16636: 16633: 16627: 16624: 16622: 16619: 16617: 16614: 16610: 16607: 16606: 16605: 16602: 16601: 16599: 16595: 16589: 16586: 16584: 16581: 16577: 16574: 16573: 16572: 16569: 16567: 16564: 16562: 16559: 16557: 16554: 16553: 16551: 16547: 16541: 16538: 16536: 16533: 16531: 16528: 16526: 16523: 16521: 16518: 16516: 16513: 16511: 16508: 16506: 16503: 16501: 16498: 16496: 16493: 16491: 16488: 16486: 16483: 16481: 16480:Quantum logic 16478: 16476: 16473: 16471: 16468: 16466: 16463: 16461: 16458: 16456: 16453: 16451: 16448: 16446: 16443: 16439: 16436: 16435: 16434: 16431: 16429: 16426: 16424: 16421: 16419: 16416: 16412: 16409: 16408: 16407: 16404: 16402: 16399: 16397: 16394: 16392: 16389: 16388: 16386: 16384: 16380: 16374: 16371: 16369: 16366: 16364: 16361: 16359: 16356: 16354: 16351: 16349: 16346: 16344: 16341: 16339: 16336: 16334: 16333:Quantum chaos 16331: 16329: 16326: 16324: 16321: 16320: 16318: 16316: 16312: 16306: 16303: 16301: 16300:Stern–Gerlach 16298: 16296: 16293: 16291: 16288: 16286: 16283: 16281: 16278: 16276: 16273: 16271: 16268: 16266: 16263: 16261: 16258: 16256: 16253: 16252: 16250: 16246: 16240: 16237: 16235: 16234:Transactional 16232: 16230: 16227: 16225: 16224:Quantum logic 16222: 16220: 16217: 16215: 16212: 16206: 16203: 16202: 16201: 16198: 16197: 16196: 16193: 16191: 16188: 16186: 16183: 16181: 16178: 16176: 16173: 16171: 16168: 16167: 16165: 16163: 16159: 16153: 16150: 16148: 16145: 16143: 16140: 16138: 16135: 16133: 16130: 16129: 16127: 16123: 16117: 16114: 16112: 16109: 16107: 16104: 16102: 16099: 16097: 16094: 16092: 16089: 16087: 16084: 16083: 16081: 16077: 16069: 16066: 16064: 16061: 16060: 16059: 16058:Wave function 16056: 16054: 16051: 16049: 16046: 16044: 16041: 16039: 16036: 16034: 16033:Superposition 16031: 16029: 16028:Quantum state 16026: 16024: 16021: 16019: 16016: 16014: 16011: 16009: 16006: 16004: 16001: 15999: 15996: 15992: 15989: 15987: 15984: 15982: 15981:Excited state 15979: 15977: 15974: 15973: 15972: 15969: 15967: 15964: 15962: 15959: 15957: 15954: 15952: 15949: 15948: 15946: 15942: 15936: 15933: 15931: 15928: 15926: 15923: 15919: 15916: 15915: 15914: 15911: 15909: 15906: 15905: 15903: 15899: 15895: 15888: 15883: 15881: 15876: 15874: 15869: 15868: 15865: 15857: 15854:Arteha, S.N. 15852: 15848: 15843: 15839: 15834: 15828: 15818:on 2020-12-19 15814: 15810: 15803: 15798: 15794: 15790: 15783: 15779: 15772: 15765: 15760: 15756: 15755: 15749: 15748: 15738: 15732: 15728: 15727: 15721: 15717: 15711: 15707: 15706: 15700: 15696: 15690: 15686: 15685: 15679: 15675: 15669: 15665: 15664: 15658: 15654: 15648: 15644: 15643: 15637: 15633: 15627: 15623: 15622: 15616: 15612: 15606: 15602: 15601: 15595: 15591: 15585: 15581: 15580: 15574: 15570: 15564: 15560: 15559: 15553: 15549: 15543: 15539: 15538: 15532: 15528: 15522: 15518: 15517: 15511: 15507: 15501: 15497: 15496: 15490: 15486: 15480: 15476: 15475: 15469: 15465: 15459: 15455: 15454: 15448: 15447: 15437: 15431: 15427: 15426: 15420: 15416: 15410: 15406: 15405: 15399: 15395: 15389: 15385: 15384: 15378: 15374: 15368: 15364: 15363: 15357: 15353: 15347: 15343: 15342: 15336: 15332: 15326: 15322: 15321: 15315: 15311: 15305: 15300: 15299: 15293: 15289: 15285: 15279: 15274: 15273: 15267: 15263: 15259: 15253: 15249: 15244: 15240: 15234: 15230: 15225: 15221: 15215: 15211: 15210: 15205: 15201: 15197: 15193: 15187: 15183: 15179: 15175: 15174: 15168: 15164: 15158: 15154: 15153: 15147: 15143: 15137: 15133: 15132: 15126: 15122: 15116: 15112: 15111: 15106: 15102: 15101: 15086: 15082: 15078: 15074: 15070: 15066: 15061: 15057: 15053: 15049: 15045: 15041: 15037: 15033: 15029: 15024: 15020: 15016: 15012: 15008: 15004: 15000: 14995: 14990: 14986: 14982: 14975: 14970: 14966: 14962: 14958: 14954: 14949: 14944: 14940: 14936: 14931: 14927: 14923: 14919: 14915: 14911: 14907: 14902: 14897: 14893: 14889: 14888:J. Math. Phys 14884: 14880: 14876: 14872: 14868: 14864: 14860: 14856: 14852: 14847: 14843: 14839: 14835: 14831: 14827: 14823: 14818: 14814: 14810: 14806: 14802: 14798: 14794: 14789: 14785: 14781: 14777: 14773: 14769: 14765: 14761: 14757: 14752: 14747: 14743: 14739: 14734: 14729: 14725: 14721: 14717: 14713: 14708: 14701: 14697: 14693: 14688: 14683: 14679: 14675: 14671: 14667: 14666: 14661: 14657: 14656:Dirac, P.A.M. 14653: 14652: 14642: 14636: 14632: 14631: 14625: 14621: 14615: 14611: 14610: 14604: 14601: 14597: 14595:9780486602691 14591: 14587: 14582: 14581: 14575: 14571: 14570: 14561: 14557: 14551: 14547: 14542: 14541: 14534: 14530: 14524: 14520: 14519: 14513: 14509: 14503: 14499: 14498: 14492: 14488: 14482: 14478: 14477: 14471: 14467: 14461: 14457: 14453: 14449: 14445: 14439: 14435: 14434: 14428: 14425: 14421: 14415: 14411: 14410: 14404: 14400: 14394: 14390: 14386: 14382: 14381: 14375: 14371: 14365: 14361: 14360: 14354: 14350: 14344: 14340: 14339: 14333: 14329: 14323: 14319: 14318: 14312: 14308: 14302: 14298: 14297: 14291: 14287: 14281: 14277: 14273: 14272:Tannoudji, C. 14269: 14265: 14259: 14255: 14251: 14250:Tannoudji, C. 14247: 14243: 14237: 14233: 14232: 14226: 14222: 14216: 14212: 14208: 14204: 14199: 14198: 14191: 14187: 14181: 14177: 14176: 14170: 14166: 14160: 14156: 14151: 14147: 14141: 14137: 14132: 14128: 14122: 14118: 14113: 14109: 14103: 14099: 14098: 14093: 14089: 14085: 14079: 14075: 14074: 14068: 14064: 14058: 14054: 14053: 14047: 14043: 14037: 14033: 14032: 14027: 14026:Dirac, P.A.M. 14023: 14022: 14009: 14005: 14000: 13995: 13991: 13987: 13983: 13976: 13968: 13964: 13960: 13956: 13953:(1): 98–106. 13952: 13948: 13940: 13936: 13932: 13928: 13924: 13920: 13916: 13912: 13904: 13900: 13896: 13892: 13888: 13884: 13880: 13876: 13869: 13865: 13861: 13857: 13853: 13849: 13841: 13837: 13833: 13829: 13825: 13821: 13814: 13805: 13800: 13796: 13792: 13788: 13784: 13783: 13778: 13774: 13768: 13759: 13754: 13750: 13746: 13745: 13737: 13730: 13721: 13716: 13712: 13708: 13704: 13700: 13699: 13694: 13687: 13681: 13679: 13672: 13666: 13662: 13661: 13653: 13645: 13639: 13635: 13628: 13619: 13614: 13610: 13606: 13602: 13598: 13591: 13584: 13576: 13570: 13566: 13561: 13560: 13551: 13543: 13537: 13533: 13528: 13527: 13518: 13510: 13504: 13500: 13496: 13490: 13482: 13476: 13472: 13468: 13467: 13459: 13451: 13445: 13441: 13436: 13435: 13429: 13428:Jackson, J.D. 13423: 13415: 13411: 13407: 13403: 13399: 13395: 13390: 13385: 13381: 13377: 13370: 13363: 13361: 13352: 13346: 13342: 13341: 13333: 13325: 13319: 13315: 13311: 13310: 13305: 13304:Wheeler, J.A. 13301: 13297: 13291: 13283: 13277: 13273: 13269: 13265: 13264: 13256: 13248: 13242: 13238: 13237: 13229: 13221: 13215: 13211: 13210: 13202: 13194: 13188: 13184: 13183: 13178: 13172: 13164: 13158: 13154: 13149: 13148: 13147:Supersymmetry 13139: 13131: 13125: 13121: 13120: 13112: 13104: 13100: 13096: 13092: 13088: 13084: 13079: 13074: 13070: 13066: 13059: 13050: 13045: 13038: 13030: 13026: 13022: 13018: 13014: 13010: 13003: 12989:on 2015-10-04 12985: 12981: 12977: 12973: 12969: 12965: 12961: 12957: 12953: 12949: 12945: 12938: 12931: 12923: 12919: 12914: 12909: 12904: 12899: 12895: 12891: 12888:(5): 211–23. 12887: 12883: 12879: 12872: 12864: 12860: 12856: 12852: 12848: 12844: 12839: 12834: 12830: 12826: 12819: 12811: 12805: 12801: 12800:Supersymmetry 12794: 12785: 12779: 12775: 12771: 12770: 12762: 12760: 12751: 12745: 12741: 12734: 12726: 12720: 12716: 12711: 12710: 12701: 12693: 12687: 12683: 12676: 12668: 12662: 12658: 12657: 12649: 12641: 12635: 12631: 12624: 12616: 12610: 12606: 12605: 12597: 12595: 12586: 12580: 12576: 12569: 12561: 12555: 12551: 12544: 12536: 12530: 12526: 12519: 12511: 12505: 12501: 12494: 12492: 12483: 12477: 12473: 12472: 12464: 12456: 12450: 12446: 12442: 12441: 12433: 12425: 12419: 12415: 12411: 12410: 12402: 12400: 12390: 12385: 12378: 12364:on 2022-03-25 12360: 12356: 12352: 12348: 12344: 12340: 12336: 12329: 12327: 12310:on 2022-03-09 12306: 12302: 12298: 12294: 12290: 12286: 12282: 12275: 12273: 12256:on 2020-12-04 12252: 12248: 12244: 12240: 12236: 12232: 12228: 12221: 12219: 12210: 12202: 12196: 12192: 12185: 12177: 12171: 12167: 12166: 12158: 12150: 12144: 12140: 12135: 12134: 12125: 12117: 12113: 12109: 12105: 12100: 12095: 12092:(1): 88–100. 12091: 12087: 12080: 12072: 12066: 12062: 12058: 12052: 12044: 12038: 12034: 12033: 12025: 12017: 12011: 12007: 12002: 12001: 11992: 11990: 11981: 11975: 11971: 11970: 11962: 11954: 11948: 11944: 11943: 11935: 11927: 11921: 11917: 11916: 11908: 11900: 11894: 11890: 11885: 11884: 11875: 11873: 11871: 11869: 11860: 11854: 11850: 11849: 11841: 11837: 11824: 11818: 11794: 11786: 11782: 11772: 11768: 11758: 11754: 11744: 11740: 11733: 11728: 11723: 11717: 11710: 11704: 11699: 11696: 11689: 11688: 11684: 11647: 11641: 11634: 11628: 11623: 11620: 11598: 11590: 11586: 11576: 11572: 11565: 11560: 11557: 11547: 11540: 11536: 11531: 11525: 11516: 11506: 11502: 11490: 11485: 11482: 11480: 11477: 11475: 11472: 11470: 11467: 11465: 11462: 11460: 11457: 11455: 11452: 11450: 11447: 11445: 11442: 11440: 11437: 11435: 11432: 11431: 11425: 11423: 11418: 11415: 11413: 11410: 11408: 11405: 11403: 11400: 11398: 11397:Spinor bundle 11395: 11393: 11390: 11388: 11385: 11384: 11376: 11373: 11371: 11368: 11366: 11363: 11361: 11358: 11357: 11351: 11350: 11338: 11334: 11330: 11326: 11323: 11319: 11315: 11312: 11308: 11307: 11301: 11299: 11295: 11280: 11265: 11261: 11257: 11242: 11227: 11221: 11211: 11209: 11205: 11201: 11197: 11193: 11189: 11185: 11181: 11177: 11172: 11168: 11164: 11160: 11147: 11143: 11142:time dilation 11139: 11135: 11131: 11127: 11124: 11120: 11116: 11112: 11108: 11105: 11101: 11100:energy levels 11097: 11093: 11090: 11086: 11083: 11079: 11075: 11072: 11068: 11064: 11060: 11056: 11053: 11049: 11045: 11042: 11041:spectral line 11038: 11037:Zeeman effect 11034: 11030: 11026: 11025:J. J. Thomson 11022: 11021: 11015: 11013: 11009: 11005: 11001: 10997: 10993: 10989: 10985: 10981: 10977: 10973: 10969: 10965: 10961: 10957: 10948: 10946: 10942: 10938: 10934: 10930: 10926: 10922: 10918: 10908: 10881: 10875: 10858: 10854: 10845: 10841: 10837: 10827: 10816: 10807: 10803: 10800: 10795: 10778: 10774: 10765: 10762: 10753: 10743: 10742: 10741: 10739: 10735: 10719: 10698: 10692: 10675: 10671: 10663: 10659: 10649: 10638: 10629: 10625: 10622: 10617: 10600: 10596: 10587: 10584: 10575: 10565: 10564: 10563: 10546: 10538: 10534: 10524: 10513: 10509: 10496: 10495: 10494: 10491: 10487: 10467: 10456: 10451: 10447: 10443: 10439: 10435: 10430: 10427: 10420: 10416: 10405: 10394: 10390: 10377: 10376: 10375: 10372: 10367: 10362: 10356: 10350: 10345: 10340: 10338: 10334: 10331:In 1926, the 10324: 10322: 10295: 10284: 10281: 10266: 10262: 10256: 10253: 10249: 10243: 10240: 10237: 10234: 10230: 10224: 10221: 10216: 10211: 10207: 10199: 10198: 10197: 10195: 10168: 10157: 10150: 10146: 10142: 10137: 10129: 10121: 10106: 10102: 10096: 10092: 10086: 10083: 10080: 10077: 10073: 10065: 10061: 10057: 10052: 10044: 10040: 10034: 10027: 10023: 10020: 10015: 10012: 10008: 10000: 9999: 9998: 9996: 9991: 9986: 9981: 9975: 9969: 9963: 9959: 9953: 9949: 9943: 9939: 9919: 9910: 9902: 9884: 9880: 9874: 9867: 9863: 9860: 9855: 9851: 9845: 9841: 9837: 9832: 9828: 9822: 9818: 9814: 9809: 9806: 9802: 9794: 9793: 9792: 9790: 9786: 9782: 9777: 9773: 9769: 9764: 9760: 9750: 9748: 9743: 9722: 9718: 9713: 9709: 9703: 9699: 9695: 9692: 9689: 9684: 9674: 9670: 9664: 9654: 9651: 9647: 9641: 9636: 9632: 9626: 9623: 9609: 9608: 9607: 9590: 9584: 9581: 9578: 9573: 9569: 9563: 9559: 9547: 9542: 9528: 9527: 9526: 9508: 9505: 9499: 9478: 9474: 9465: 9460: 9431: 9425: 9413: 9412: 9411: 9409: 9405: 9404: 9399: 9394: 9383: 9381: 9377: 9357: 9339: 9330: 9323: 9314: 9309: 9288: 9287: 9286: 9284: 9249: 9246: 9241: 9220: 9219: 9218: 9215: 9212: 9208: 9197: 9195: 9175: 9171: 9167: 9163: 9149: 9139: 9137: 9133: 9111: 9102: 9099: 9086: 9082: 9071: 9068: 9055: 9051: 9044: 9039: 9033: 9030: 9019: 9012: 9011: 9010: 9008: 9004: 9000: 8996: 8975: 8967: 8959: 8956: 8943: 8940: 8937: 8934: 8930: 8927: 8923: 8915: 8908: 8900: 8897: 8884: 8881: 8878: 8875: 8872: 8868: 8865: 8861: 8851: 8843: 8840: 8827: 8824: 8821: 8817: 8814: 8810: 8801: 8794: 8786: 8783: 8772: 8760: 8751: 8748: 8735: 8732: 8729: 8726: 8722: 8719: 8715: 8704: 8701: 8688: 8685: 8682: 8679: 8676: 8673: 8669: 8666: 8662: 8654: 8644: 8641: 8628: 8625: 8622: 8619: 8616: 8612: 8609: 8605: 8594: 8591: 8578: 8575: 8572: 8568: 8565: 8561: 8550: 8547: 8534: 8531: 8528: 8525: 8521: 8518: 8514: 8503: 8500: 8487: 8484: 8481: 8478: 8475: 8472: 8468: 8465: 8461: 8453: 8443: 8440: 8427: 8424: 8421: 8418: 8415: 8411: 8408: 8404: 8393: 8390: 8377: 8374: 8371: 8367: 8364: 8360: 8353: 8348: 8342: 8339: 8328: 8321: 8320: 8319: 8315: 8308: 8301: 8295: 8289: 8281: 8274: 8269: 8265: 8244: 8236: 8228: 8225: 8212: 8209: 8206: 8203: 8199: 8189: 8181: 8178: 8165: 8162: 8159: 8156: 8153: 8150: 8146: 8138: 8131: 8123: 8120: 8107: 8104: 8101: 8098: 8095: 8091: 8081: 8073: 8070: 8057: 8054: 8051: 8047: 8038: 8033: 8028: 8020: 8017: 8006: 7994: 7985: 7982: 7969: 7966: 7963: 7960: 7956: 7945: 7942: 7929: 7926: 7923: 7920: 7917: 7914: 7910: 7902: 7892: 7889: 7876: 7873: 7870: 7867: 7864: 7860: 7849: 7846: 7833: 7830: 7827: 7823: 7816: 7811: 7805: 7802: 7791: 7784: 7783: 7782: 7780: 7776: 7772: 7771:spinor fields 7767: 7765: 7764:Lorentz group 7761: 7757: 7753: 7711: 7693: 7687: 7670: 7661: 7651: 7650: 7649: 7646: 7628: 7618: 7614: 7608: 7604: 7597: 7592: 7588: 7560: 7557: 7553: 7548: 7527: 7521: 7517: 7513: 7509: 7505: 7481: 7471: 7467: 7461: 7457: 7450: 7445: 7441: 7421: 7415: 7398: 7362: 7345: 7336: 7326: 7325: 7324: 7322: 7316: 7310: 7306: 7296: 7294: 7274: 7271:In 1929, the 7269: 7267: 7263: 7244: 7238: 7235: 7231: 7227: 7224: 7219: 7215: 7209: 7205: 7201: 7196: 7192: 7186: 7182: 7178: 7173: 7168: 7162: 7158: 7154: 7149: 7145: 7140: 7131: 7130: 7129: 7126: 7122: 7118: 7114: 7110: 7107: 7102: 7077: 7073: 7067: 7064: 7060: 7056: 7053: 7048: 7043: 7037: 7033: 7029: 7024: 7020: 7015: 7009: 7003: 6999: 6993: 6990: 6987: 6983: 6979: 6976: 6973: 6969: 6963: 6959: 6955: 6950: 6946: 6941: 6933: 6932: 6931: 6926: 6922: 6918: 6915: 6911: 6907: 6903: 6898: 6892: 6887: 6883: 6879: 6878:Weyl equation 6860: 6856: 6853: 6848: 6844: 6838: 6828: 6819: 6815: 6810: 6806: 6803: 6798: 6794: 6788: 6778: 6769: 6765: 6756: 6753: 6748: 6744: 6739: 6721: 6713: 6708: 6700: 6692: 6687: 6683: 6680: 6675: 6671: 6666: 6648: 6640: 6635: 6627: 6619: 6611: 6610: 6609: 6606: 6604: 6600: 6596: 6592: 6587: 6581: 6577: 6572: 6568: 6564: 6560: 6559: 6553: 6549: 6548:time-reversed 6544: 6538: 6532: 6528: 6522: 6514: 6492: 6486: 6479: 6475: 6465: 6461:The 2-spinor 6443: 6432: 6428: 6415: 6411: 6398: 6394: 6381: 6377: 6370: 6365: 6360: 6352: 6348: 6338: 6334: 6327: 6322: 6319: 6312: 6311: 6310: 6307: 6302: 6293: 6286: 6279: 6275: 6272: 6265: 6258: 6251: 6247: 6243: 6239: 6235: 6231: 6212: 6209: 6206: 6202: 6196: 6192: 6188: 6185: 6177: 6173: 6169: 6166: 6161: 6151: 6139: 6135: 6130: 6121: 6117: 6111: 6107: 6103: 6100: 6088: 6085: 6064: 6056: 6052: 6046: 6042: 6038: 6035: 6031: 6027: 6024: 6021: 6015: 6000: 5996: 5988: 5987: 5986: 5983: 5964: 5942: 5920: 5908: 5894: 5887: 5882: 5879: 5872: 5871: 5870: 5868: 5863: 5858: 5839: 5836: 5833: 5828: 5812: 5809: 5798: 5782: 5779: 5775: 5770: 5761: 5750: 5746: 5740: 5724: 5721: 5710: 5694: 5691: 5687: 5681: 5677: 5674: 5670: 5666: 5663: 5660: 5654: 5639: 5635: 5627: 5626: 5625: 5623: 5619: 5615: 5611: 5610: 5603: 5564: 5561: 5558: 5555: 5550: 5537: 5534: 5509: 5506: 5502: 5497: 5488: 5477: 5472: 5459: 5456: 5431: 5428: 5424: 5419: 5416: 5412: 5408: 5405: 5402: 5396: 5381: 5377: 5369: 5368: 5367: 5349: 5344: 5342: 5338: 5333: 5331: 5310: 5297: 5292: 5273: 5270: 5267: 5263: 5257: 5249: 5246: 5238: 5229: 5225: 5221: 5218: 5213: 5203: 5186: 5182: 5178: 5175: 5170: 5160: 5148: 5139: 5134: 5124: 5120: 5116: 5110: 5107: 5102: 5089: 5086: 5062: 5058: 5054: 5051: 5046: 5038: 5035: 5032: 5023: 5008: 5007: 5006: 4998: 4981: 4972: 4969: 4960: 4954: 4950: 4946: 4943: 4940: 4937: 4934: 4931: 4924: 4923: 4922: 4920: 4915: 4907: 4903: 4902:four-momentum 4898: 4873: 4869: 4865: 4862: 4857: 4847: 4835: 4825: 4808: 4805: 4770: 4766: 4763: 4760: 4751: 4736: 4726: 4725: 4724: 4720: 4716: 4712: 4708: 4703: 4699: 4692: 4688: 4684: 4680: 4675: 4671: 4667: 4657: 4655: 4636: 4632: 4629: 4624: 4620: 4614: 4601: 4598: 4590: 4584: 4578: 4570: 4557: 4556: 4555: 4553: 4549: 4544: 4539: 4535: 4531: 4526: 4523: 4519: 4512: 4507: 4506:four-gradient 4479: 4475: 4469: 4461: 4458: 4455: 4450: 4440: 4436: 4426: 4423: 4415: 4409: 4404: 4400: 4390: 4384: 4380: 4373: 4370: 4367: 4359: 4355: 4350: 4343: 4340: 4332: 4326: 4323: 4314: 4309: 4302: 4292: 4288: 4278: 4275: 4269: 4261: 4250: 4246: 4241: 4232: 4228: 4224: 4221: 4213: 4207: 4204: 4197: 4196: 4195: 4193: 4189: 4184: 4179: 4178:Dirac adjoint 4174: 4171: 4166: 4160: 4141: 4136: 4132: 4126: 4122: 4116: 4112: 4108: 4103: 4099: 4090: 4080: 4076: 4070: 4066: 4062: 4053: 4050: 4045: 4041: 4037: 4034: 4027: 4026: 4025: 4023: 4019: 4016: 4012: 4007: 4003: 3998: 3991: 3987: 3983: 3978: 3972: 3968: 3964: 3959: 3955: 3951: 3941: 3940:for details. 3939: 3935: 3931: 3927: 3921: 3915: 3909: 3903: 3899: 3895: 3892: 3888: 3883: 3882:negative mass 3879: 3858: 3854: 3850: 3847: 3844: 3827: 3819: 3816: 3807: 3796: 3793: 3790: 3786: 3780: 3776: 3772: 3769: 3766: 3749: 3741: 3738: 3729: 3722: 3714: 3713: 3712: 3695: 3691: 3688: 3683: 3679: 3675: 3670: 3665: 3661: 3653: 3652: 3651: 3649: 3630: 3624: 3620: 3614: 3610: 3606: 3603: 3598: 3594: 3588: 3584: 3579: 3574: 3570: 3566: 3563: 3560: 3557: 3552: 3548: 3540: 3539: 3538: 3535: 3531: 3526: 3522: 3517: 3507: 3500: 3493: 3489: 3469: 3465: 3462: 3459: 3455: 3449: 3445: 3441: 3438: 3435: 3418: 3410: 3407: 3398: 3391: 3386: 3380: 3376: 3372: 3369: 3366: 3349: 3341: 3338: 3329: 3322: 3314: 3313: 3312: 3310: 3307: 3287: 3283: 3278: 3268: 3264: 3260: 3254: 3251: 3234: 3215: 3211: 3207: 3204: 3199: 3189: 3178: 3177: 3176: 3174: 3164: 3130: 3113: 3110: 3107: 3090: 3084: 3075: 3065: 3064: 3063: 3060: 3055: 3051: 3046: 3041: 3039: 3038:automatically 3018: 3001: 2993: 2990: 2985: 2975: 2964: 2963: 2962: 2959: 2955: 2951: 2947: 2946:spin operator 2942: 2937: 2932: 2912: 2908: 2903: 2899: 2895: 2892: 2889: 2885: 2880: 2870: 2865: 2839: 2835: 2831: 2825: 2822: 2817: 2796: 2795: 2794: 2792: 2791:Bohr magneton 2787: 2777: 2773: 2768: 2764: 2758: 2752: 2748: 2740: 2736: 2732: 2725: 2718: 2714: 2705: 2700: 2699: 2694:and can even 2693: 2689: 2685: 2680: 2675: 2656: 2648: 2638: 2634: 2630: 2624: 2607: 2588: 2584: 2578: 2575: 2569: 2554: 2542: 2532: 2528: 2524: 2518: 2501: 2482: 2478: 2472: 2463: 2457: 2448: 2438: 2437: 2436: 2434: 2411: 2408: 2397: 2394: 2388: 2385: 2366: 2346: 2337: 2327: 2326: 2325: 2323: 2317: 2313: 2309: 2305: 2300: 2296: 2292: 2288: 2284: 2280: 2274: 2263: 2259: 2255: 2251: 2246: 2243: 2237: 2231: 2227: 2222: 2215: 2209: 2208:column vector 2204: 2199: 2198:square matrix 2194: 2190: 2182: 2156: 2153: 2140: 2137: 2124: 2120: 2107: 2098: 2095: 2082: 2078: 2070: 2069: 2068: 2066: 2065:Lorentz group 2061: 2057: 2052: 2044: 2038: 2034: 2030: 2026: 2021: 2018: 2017:orthochronous 2013: 2010: 2006: 2003: 1997: 1993: 1989: 1985: 1981: 1980:four-momentum 1975: 1971: 1967: 1963: 1959: 1955: 1951: 1947: 1943: 1938: 1931: 1924: 1917: 1913: 1906: 1899: 1892: 1888: 1884: 1880: 1870: 1868: 1863: 1861: 1857: 1853: 1849: 1844: 1827: 1823: 1817: 1805: 1793: 1787: 1784: 1772: 1768: 1763: 1760: 1754: 1751: 1747: 1738: 1737: 1736: 1732: 1728: 1723: 1718: 1713: 1708: 1705: 1700: 1695: 1691: 1672: 1662: 1659: 1656: 1638: 1631: 1616: 1613: 1604: 1594: 1593: 1592: 1590: 1587: 1583: 1564: 1558: 1548: 1544: 1540: 1534: 1526: 1516: 1512: 1508: 1503: 1499: 1491: 1490: 1489: 1468: 1458: 1455: 1448: 1444: 1439: 1435: 1430: 1426: 1422: 1417: 1413: 1409: 1404: 1402: 1398: 1394: 1393:colour charge 1390: 1386: 1382: 1378: 1374: 1370: 1366: 1362: 1358: 1354: 1350: 1346: 1341: 1337: 1333: 1329: 1325: 1319: 1315: 1311: 1307: 1304: = 1303: 1298: 1293: 1286: 1281: 1278:and even for 1277: 1272: 1267:. The number 1265: 1261: 1256: 1253: 1247: 1243: 1240: 1236: 1230: 1226: 1222: 1218: 1214: 1209: 1205: 1186: 1177: 1171: 1168: 1162: 1147: 1140: 1139: 1138: 1136: 1132: 1128: 1123: 1121: 1111: 1109: 1105: 1101: 1100:natural units 1097: 1093: 1089: 1085: 1081: 1077: 1072: 1070: 1066: 1062: 1060: 1056: 1052: 1048: 1047: 1044:relativistic 1040: 1038: 1034: 1030: 1026: 1022: 1018: 1015: 1003: 999: 995: 990: 988: 984: 980: 976: 972: 968: 964: 960: 956: 952: 948: 944: 941:, as well as 940: 936: 932: 928: 924: 920: 916: 912: 908: 904: 901: 897: 893: 889: 877: 872: 870: 865: 863: 858: 857: 855: 854: 846: 843: 841: 838: 836: 833: 831: 828: 826: 823: 821: 818: 816: 813: 811: 808: 806: 803: 801: 798: 796: 793: 791: 788: 786: 783: 781: 778: 776: 773: 771: 768: 766: 763: 761: 758: 756: 753: 751: 748: 746: 743: 741: 738: 736: 733: 731: 728: 726: 723: 721: 718: 716: 713: 711: 708: 706: 703: 701: 698: 696: 693: 691: 688: 686: 683: 681: 678: 676: 673: 671: 668: 666: 663: 661: 658: 656: 653: 651: 648: 646: 643: 641: 638: 636: 633: 631: 628: 626: 623: 621: 618: 617: 610: 609: 602: 599: 597: 594: 592: 589: 587: 584: 582: 579: 577: 576:Quantum chaos 574: 572: 569: 567: 564: 562: 559: 557: 554: 553: 547: 546: 538: 535: 533: 532:Transactional 530: 528: 525: 523: 522:Quantum logic 520: 518: 515: 513: 510: 504: 501: 500: 499: 496: 495: 494: 491: 489: 486: 484: 481: 479: 476: 474: 471: 469: 466: 465: 461: 456: 455: 447: 444: 442: 439: 437: 434: 432: 429: 427: 424: 423: 416: 415: 407: 404: 402: 399: 397: 394: 392: 389: 387: 384: 382: 379: 378: 374: 371: 370: 364: 363: 355: 352: 350: 347: 345: 342: 341: 335: 332: 331: 330: 327: 326: 322: 319: 317: 314: 312: 309: 307: 304: 302: 299: 297: 294: 292: 289: 287: 284: 282: 279: 277: 274: 273: 266: 265: 255: 252: 251: 250: 249:Wave function 247: 245: 242: 240: 237: 235: 232: 230: 229:Superposition 227: 225: 222: 220: 217: 215: 212: 210: 207: 205: 202: 200: 197: 195: 192: 190: 187: 186: 179: 178: 170: 167: 165: 162: 161: 157: 154: 152: 149: 147: 144: 143: 137: 136: 131: 128: 126: 123: 121: 118: 117: 116: 115: 111: 78: 72: 55: 52: 48: 40: 33: 32: 29: 26: 25: 21: 20: 16587: 16363:Quantum mind 16275:Franck–Hertz 16137:Klein–Gordon 16086:Formulations 16079:Formulations 16008:Interference 15998:Entanglement 15976:Ground state 15971:Energy level 15944:Fundamentals 15908:Introduction 15820:. Retrieved 15813:the original 15771:the original 15753: 15725: 15704: 15683: 15662: 15641: 15620: 15599: 15582:. Springer. 15578: 15557: 15536: 15515: 15494: 15473: 15452: 15424: 15403: 15382: 15361: 15340: 15319: 15297: 15292:Weinberg, S. 15271: 15266:Weinberg, S. 15247: 15228: 15208: 15172: 15151: 15130: 15109: 15068: 15064: 15031: 15028:Eur. J. Phys 15027: 14987:(1): 51–52. 14984: 14980: 14938: 14935:Phys. Rev. D 14934: 14891: 14887: 14854: 14850: 14825: 14822:Phys. Rev. D 14821: 14796: 14792: 14759: 14756:Phys. Rev. A 14755: 14715: 14711: 14669: 14663: 14629: 14608: 14599: 14579: 14559: 14539: 14517: 14496: 14475: 14455: 14432: 14423: 14412:. Springer. 14408: 14383:. Springer. 14379: 14362:. Springer. 14358: 14337: 14316: 14295: 14275: 14253: 14234:. Springer. 14230: 14210: 14196: 14174: 14154: 14135: 14116: 14100:. Springer. 14096: 14076:. Springer. 14072: 14051: 14030: 13989: 13985: 13975: 13950: 13946: 13914: 13910: 13878: 13874: 13851: 13847: 13823: 13819: 13813: 13786: 13780: 13767: 13748: 13742: 13729: 13702: 13696: 13686: 13677: 13675: 13659: 13652: 13633: 13627: 13600: 13596: 13583: 13558: 13550: 13525: 13517: 13498: 13495:Atkins, P.W. 13489: 13465: 13458: 13433: 13422: 13382:(1): 51–52. 13379: 13375: 13339: 13332: 13308: 13300:Thorne, K.S. 13296:Misner, C.W. 13290: 13262: 13255: 13235: 13228: 13208: 13201: 13181: 13177:Weinberg, S. 13171: 13146: 13138: 13118: 13111: 13068: 13064: 13058: 13037: 13012: 13008: 13002: 12991:. Retrieved 12984:the original 12947: 12943: 12930: 12885: 12881: 12871: 12828: 12824: 12818: 12799: 12793: 12768: 12739: 12733: 12708: 12700: 12681: 12675: 12655: 12648: 12629: 12623: 12603: 12574: 12568: 12543: 12524: 12518: 12499: 12470: 12463: 12439: 12432: 12408: 12377: 12366:. Retrieved 12359:the original 12338: 12334: 12325: 12312:. Retrieved 12305:the original 12284: 12280: 12271: 12258:. Retrieved 12251:the original 12230: 12226: 12217: 12209: 12184: 12164: 12157: 12132: 12124: 12089: 12085: 12079: 12060: 12051: 12031: 12024: 11999: 11968: 11961: 11941: 11934: 11914: 11907: 11882: 11847: 11840: 11822: 11817: 11683: 11546: 11529: 11520: 11511: 11505: 11488: 11474:Gauge theory 11421: 11278: 11263: 11240: 11225: 11223: 11179: 11175: 11156: 11078:Stark effect 10954: 10937:spectroscopy 10920: 10914: 10896: 10737: 10717: 10713: 10561: 10489: 10485: 10482: 10370: 10360: 10354: 10348: 10341: 10330: 10318: 10191: 9994: 9989: 9979: 9973: 9967: 9961: 9957: 9951: 9947: 9941: 9937: 9934: 9775: 9771: 9767: 9763:pseudovector 9756: 9741: 9737: 9605: 9524: 9401: 9392: 9389: 9375: 9373: 9282: 9280: 9213: 9210: 9206: 9203: 9147: 9140: 9128: 9006: 9002: 8998: 8994: 8992: 8313: 8306: 8299: 8294:antiparticle 8287: 8279: 8276:, there are 8272: 8267: 8261: 7768: 7733: 7730:Higher spins 7647: 7578:case) times 7558: 7555: 7551: 7546: 7528: 7519: 7515: 7511: 7507: 7503: 7501: 7318: 7270: 7259: 7124: 7097: 7094: 6899: 6890: 6875: 6607: 6585: 6579: 6575: 6563:Dirac spinor 6558:antiparticle 6555: 6542: 6536: 6530: 6526: 6520: 6512: 6490: 6484: 6477: 6473: 6463: 6460: 6305: 6291: 6284: 6277: 6273: 6270: 6263: 6256: 6249: 6245: 6241: 6237: 6229: 6227: 5984: 5962: 5940: 5937: 5867:spinor field 5861: 5854: 5607: 5605: 5345: 5340: 5334: 5329: 5308: 5290: 5288: 5004: 4996: 4910: 4893: 4890: 4718: 4714: 4710: 4701: 4697: 4690: 4686: 4682: 4673: 4663: 4651: 4547: 4542: 4527: 4521: 4517: 4510: 4499: 4190:, while the 4182: 4172: 4169: 4164: 4156: 4014: 4010: 4005: 3996: 3989: 3985: 3981: 3970: 3966: 3957: 3954:wavefunction 3947: 3919: 3913: 3907: 3901: 3897: 3893: 3890: 3886: 3875: 3710: 3645: 3533: 3529: 3515: 3505: 3498: 3491: 3487: 3484: 3305: 3302: 3170: 3162: 3058: 3042: 3037: 3035: 2957: 2940: 2930: 2927: 2782: 2778: 2771: 2766: 2762: 2756: 2750: 2746: 2738: 2734: 2730: 2723: 2716: 2712: 2703: 2695: 2687: 2678: 2674:power series 2671: 2432: 2429: 2315: 2311: 2307: 2298: 2294: 2290: 2276: 2254:group theory 2241: 2235: 2229: 2220: 2213: 2202: 2192: 2188: 2180: 2177: 2059: 2047: 2036: 2032: 2028: 2024: 2014: 2001: 1995: 1991: 1987: 1983: 1973: 1969: 1965: 1953: 1939: 1929: 1922: 1915: 1911: 1904: 1897: 1890: 1886: 1876: 1864: 1845: 1842: 1730: 1726: 1721: 1716: 1709: 1698: 1687: 1579: 1437: 1428: 1415: 1405: 1342: 1335: 1331: 1327: 1323: 1317: 1313: 1309: 1305: 1301: 1296: 1291: 1284: 1270: 1263: 1257: 1251: 1245: 1228: 1224: 1220: 1217:wavefunction 1207: 1201: 1129:is that the 1124: 1117: 1073: 1063: 1059:annihilation 1051:field quanta 1050: 1043: 1041: 991: 974: 954: 922: 895: 891: 885: 555: 431:Klein–Gordon 367:Formulations 204:Energy level 199:Entanglement 182:Fundamentals 169:Interference 120:Introduction 16621:EPR paradox 16401:Quantum bus 16270:Double-slit 16248:Experiments 16214:Many-worlds 16152:Schrödinger 16116:Phase space 16106:Schrödinger 16096:Interaction 16053:Uncertainty 16023:Nonlocality 16018:Measurement 16013:Decoherence 16003:Hamiltonian 15105:Ohlsson, T. 15065:Am. J. Phys 14718:(4): 1465. 14452:Bethe, H.A. 13309:Gravitation 11178:). QM does 11018:Experiments 10966:. In 1916, 10941:diffraction 9791:formalism: 9765:definition 9132:Élie Cartan 8296:(there are 6908:and to the 6906:quaternions 6599:Darwin term 6593:, electron 6523:4-momentum 6482:and charge 5309:independent 5291:irreducible 4723:, this is: 3525:anticommute 2684:derivatives 1848:Schrödinger 1447:dot product 820:von Neumann 805:Schrödinger 581:EPR paradox 512:Many-worlds 446:Schrödinger 401:Schrödinger 396:Phase-space 386:Interaction 291:Double-slit 269:Experiments 244:Uncertainty 214:Nonlocality 209:Measurement 194:Decoherence 164:Hamiltonian 16659:Categories 16549:Extensions 16383:Technology 16229:Relational 16180:Copenhagen 16091:Heisenberg 16038:Tunnelling 15901:Background 15822:2020-11-17 15200:Wu, Ta-you 15071:(6): 551. 14793:J. Phys. A 14712:J. Phys. A 12993:2013-04-14 12368:2013-04-14 12328:spin. III" 12314:2013-04-14 12260:2014-08-24 12099:2209.03937 11832:References 11260:Retherford 11220:Lamb shift 11067:Rutherford 11012:G. Thomson 10988:de Broglie 10968:Sommerfeld 10945:scattering 10194:Hodge dual 9398:Lagrangian 7313:See also: 6921:commutator 6917:Lie groups 5982:) states. 5312:spin 4194:does not: 3961:gives the 1852:Heisenberg 1387:and their 1065:Paul Dirac 1005:spin 1002:elementary 994:antimatter 915:velocities 815:Sommerfeld 730:Heisenberg 725:Gutzwiller 665:de Broglie 613:Scientists 527:Relational 478:Copenhagen 381:Heisenberg 239:Tunnelling 140:Background 16255:Bell test 16125:Equations 15951:Born rule 15056:122527454 15019:119533324 14926:123341332 14879:123341332 14728:CiteSeerX 14696:Pauli, W. 14207:Dyson, F. 14092:Pauli, W. 14008:0950-1207 13947:Phys. Rev 13911:Phys. Rev 13875:Phys. Rev 13848:Phys. Rev 13820:Phys. Rev 13597:Phys. Rev 13414:119533324 13078:0901.4200 13049:0901.4199 12980:121773411 12863:119147261 12838:1110.6878 12414:1193–1194 12389:1208.0644 12335:Phys. Rev 12281:Phys. Rev 12227:Phys. Rev 12116:212835814 11697:ψ 11642:η 11635:χ 11621:ψ 11558:ψ 11498:Footnotes 11329:Schwinger 11294:microwave 11119:positrons 11076:1913 The 11065:, led by 10970:explains 10921:necessary 10869:^ 10866:μ 10859:⋅ 10828:× 10804:− 10789:^ 10786:μ 10779:⋅ 10766:− 10757:^ 10686:^ 10683:μ 10676:⋅ 10650:× 10626:− 10611:^ 10608:μ 10601:⋅ 10588:− 10579:^ 10525:× 10488:<< 10431:− 10406:× 10296:∧ 10285:⋆ 10273:⇌ 10267:δ 10257:γ 10254:β 10244:δ 10241:γ 10238:β 10235:α 10231:ε 10212:α 10169:∧ 10158:⋆ 10130:∧ 10113:⇌ 10107:δ 10097:γ 10087:δ 10084:γ 10081:β 10078:α 10074:ε 10045:β 10035:α 10016:β 10013:α 9911:∧ 9894:⇌ 9885:β 9875:α 9856:α 9846:β 9838:− 9833:β 9823:α 9810:β 9807:α 9719:ψ 9714:∗ 9710:ψ 9693:− 9690:ψ 9685:ν 9681:∂ 9675:∗ 9671:ψ 9665:μ 9661:∂ 9655:ν 9652:μ 9648:η 9633:ℏ 9627:− 9591:ψ 9579:− 9574:μ 9564:μ 9560:γ 9551:¯ 9548:ψ 9500:ψ 9497:∂ 9485:∂ 9479:− 9466:ψ 9461:μ 9457:∂ 9450:∂ 9438:∂ 9426:μ 9422:∂ 9351:^ 9334:^ 9318:ℏ 9304:^ 9263:^ 9236:^ 9087:− 9083:ψ 9052:ψ 9020:ψ 8968:⋆ 8941:− 8935:σ 8928:− 8924:ψ 8916:⋯ 8909:⋆ 8882:− 8873:σ 8862:ψ 8852:⋆ 8822:σ 8811:ψ 8795:† 8773:ψ 8767:⇌ 8733:− 8727:σ 8720:− 8716:ψ 8680:− 8674:σ 8667:− 8663:ψ 8655:⋮ 8626:− 8617:σ 8610:− 8606:ψ 8573:σ 8566:− 8562:ψ 8532:− 8526:σ 8515:ψ 8479:− 8473:σ 8462:ψ 8454:⋮ 8425:− 8416:σ 8405:ψ 8372:σ 8361:ψ 8329:ψ 8304:possible 8237:⋆ 8210:− 8204:σ 8200:ψ 8190:⋆ 8157:− 8151:σ 8147:ψ 8139:⋯ 8132:⋆ 8105:− 8096:σ 8092:ψ 8082:⋆ 8052:σ 8048:ψ 8029:† 8007:ψ 8001:⇌ 7967:− 7961:σ 7957:ψ 7921:− 7915:σ 7911:ψ 7903:⋮ 7874:− 7865:σ 7861:ψ 7828:σ 7824:ψ 7792:ψ 7779:functions 7705:^ 7688:⋅ 7682:^ 7665:^ 7598:− 7451:− 7433:^ 7416:⋅ 7410:^ 7375:^ 7363:⋅ 7357:^ 7340:^ 7266:vierbeins 7239:β 7236:α 7232:η 7220:α 7216:γ 7210:β 7206:γ 7197:β 7193:γ 7187:α 7183:γ 7163:β 7159:γ 7150:α 7146:γ 7074:σ 7061:δ 7034:σ 7021:σ 7000:σ 6984:ε 6960:σ 6947:σ 6882:neutrinos 6849:− 6845:ψ 6839:μ 6832:^ 6820:μ 6816:σ 6795:ψ 6789:μ 6782:^ 6770:μ 6766:σ 6761:⇌ 6749:− 6745:ψ 6733:^ 6722:⋅ 6718:σ 6714:− 6704:^ 6672:ψ 6660:^ 6649:⋅ 6645:σ 6631:^ 6436:↓ 6433:− 6429:ψ 6419:↑ 6416:− 6412:ψ 6402:↓ 6395:ψ 6385:↑ 6378:ψ 6353:− 6349:ψ 6335:ψ 6320:ψ 6207:ψ 6186:− 6178:μ 6167:− 6162:μ 6155:^ 6140:μ 6136:γ 6126:⇌ 6122:ψ 6101:− 6086:− 6080:^ 6065:⋅ 6061:γ 6043:γ 6036:ψ 6028:ϕ 6022:− 6013:∂ 6009:∂ 6004:ℏ 5965:= − 5913:↓ 5909:ψ 5899:↑ 5895:ψ 5880:ψ 5840:ϕ 5810:− 5799:⋅ 5795:σ 5765:^ 5755:⇔ 5751:ψ 5722:− 5711:⋅ 5707:σ 5675:ψ 5667:ϕ 5661:− 5652:∂ 5648:∂ 5643:ℏ 5562:ϕ 5535:− 5529:^ 5492:^ 5482:⇔ 5478:ψ 5457:− 5451:^ 5417:ψ 5409:ϕ 5403:− 5394:∂ 5390:∂ 5385:ℏ 5268:ψ 5239:− 5230:μ 5219:− 5214:μ 5207:^ 5187:μ 5176:− 5171:μ 5164:^ 5144:⇌ 5140:ψ 5108:ψ 5087:− 5081:^ 5052:ψ 5039:ϕ 5033:− 5027:^ 4970:≈ 4944:≈ 4941:ϕ 4935:− 4874:μ 4863:− 4858:μ 4851:^ 4841:→ 4836:μ 4829:^ 4818:⇌ 4806:− 4800:^ 4789:→ 4783:^ 4767:ϕ 4761:− 4755:^ 4746:→ 4740:^ 4625:μ 4615:μ 4611:∂ 4606:⇌ 4591:⋅ 4588:∇ 4576:∂ 4571:ρ 4568:∂ 4480:∗ 4476:ψ 4470:μ 4466:∂ 4462:ψ 4459:− 4456:ψ 4451:μ 4447:∂ 4441:∗ 4437:ψ 4419:ℏ 4405:μ 4396:⇌ 4385:∗ 4381:ψ 4377:∇ 4374:ψ 4371:− 4368:ψ 4365:∇ 4360:∗ 4356:ψ 4336:ℏ 4327:− 4300:∂ 4293:∗ 4289:ψ 4285:∂ 4279:ψ 4276:− 4267:∂ 4262:ψ 4259:∂ 4251:∗ 4247:ψ 4217:ℏ 4205:ρ 4142:ψ 4137:μ 4133:γ 4123:γ 4117:† 4113:ψ 4104:μ 4095:⇌ 4091:ψ 4087:γ 4077:γ 4071:† 4067:ψ 4051:ψ 4046:† 4042:ψ 4035:ρ 3938:Dirac sea 3848:β 3839:^ 3828:⋅ 3824:α 3811:^ 3801:⇔ 3791:ψ 3770:β 3767:− 3761:^ 3750:⋅ 3746:α 3739:− 3733:^ 3680:β 3662:α 3621:α 3611:α 3607:− 3595:α 3585:α 3571:α 3567:β 3564:− 3558:β 3549:α 3460:ψ 3439:β 3430:^ 3419:⋅ 3415:α 3402:^ 3370:β 3367:− 3361:^ 3350:⋅ 3346:α 3339:− 3333:^ 3284:ψ 3252:ψ 3246:^ 3235:⋅ 3229:^ 3205:ψ 3193:^ 3142:^ 3125:^ 3094:^ 3079:^ 3045:rest mass 3012:^ 3009:μ 3002:⋅ 2994:− 2979:^ 2909:σ 2900:μ 2893:− 2876:μ 2859:^ 2846:ℏ 2836:μ 2826:− 2811:^ 2808:μ 2698:causality 2657:ψ 2619:^ 2608:⋅ 2602:^ 2576:ψ 2567:∂ 2563:∂ 2558:ℏ 2551:⇒ 2513:^ 2502:⋅ 2496:^ 2467:^ 2452:^ 2378:^ 2367:⋅ 2361:^ 2341:^ 2302:plus the 2283:potential 2200:. 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10796:S 10771:B 10763:= 10754:H 10729:2 10725:1 10716:( 10699:, 10693:S 10672:) 10664:2 10660:c 10654:v 10646:E 10639:+ 10635:B 10630:( 10623:= 10618:S 10593:B 10585:= 10576:H 10547:, 10539:2 10535:c 10529:v 10521:E 10514:= 10506:B 10490:c 10486:v 10468:. 10457:2 10452:) 10448:c 10444:/ 10440:v 10436:( 10428:1 10421:2 10417:c 10410:v 10402:E 10395:= 10387:B 10361:B 10355:E 10349:v 10304:) 10300:P 10292:M 10288:( 10282:= 10278:W 10263:p 10250:M 10225:2 10222:1 10217:= 10208:W 10177:) 10173:P 10165:W 10161:( 10151:2 10147:m 10143:1 10138:+ 10134:P 10126:X 10122:= 10118:J 10103:p 10093:W 10066:2 10062:m 10058:1 10053:+ 10048:] 10041:P 10032:[ 10028:X 10024:2 10021:= 10009:J 9990:m 9980:M 9974:M 9968:M 9962:L 9958:M 9952:L 9948:M 9942:L 9938:M 9920:, 9915:P 9907:X 9903:= 9899:M 9888:] 9881:P 9872:[ 9868:X 9864:2 9861:= 9852:P 9842:X 9829:P 9819:X 9815:= 9803:M 9776:p 9772:r 9768:L 9742:ψ 9723:. 9704:2 9700:c 9696:m 9642:m 9637:2 9624:= 9619:L 9588:) 9585:c 9582:m 9570:P 9556:( 9543:= 9538:L 9509:0 9506:= 9490:L 9475:) 9469:) 9453:( 9443:L 9432:( 9393:ψ 9376:c 9358:] 9347:r 9340:, 9331:H 9324:[ 9315:i 9310:= 9300:v 9259:p 9250:m 9247:1 9242:= 9232:v 9214:v 9211:m 9207:p 9187:2 9184:/ 9181:1 9155:2 9152:/ 9148:ħ 9112:) 9106:) 9103:t 9100:, 9096:r 9092:( 9075:) 9072:t 9069:, 9065:r 9061:( 9056:+ 9045:( 9040:= 9037:) 9034:t 9031:, 9027:r 9023:( 9007:s 9003:s 8999:s 8976:] 8963:) 8960:t 8957:, 8953:r 8949:( 8944:s 8938:= 8931:, 8904:) 8901:t 8898:, 8894:r 8890:( 8885:1 8879:s 8876:= 8869:, 8866:+ 8847:) 8844:t 8841:, 8837:r 8833:( 8828:s 8825:= 8818:, 8815:+ 8802:[ 8790:) 8787:t 8784:, 8780:r 8776:( 8761:] 8755:) 8752:t 8749:, 8745:r 8741:( 8736:s 8730:= 8723:, 8708:) 8705:t 8702:, 8698:r 8694:( 8689:1 8686:+ 8683:s 8677:= 8670:, 8648:) 8645:t 8642:, 8638:r 8634:( 8629:1 8623:s 8620:= 8613:, 8598:) 8595:t 8592:, 8588:r 8584:( 8579:s 8576:= 8569:, 8554:) 8551:t 8548:, 8544:r 8540:( 8535:s 8529:= 8522:, 8519:+ 8507:) 8504:t 8501:, 8497:r 8493:( 8488:1 8485:+ 8482:s 8476:= 8469:, 8466:+ 8447:) 8444:t 8441:, 8437:r 8433:( 8428:1 8422:s 8419:= 8412:, 8409:+ 8397:) 8394:t 8391:, 8387:r 8383:( 8378:s 8375:= 8368:, 8365:+ 8354:[ 8349:= 8346:) 8343:t 8340:, 8336:r 8332:( 8314:s 8307:σ 8300:s 8298:2 8288:s 8286:2 8280:s 8278:2 8273:s 8245:] 8232:) 8229:t 8226:, 8222:r 8218:( 8213:s 8207:= 8185:) 8182:t 8179:, 8175:r 8171:( 8166:1 8163:+ 8160:s 8154:= 8127:) 8124:t 8121:, 8117:r 8113:( 8108:1 8102:s 8099:= 8077:) 8074:t 8071:, 8067:r 8063:( 8058:s 8055:= 8039:[ 8034:= 8024:) 8021:t 8018:, 8014:r 8010:( 7995:] 7989:) 7986:t 7983:, 7979:r 7975:( 7970:s 7964:= 7949:) 7946:t 7943:, 7939:r 7935:( 7930:1 7927:+ 7924:s 7918:= 7896:) 7893:t 7890:, 7886:r 7882:( 7877:1 7871:s 7868:= 7853:) 7850:t 7847:, 7843:r 7839:( 7834:s 7831:= 7817:[ 7812:= 7809:) 7806:t 7803:, 7799:r 7795:( 7745:2 7742:/ 7739:1 7712:E 7701:p 7694:c 7678:S 7671:= 7662:h 7629:2 7625:) 7619:2 7615:c 7609:0 7605:m 7601:( 7593:2 7589:E 7573:2 7570:/ 7567:1 7559:p 7556:c 7552:σ 7547:c 7540:2 7537:/ 7534:1 7523:0 7520:m 7516:E 7512:s 7508:S 7504:p 7482:2 7478:) 7472:2 7468:c 7462:0 7458:m 7454:( 7446:2 7442:E 7429:p 7422:c 7406:S 7399:= 7392:| 7387:p 7382:| 7371:p 7353:S 7346:= 7337:h 7286:2 7283:/ 7280:1 7245:, 7228:2 7225:= 7202:+ 7179:= 7174:+ 7169:] 7155:, 7141:[ 7125:η 7098:ε 7078:0 7068:b 7065:a 7057:2 7054:= 7049:+ 7044:] 7038:b 7030:, 7025:a 7016:[ 7010:, 7004:c 6994:c 6991:b 6988:a 6980:i 6977:2 6974:= 6970:] 6964:b 6956:, 6951:a 6942:[ 6928:+ 6894:0 6891:σ 6861:, 6857:0 6854:= 6829:P 6811:, 6807:0 6804:= 6799:+ 6779:P 6757:0 6754:= 6740:) 6729:p 6709:c 6701:E 6693:( 6688:, 6684:0 6681:= 6676:+ 6667:) 6656:p 6641:+ 6636:c 6628:E 6620:( 6586:ϕ 6580:0 6576:A 6543:q 6541:− 6533:) 6531:p 6527:E 6516:− 6513:ψ 6504:2 6501:/ 6498:1 6491:σ 6485:q 6480:) 6478:p 6474:E 6472:( 6467:+ 6464:ψ 6444:) 6399:+ 6382:+ 6371:( 6366:= 6361:) 6339:+ 6328:( 6323:= 6297:) 6295:3 6288:2 6281:1 6274:α 6271:β 6267:3 6264:γ 6260:2 6257:γ 6253:1 6250:γ 6246:γ 6242:β 6238:γ 6213:0 6210:= 6203:] 6197:2 6193:c 6189:m 6183:) 6174:A 6170:q 6152:P 6145:( 6131:[ 6118:] 6112:2 6108:c 6104:m 6097:) 6093:A 6089:q 6076:p 6069:( 6057:c 6053:[ 6047:0 6039:= 6032:) 6025:q 6016:t 6001:i 5997:( 5976:2 5973:/ 5970:1 5963:σ 5954:2 5951:/ 5948:1 5941:σ 5921:) 5888:( 5883:= 5837:q 5834:+ 5829:2 5824:) 5821:) 5817:A 5813:q 5806:p 5802:( 5791:( 5783:m 5780:2 5776:1 5771:= 5762:H 5747:] 5741:2 5736:) 5733:) 5729:A 5725:q 5718:p 5714:( 5703:( 5695:m 5692:2 5688:1 5682:[ 5678:= 5671:) 5664:q 5655:t 5640:i 5636:( 5592:2 5589:/ 5586:1 5565:. 5559:q 5556:+ 5551:2 5546:) 5542:A 5538:q 5525:p 5518:( 5510:m 5507:2 5503:1 5498:= 5489:H 5473:2 5468:) 5464:A 5460:q 5447:p 5440:( 5432:m 5429:2 5425:1 5420:= 5413:) 5406:q 5397:t 5382:i 5378:( 5361:2 5358:/ 5355:1 5341:π 5323:2 5320:/ 5317:1 5271:= 5264:] 5258:2 5253:) 5250:c 5247:m 5244:( 5235:) 5226:A 5222:q 5204:P 5197:( 5192:) 5183:A 5179:q 5161:P 5154:( 5149:[ 5135:2 5131:) 5125:2 5121:c 5117:m 5114:( 5111:+ 5103:2 5098:) 5094:A 5090:q 5077:p 5070:( 5063:2 5059:c 5055:= 5047:2 5042:) 5036:q 5024:E 5018:( 4982:, 4977:v 4973:m 4966:p 4961:, 4955:2 4951:c 4947:m 4938:e 4932:E 4913:μ 4911:A 4896:μ 4894:P 4870:A 4866:q 4848:P 4826:P 4813:A 4809:q 4796:p 4779:p 4771:, 4764:q 4752:E 4737:E 4721:) 4719:t 4715:r 4713:( 4711:ϕ 4702:A 4698:B 4693:) 4691:t 4687:r 4685:( 4683:A 4674:q 4637:, 4633:0 4630:= 4621:J 4602:0 4599:= 4595:J 4585:+ 4579:t 4543:ψ 4522:t 4518:ψ 4516:∂ 4511:ψ 4502:∂ 4485:) 4433:( 4427:m 4424:2 4416:i 4410:= 4401:J 4391:) 4351:( 4344:m 4341:2 4333:i 4324:= 4320:j 4315:, 4310:) 4303:t 4270:t 4242:( 4233:2 4229:c 4225:m 4222:2 4214:i 4208:= 4183:J 4173:γ 4170:ψ 4165:ψ 4127:0 4109:= 4100:J 4081:0 4063:= 4059:j 4054:, 4038:= 4006:j 3997:ρ 3992:) 3990:t 3986:r 3984:( 3982:ψ 3973:| 3971:ψ 3967:ρ 3958:ψ 3920:ψ 3914:β 3908:α 3898:p 3894:α 3891:c 3887:E 3859:2 3855:c 3851:m 3845:+ 3835:p 3820:c 3817:= 3808:H 3797:0 3794:= 3787:) 3781:2 3777:c 3773:m 3757:p 3742:c 3730:E 3723:( 3696:, 3692:I 3689:= 3684:2 3676:= 3671:2 3666:i 3631:, 3625:i 3615:j 3604:= 3599:j 3589:i 3580:, 3575:i 3561:= 3553:i 3534:j 3530:i 3516:β 3511:) 3509:3 3506:α 3502:2 3499:α 3495:1 3492:α 3488:α 3470:, 3466:0 3463:= 3456:) 3450:2 3446:c 3442:m 3436:+ 3426:p 3411:c 3408:+ 3399:E 3392:( 3387:) 3381:2 3377:c 3373:m 3357:p 3342:c 3330:E 3323:( 3288:, 3279:2 3275:) 3269:2 3265:c 3261:m 3258:( 3255:+ 3242:p 3225:p 3216:2 3212:c 3208:= 3200:2 3190:E 3148:) 3138:S 3131:, 3121:p 3114:, 3111:t 3108:, 3104:r 3100:( 3091:H 3085:= 3076:H 3059:σ 3019:S 2998:B 2991:= 2986:B 2976:H 2958:B 2941:S 2931:g 2913:, 2904:B 2896:g 2890:= 2886:| 2881:S 2871:| 2866:, 2855:S 2840:B 2832:g 2823:= 2818:S 2785:B 2783:μ 2772:t 2763:r 2761:| 2759:( 2757:ψ 2747:r 2745:| 2739:t 2735:r 2733:( 2731:ψ 2724:t 2720:0 2717:r 2715:( 2713:ψ 2707:0 2704:r 2679:ψ 2649:2 2645:) 2639:2 2635:c 2631:m 2628:( 2625:+ 2615:p 2598:p 2589:2 2585:c 2579:= 2570:t 2555:i 2543:2 2539:) 2533:2 2529:c 2525:m 2522:( 2519:+ 2509:p 2492:p 2483:2 2479:c 2473:= 2464:E 2458:= 2449:H 2415:) 2412:t 2409:, 2405:r 2401:( 2398:V 2395:+ 2389:m 2386:2 2374:p 2357:p 2347:= 2338:H 2318:) 2316:t 2312:r 2310:( 2308:V 2299:m 2295:p 2293:· 2291:p 2242:σ 2236:σ 2230:s 2221:σ 2214:s 2203:ψ 2193:s 2189:s 2181:D 2163:) 2160:) 2157:t 2154:, 2150:r 2146:( 2141:1 2130:( 2117:) 2111:( 2108:D 2102:) 2099:t 2096:, 2092:r 2088:( 2060:D 2050:σ 2048:ψ 2039:) 2037:t 2033:r 2029:t 2025:r 2023:( 1998:) 1996:p 1992:c 1990:/ 1988:E 1984:P 1976:) 1974:r 1966:X 1933:3 1930:σ 1926:2 1923:σ 1919:1 1916:σ 1912:t 1908:3 1905:r 1901:2 1898:r 1894:1 1891:r 1889:( 1887:ψ 1828:, 1824:A 1818:t 1806:+ 1803:] 1794:H 1788:, 1785:A 1782:[ 1773:i 1769:1 1764:= 1761:A 1755:t 1752:d 1748:d 1733:) 1731:t 1729:( 1727:A 1717:ψ 1699:ψ 1673:, 1663:i 1657:= 1647:p 1639:, 1632:t 1617:i 1614:= 1605:E 1565:. 1559:2 1555:) 1549:2 1545:c 1541:m 1538:( 1535:+ 1531:p 1523:p 1517:2 1513:c 1509:= 1504:2 1500:E 1473:p 1465:p 1459:= 1456:p 1438:p 1429:E 1416:m 1338:) 1336:σ 1332:t 1328:r 1326:( 1324:ψ 1318:s 1314:s 1310:s 1306:s 1302:σ 1297:z 1292:s 1290:2 1285:s 1271:s 1269:2 1264:s 1252:t 1246:r 1231:) 1229:t 1225:r 1223:( 1221:ψ 1208:Ĥ 1178:H 1172:= 1163:t 1148:i 1012:2 1008:1 923:c 894:( 875:e 868:t 861:v 89:| 79:H 73:= 63:| 56:t 53:d 49:d 41:i

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Knowledge

Relativistic quantum mechanics

Index