5764:
5213:
5759:{\displaystyle {\begin{aligned}{\hat {H}}&=-{\frac {\hbar ^{2}}{2m}}\left({\frac {\partial ^{2}}{\partial x^{2}}}+{\frac {\partial ^{2}}{\partial y^{2}}}+{\frac {\partial ^{2}}{\partial z^{2}}}\right)+{\frac {m\omega ^{2}}{2}}\left(x^{2}+y^{2}+z^{2}\right)\\&=\left(-{\frac {\hbar ^{2}}{2m}}{\frac {\partial ^{2}}{\partial x^{2}}}+{\frac {m\omega ^{2}}{2}}x^{2}\right)+\left(-{\frac {\hbar ^{2}}{2m}}{\frac {\partial ^{2}}{\partial y^{2}}}+{\frac {m\omega ^{2}}{2}}y^{2}\right)+\left(-{\frac {\hbar ^{2}}{2m}}{\frac {\partial ^{2}}{\partial z^{2}}}+{\frac {m\omega ^{2}}{2}}z^{2}\right)\end{aligned}}}
7407:
1504:
2794:
6968:
1060:
2392:
11969:
7402:{\displaystyle {\begin{aligned}{\hat {H}}&=-{\frac {\hbar ^{2}}{2}}\sum _{j=1}^{N}{\frac {1}{m_{j}}}\nabla _{j}^{2}+{\frac {1}{8\pi \varepsilon _{0}}}\sum _{j=1}^{N}\sum _{i\neq j}{\frac {q_{i}q_{j}}{|\mathbf {r} _{i}-\mathbf {r} _{j}|}}\\&=\sum _{j=1}^{N}\left(-{\frac {\hbar ^{2}}{2m_{j}}}\nabla _{j}^{2}+{\frac {1}{8\pi \varepsilon _{0}}}\sum _{i\neq j}{\frac {q_{i}q_{j}}{|\mathbf {r} _{i}-\mathbf {r} _{j}|}}\right)\\\end{aligned}}}
1499:{\displaystyle {\begin{aligned}KE&=-{\frac {\hbar ^{2}}{2m}}\int _{-\infty }^{+\infty }\psi ^{*}\left({\frac {d^{2}\psi }{dx^{2}}}\right)\,dx\\&=-{\frac {\hbar ^{2}}{2m}}\left({\left_{-\infty }^{+\infty }}-\int _{-\infty }^{+\infty }\left({\frac {d\psi }{dx}}\right)\left({\frac {d\psi }{dx}}\right)^{*}\,dx\right)\\&={\frac {\hbar ^{2}}{2m}}\int _{-\infty }^{+\infty }\left|{\frac {d\psi }{dx}}\right|^{2}\,dx\geq 0\end{aligned}}}
11181:
3419:
2789:{\displaystyle {\begin{aligned}{\hat {H}}&=\sum _{n=1}^{N}{\hat {T}}_{n}+{\hat {V}}\\&=\sum _{n=1}^{N}{\frac {\mathbf {\hat {p}} _{n}\cdot \mathbf {\hat {p}} _{n}}{2m_{n}}}+V(\mathbf {r} _{1},\mathbf {r} _{2},\ldots ,\mathbf {r} _{N},t)\\&=-{\frac {\hbar ^{2}}{2}}\sum _{n=1}^{N}{\frac {1}{m_{n}}}\nabla _{n}^{2}+V(\mathbf {r} _{1},\mathbf {r} _{2},\ldots ,\mathbf {r} _{N},t)\end{aligned}}}
2802:. Since the potential energy depends on the spatial arrangement of the particles, the kinetic energy will also depend on the spatial configuration to conserve energy. The motion due to any one particle will vary due to the motion of all the other particles in the system. For this reason cross terms for kinetic energy may appear in the Hamiltonian; a mix of the gradients for two particles:
997:
3122:
10890:
6691:
1740:
10111:
789:
6001:
2363:
3423:
where the sum is taken over all particles and their corresponding potentials; the result is that the
Hamiltonian of the system is the sum of the separate Hamiltonians for each particle. This is an idealized situation—in practice the particles are almost always influenced by some potential, and there
8500:
174:
of the system; that is, the sum of the kinetic and potential energies of all particles associated with the system. The
Hamiltonian takes different forms and can be simplified in some cases by taking into account the concrete characteristics of the system under analysis, such as single or several
9924:
variables. We can treat them as coordinates which specify the state of the system, like the position and momentum coordinates which specify a classical system. Like classical coordinates, they are generally not constant in time, and their time dependence gives rise to the time dependence of the
772:
10429:
481:
10745:
2114:
9416:
6959:
8677:
In many systems, two or more energy eigenstates have the same energy. A simple example of this is a free particle, whose energy eigenstates have wavefunctions that are propagating plane waves. The energy of each of these plane waves is inversely proportional to the square of its
3414:{\displaystyle {\begin{aligned}{\hat {H}}&=-{\frac {\hbar ^{2}}{2}}\sum _{i=1}^{N}{\frac {1}{m_{i}}}\nabla _{i}^{2}+\sum _{i=1}^{N}V_{i}\\&=\sum _{i=1}^{N}\left(-{\frac {\hbar ^{2}}{2m_{i}}}\nabla _{i}^{2}+V_{i}\right)\\&=\sum _{i=1}^{N}{\hat {H}}_{i}\end{aligned}}}
4154:
Following are expressions for the
Hamiltonian in a number of situations. Typical ways to classify the expressions are the number of particles, number of dimensions, and the nature of the potential energy function—importantly space and time dependence. Masses are denoted by
6500:
10538:
3113:
3424:
are many-body interactions. One illustrative example of a two-body interaction where this form would not apply is for electrostatic potentials due to charged particles, because they interact with each other by
Coulomb interaction (electrostatic force), as shown below.
4927:
10636:
5797:
1513:
2222:
7672:
9933:
8371:
4506:
8668:
3591:
654:
338:
5033:
4294:
1963:
5204:
3763:
1968:
4141:
From a mathematically rigorous point of view, care must be taken with the above assumptions. Operators on infinite-dimensional
Hilbert spaces need not have eigenvalues (the set of eigenvalues does not necessarily coincide with the
992:{\displaystyle {\begin{aligned}{\hat {H}}&={\hat {T}}+{\hat {V}}\\&={\frac {\mathbf {\hat {p}} \cdot \mathbf {\hat {p}} }{2m}}+V(\mathbf {r} ,t)\\&=-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}+V(\mathbf {r} ,t)\end{aligned}}}
9267:
8942:
6814:
2942:
simply a sum of the separate potentials (and certainly not a product, as this is dimensionally incorrect). The potential energy function can only be written as above: a function of all the spatial positions of each particle.
10288:
8058:
9517:
6440:
8273:
7897:
2951:
4590:
2867:
4692:
1863:
10199:
independent degrees of freedom, since the variable has a real part and an imaginary part. We now perform the following trick: instead of using the real and imaginary parts as the independent variables, we use
7816:
4106:
10885:{\displaystyle {\frac {\partial \langle H\rangle }{\partial \pi _{n}}}={\frac {\partial a_{n}}{\partial t}},\quad {\frac {\partial \langle H\rangle }{\partial a_{n}}}=-{\frac {\partial \pi _{n}}{\partial t}}}
4811:
3127:
2397:
794:
4365:
10442:
9808:
10547:
9660:
5218:
7764:
2946:
For non-interacting particles, i.e. particles which do not interact mutually and move independently, the potential of the system is the sum of the separate potential energy for each particle, that is
7584:
7533:
10154:
9726:
6444:
However, this is only the potential for one point charge due to another. If there are many charged particles, each charge has a potential energy due to every other point charge (except itself). For
1508:
Hence the expectation value of kinetic energy is always non-negative. This result can be used to calculate the expectation value of the total energy which is given for a normalized wavefunction as:
9156:
9880:
9598:
8362:
6973:
1065:
1965:
is the potential energy function, now a function of the spatial configuration of the system and time (a particular set of spatial positions at some instant of time defines a configuration) and
266:
7593:
10736:
329:
3475:
6686:{\displaystyle V_{j}={\frac {1}{2}}\sum _{i\neq j}q_{i}\phi (\mathbf {r} _{i})={\frac {1}{8\pi \varepsilon _{0}}}\sum _{i\neq j}{\frac {q_{i}q_{j}}{|\mathbf {r} _{i}-\mathbf {r} _{j}|}}}
560:
4802:
4412:
1735:{\displaystyle E=KE+\langle V(x)\rangle =KE+\int _{-\infty }^{+\infty }V(x)|\psi (x)|^{2}\,dx\geq V_{\text{min}}(x)\int _{-\infty }^{+\infty }|\psi (x)|^{2}\,dx\geq V_{\text{min}}(x)}
8574:
6731:
3502:
8309:
4936:
4009:
1052:
It can be shown that the expectation value of the
Hamiltonian which gives the energy expectation value will always be greater than or equal to the minimum potential of the system.
1039:
7701:
8182:
8565:
4213:
1870:
10972:
3868:
6787:
5102:
3674:
2213:
6214:
6176:
6138:
10278:
8204:
8153:
7976:
7723:
7485:
7439:
5996:{\displaystyle {\hat {H}}=-{\frac {\hbar ^{2}}{2I_{xx}}}{\hat {J}}_{x}^{2}-{\frac {\hbar ^{2}}{2I_{yy}}}{\hat {J}}_{y}^{2}-{\frac {\hbar ^{2}}{2I_{zz}}}{\hat {J}}_{z}^{2}}
5057:
154:
9023:
8816:
2161:
645:
8992:
8785:
105:
8823:
6492:
4046:
10946:
10665:
10234:
10193:
9918:
3922:
5095:
4717:
4405:
615:
584:
6093:
6063:
6033:
10919:
8522:
8128:
7926:
6758:
6489:
6344:
6317:
2358:{\displaystyle \nabla _{n}^{2}={\frac {\partial ^{2}}{\partial x_{n}^{2}}}+{\frac {\partial ^{2}}{\partial y_{n}^{2}}}+{\frac {\partial ^{2}}{\partial z_{n}^{2}}},}
7991:
3647:
10106:{\displaystyle \langle H(t)\rangle \mathrel {\stackrel {\mathrm {def} }{=}} \langle \psi (t)|H|\psi (t)\rangle =\sum _{nn'}a_{n'}^{*}a_{n}\langle n'|H|n\rangle }
9429:
9258:
6363:
9685:
9543:
9218:
9198:
9178:
9094:
9074:
9043:
8964:
8754:
8720:
8700:
8105:
8085:
7954:
7459:
6807:
6462:
6278:
6258:
6238:
4760:
4740:
4193:
4173:
4128:
3981:
3793:
3667:
3621:
3495:
2936:
2916:
2889:
2385:
2181:
2134:
1774:
505:
125:
8215:
7838:
11185:
8495:{\displaystyle {\hat {T}}={\frac {1}{2}}m{\dot {\mathbf {r} }}\cdot {\dot {\mathbf {r} }}={\frac {1}{2m}}\left(\mathbf {\hat {p}} -q\mathbf {A} \right)^{2}}
12213:
4515:
2807:
4619:
7768:
Since the particle is stationary, there is no translational kinetic energy of the dipole, so the
Hamiltonian of the dipole is just the potential energy:
7588:
Since the particle is stationary, there is no translational kinetic energy of the dipole, so the
Hamiltonian of the dipole is just the potential energy:
1781:
767:{\displaystyle \nabla ^{2}={\frac {\partial ^{2}}{{\partial x}^{2}}}+{\frac {\partial ^{2}}{{\partial y}^{2}}}+{\frac {\partial ^{2}}{{\partial z}^{2}}}}
7773:
4053:
476:{\displaystyle {\hat {T}}={\frac {\mathbf {\hat {p}} \cdot \mathbf {\hat {p}} }{2m}}={\frac {{\hat {p}}^{2}}{2m}}=-{\frac {\hbar ^{2}}{2m}}\nabla ^{2},}
11213:
3925:
4303:
9733:
51:, is the set of possible outcomes obtainable from a measurement of the system's total energy. Due to its close relation to the energy spectrum and
9600:, which need not necessarily be eigenstates of the energy. For simplicity, we assume that they are discrete, and that they are orthonormal, i.e.,
5777:—i.e., system of particles which can rotate freely about any axes, not bound in any potential (such as free molecules with negligible vibrational
2109:{\displaystyle {\hat {T}}_{n}={\frac {\mathbf {\hat {p}} _{n}\cdot \mathbf {\hat {p}} _{n}}{2m_{n}}}=-{\frac {\hbar ^{2}}{2m_{n}}}\nabla _{n}^{2}}
11766:
9045:
has at least one pair of degenerate energy eigenkets. In the case of the free particle, the unitary operator which produces the symmetry is the
7730:
7542:
7492:
11414:
56:
12161:
9411:{\displaystyle {\frac {\partial }{\partial t}}\langle \psi (t)|G|\psi (t)\rangle ={\frac {1}{i\hbar }}\langle \psi (t)||\psi (t)\rangle =0.}
9101:
1041:. This is the approach commonly taken in introductory treatments of quantum mechanics, using the formalism of Schrödinger's wave mechanics.
6954:{\displaystyle V={\frac {1}{8\pi \varepsilon _{0}}}\sum _{j=1}^{N}\sum _{i\neq j}{\frac {q_{i}q_{j}}{|\mathbf {r} _{i}-\mathbf {r} _{j}|}}}
9817:
8320:
10424:{\displaystyle {\frac {\partial \langle H\rangle }{\partial a_{n'}^{*}}}=\sum _{n}a_{n}\langle n'|H|n\rangle =\langle n'|H|\psi \rangle }
9605:
12005:
5778:
209:
11937:
10672:
9664:
Note that these basis states are assumed to be independent of time. We will assume that the
Hamiltonian is also independent of time.
4208:
The particle is not bound by any potential energy, so the potential is zero and this
Hamiltonian is the simplest. For one dimension:
273:
8722:
direction, but if they have the same wavelength, then their energies will be the same. When this happens, the states are said to be
11263:
2918:
interacting particles, i.e. particles which interact mutually and constitute a many-body situation, the potential energy function
175:
particles in the system, interaction between particles, kind of potential energy, time varying potential or time independent one.
11949:
518:
4767:
3108:{\displaystyle V=\sum _{i=1}^{N}V(\mathbf {r} _{i},t)=V(\mathbf {r} _{1},t)+V(\mathbf {r} _{2},t)+\cdots +V(\mathbf {r} _{N},t)}
11633:
11206:
12119:
11567:
11135:
10118:
9690:
9046:
11490:
1044:
One can also make substitutions to certain variables to fit specific cases, such as some involving electromagnetic fields.
9567:
4922:{\displaystyle {\hat {H}}=-{\frac {\hbar ^{2}}{2m}}{\frac {\partial ^{2}}{\partial x^{2}}}+{\frac {m\omega ^{2}}{2}}x^{2}}
12239:
3799:
that are not defined everywhere may not make mathematical sense. Rigorously, to take functions of unbounded operators, a
8529:
2891:
denotes the mass of the collection of particles resulting in this extra kinetic energy. Terms of this form are known as
4015:
for the Hilbert space. The spectrum of allowed energy levels of the system is given by the set of eigenvalues, denoted
3823:
10533:{\displaystyle {\frac {\partial \langle H\rangle }{\partial a_{n'}^{*}}}=i\hbar {\frac {\partial a_{n'}}{\partial t}}}
5208:
Writing the Hamiltonian out in full shows it is simply the sum of the one-dimensional Hamiltonians in each direction:
165:
11236:
11199:
11160:
11106:
11081:
10631:{\displaystyle {\frac {\partial \langle H\rangle }{\partial a_{n}}}=-i\hbar {\frac {\partial a_{n}^{*}}{\partial t}}}
12289:
11346:
3441:
12294:
11904:
11613:
11608:
7979:
3808:
3811:
suffices. We note again, however, that for common calculations the physicists' formulation is quite sufficient.
12284:
12264:
12208:
11998:
11916:
11588:
8207:
3804:
6100:
11894:
11671:
11593:
11376:
11246:
6698:
3772:
8280:
11628:
11562:
11557:
11528:
11241:
11044:
3986:
3600:
1007:
12171:
10434:
9545:
is conserved for any state of the system. In the case of the free particle, the conserved quantity is the
7684:
11696:
11603:
8158:
7667:{\displaystyle {\hat {H}}=-\mathbf {\hat {d}} \cdot \mathbf {E} =-q\mathbf {\hat {r}} \cdot \mathbf {E} }
9225:
12279:
12274:
11972:
11734:
11542:
11513:
10951:
9053:
4611:
6763:
12166:
11991:
11756:
11623:
11547:
11508:
11465:
11439:
11396:
11289:
2186:
1744:
which complete the proof. Similarly, the condition can be generalized to any higher dimensions using
6183:
6145:
6107:
4146:). However, all routine quantum mechanical calculations can be done using the physical formulation.
3771:
operator on the right hand side of the Schrödinger equation is usually defined by the corresponding
12269:
12187:
11823:
11803:
11793:
11783:
11739:
11314:
10989:
8730:
6289:
10242:
8187:
8136:
7959:
7706:
7468:
7422:
5040:
130:
12077:
11518:
11444:
8997:
8790:
4501:{\displaystyle {\hat {H}}=-{\frac {\hbar ^{2}}{2m}}{\frac {\partial ^{2}}{\partial x^{2}}}+V_{0}}
4143:
2139:
623:
52:
40:
11932:
11480:
8969:
8762:
8663:{\displaystyle {\hat {H}}={\frac {1}{2m}}\left(-i\hbar \nabla -q\mathbf {A} \right)^{2}+q\phi .}
3596:
3586:{\displaystyle H\left|\psi (t)\right\rangle =i\hbar {d \over \ dt}\left|\psi (t)\right\rangle .}
3433:
781:
81:
12035:
11843:
11618:
11598:
11523:
11391:
11014:
9557:
7417:
4018:
590:
63:
11434:
10924:
10643:
10203:
10162:
9887:
5028:{\displaystyle {\hat {H}}=-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}+{\frac {m\omega ^{2}}{2}}r^{2}}
3892:
12203:
11868:
11381:
11361:
10984:
9561:
5062:
4702:
4377:
777:
648:
600:
569:
189:
71:
8155:, there are two parts to the Hamiltonian to substitute for. The canonical momentum operator
6068:
6038:
6008:
11899:
11828:
11773:
11503:
11326:
11284:
10999:
10897:
9421:
8756:
8507:
8113:
7904:
6736:
6467:
6322:
6295:
4289:{\displaystyle {\hat {H}}=-{\frac {\hbar ^{2}}{2m}}{\frac {\partial ^{2}}{\partial x^{2}}}}
3945:
1958:{\displaystyle {\hat {V}}=V(\mathbf {r} _{1},\mathbf {r} _{2},\ldots ,\mathbf {r} _{N},t),}
9928:
The expectation value of the Hamiltonian of this state, which is also the mean energy, is
8:
11884:
11853:
11798:
11778:
11686:
11643:
11498:
11424:
11351:
11341:
11253:
7462:
5199:{\displaystyle r^{2}=\mathbf {r} \cdot \mathbf {r} =|\mathbf {r} |^{2}=x^{2}+y^{2}+z^{2}}
3800:
3758:{\displaystyle \left|\psi (t)\right\rangle =e^{-iHt/\hbar }\left|\psi (0)\right\rangle .}
3626:
193:
67:
9049:, which rotates the wavefunctions by some angle while otherwise preserving their shape.
12014:
11944:
11813:
11711:
11419:
11366:
11258:
11124:
10994:
10281:
9670:
9528:
9203:
9183:
9163:
9079:
9059:
9028:
8949:
8739:
8705:
8685:
8090:
8070:
7939:
7933:
7929:
7444:
6792:
6447:
6263:
6243:
6223:
4745:
4725:
4178:
4158:
4131:
4113:
3966:
3796:
3778:
3768:
3652:
3606:
3480:
2921:
2901:
2874:
2370:
2166:
2119:
1759:
1745:
490:
110:
28:
12114:
12109:
11954:
11863:
11833:
11761:
11724:
11719:
11701:
11666:
11656:
11371:
11336:
11319:
11222:
11156:
11131:
11102:
11077:
11070:
11024:
10237:
6346:(i.e., those that have no spatial extent independently), in three dimensions, is (in
6096:
4697:
4595:
4012:
2799:
780:, it is the form it most commonly takes. Combining these yields the form used in the
563:
74:, which was historically important to the development of quantum physics. Similar to
20:
12218:
12040:
11681:
11676:
11533:
11429:
11019:
9546:
8312:
8131:
8108:
6355:
6217:
4614:
in one dimension, the potential varies with position (but not time), according to:
3933:
3873:
332:
201:
36:
9564:
have a direct analogy in quantum mechanics. Suppose we have a set of basis states
8937:{\displaystyle UH|a\rangle =UE_{a}|a\rangle =E_{a}(U|a\rangle )=H\;(U|a\rangle ).}
6493:
Electrostatic potential energy stored in a configuration of discrete point charges
12234:
12135:
12082:
11911:
11838:
11818:
11788:
11751:
11746:
11651:
11475:
4720:
3649:), we can solve it to obtain the state at any subsequent time. In particular, if
75:
9420:
In obtaining this result, we have used the Schrödinger equation, as well as its
12145:
12045:
11889:
11858:
11848:
11470:
11460:
11294:
11039:
11034:
11029:
11009:
9921:
9522:
8734:
7983:
6351:
4599:
3878:
484:
197:
32:
8672:
12258:
11808:
11661:
11552:
11386:
11356:
11309:
11004:
8053:{\displaystyle {\hat {H}}={\frac {g_{s}e}{2m}}\mathbf {S} \cdot \mathbf {B} }
5789:
3953:
1002:
107:, where the hat indicates that it is an operator. It can also be written as
11691:
11304:
11299:
9512:{\displaystyle \langle \psi (t)|H=-i\hbar {d \over \ dt}\langle \psi (t)|.}
6435:{\displaystyle V={\frac {q_{1}q_{2}}{4\pi \varepsilon _{0}|\mathbf {r} |}}}
3960:
3929:
3815:
10437:
and using the orthonormality of the basis states, this further reduces to
11729:
8268:{\displaystyle \mathbf {\hat {p}} =m{\dot {\mathbf {r} }}+q\mathbf {A} ,}
7892:{\displaystyle {\boldsymbol {\mu }}_{S}={\frac {g_{s}e}{2m}}\mathbf {S} }
5786:
5782:
5774:
4135:
594:
512:
4407:(no dependence on space or time), in one dimension, the Hamiltonian is:
12061:
11983:
8679:
3949:
3885:
3623:
is also called the Hamiltonian. Given the state at some initial time (
11583:
11279:
4585:{\displaystyle {\hat {H}}=-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}+V_{0}}
2862:{\displaystyle -{\frac {\hbar ^{2}}{2M}}\nabla _{i}\cdot \nabla _{j}}
618:
11191:
4687:{\displaystyle V={\frac {k}{2}}x^{2}={\frac {m\omega ^{2}}{2}}x^{2}}
3889:
of a closed quantum system. If the Hamiltonian is time-independent,
2895:, and appear in the Hamiltonian of many-electron atoms (see below).
1001:
which allows one to apply the Hamiltonian to systems described by a
7821:
6347:
3438:
The Hamiltonian generates the time evolution of quantum states. If
1858:{\displaystyle {\hat {H}}=\sum _{n=1}^{N}{\hat {T}}_{n}+{\hat {V}}}
10280:. With this choice of independent variables, we can calculate the
7811:{\displaystyle {\hat {H}}=-{\boldsymbol {\mu }}\cdot \mathbf {B} }
4101:{\displaystyle H\left|a\right\rangle =E_{a}\left|a\right\rangle .}
11180:
11072:
Quantum Physics of Atoms, Molecules, Solids, Nuclei and Particles
31:
corresponding to the total energy of that system, including both
9052:
The existence of a symmetry operator implies the existence of a
8062:
4360:{\displaystyle {\hat {H}}=-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}}
3952:, the Hamiltonian is typically implemented as an operator on a
171:
10894:
which is precisely the form of Hamilton's equations, with the
3795:. One might notice that taking polynomials or power series of
9803:{\displaystyle |\psi (t)\rangle =\sum _{n}a_{n}(t)|n\rangle }
1055:
Consider computing the expectation value of kinetic energy:
8702:
direction is a different state from one propagating in the
8673:
Energy eigenket degeneracy, symmetry, and conservation laws
2367:
Combining these yields the Schrödinger Hamiltonian for the
508:
3803:
is required. In the case of the exponential function, the
7759:{\displaystyle V=-{\boldsymbol {\mu }}\cdot \mathbf {B} }
6789:. The total potential of the system is then the sum over
587:
7579:{\displaystyle \mathbf {\hat {d}} =q\mathbf {\hat {r}} }
7528:{\displaystyle V=-\mathbf {\hat {d}} \cdot \mathbf {E} }
166:
Operator (physics) § Operators in quantum mechanics
10149:{\displaystyle \left|\psi \left(t\right)\right\rangle }
9721:{\displaystyle \left|\psi \left(t\right)\right\rangle }
7411:
9151:{\displaystyle U=I-i\varepsilon G+O(\varepsilon ^{2})}
8818:
is an energy eigenket with the same eigenvalue, since
7676:
192:, the Hamiltonian is commonly expressed as the sum of
10954:
10927:
10900:
10748:
10675:
10646:
10550:
10445:
10291:
10245:
10206:
10165:
10121:
9936:
9890:
9820:
9736:
9693:
9673:
9608:
9570:
9531:
9432:
9270:
9228:
9206:
9186:
9166:
9104:
9082:
9062:
9031:
9000:
8972:
8952:
8826:
8793:
8765:
8742:
8708:
8688:
8577:
8532:
8510:
8374:
8323:
8283:
8218:
8190:
8161:
8139:
8116:
8093:
8073:
7994:
7962:
7942:
7907:
7841:
7833:
particle, the corresponding spin magnetic moment is:
7776:
7733:
7709:
7703:
in a uniform, magnetostatic field (time-independent)
7687:
7596:
7545:
7495:
7471:
7447:
7425:
6971:
6817:
6795:
6766:
6739:
6701:
6503:
6470:
6450:
6366:
6325:
6298:
6266:
6246:
6226:
6186:
6148:
6110:
6099:
components (technically the diagonal elements of the
6071:
6041:
6011:
5800:
5216:
5105:
5065:
5043:
4939:
4814:
4770:
4748:
4728:
4705:
4622:
4518:
4415:
4380:
4306:
4216:
4181:
4161:
4116:
4056:
4021:
3989:
3969:
3895:
3826:
3781:
3677:
3655:
3629:
3609:
3505:
3483:
3444:
3125:
3117:
The general form of the Hamiltonian in this case is:
2954:
2924:
2904:
2877:
2810:
2395:
2373:
2225:
2189:
2169:
2142:
2122:
1971:
1873:
1784:
1762:
1516:
1063:
1010:
792:
776:
Although this is not the technical definition of the
657:
626:
603:
572:
521:
493:
341:
276:
212:
133:
113:
84:
55:
of a system, it is of fundamental importance in most
9875:{\displaystyle a_{n}(t)=\langle n|\psi (t)\rangle .}
9593:{\displaystyle \left\{\left|n\right\rangle \right\}}
8357:{\displaystyle \mathbf {\hat {p}} =-i\hbar \nabla ,}
6283:
16:
Quantum operator for the sum of energies of a system
11123:
11069:
10966:
10940:
10913:
10884:
10730:
10659:
10630:
10532:
10423:
10272:
10228:
10187:
10148:
10105:
9912:
9874:
9802:
9728:, can be expanded in terms of these basis states:
9720:
9679:
9655:{\displaystyle \langle n'|n\rangle =\delta _{nn'}}
9654:
9592:
9537:
9511:
9410:
9252:
9212:
9192:
9172:
9150:
9088:
9068:
9037:
9017:
8986:
8958:
8936:
8810:
8779:
8748:
8714:
8694:
8662:
8559:
8516:
8494:
8356:
8303:
8267:
8198:
8176:
8147:
8122:
8099:
8079:
8052:
7970:
7948:
7920:
7891:
7810:
7758:
7717:
7695:
7666:
7578:
7527:
7479:
7453:
7433:
7401:
6953:
6801:
6781:
6752:
6725:
6685:
6483:
6456:
6434:
6338:
6311:
6272:
6252:
6232:
6208:
6170:
6132:
6087:
6057:
6027:
5995:
5758:
5198:
5089:
5051:
5027:
4921:
4796:
4754:
4734:
4711:
4686:
4584:
4500:
4399:
4359:
4288:
4187:
4167:
4149:
4122:
4100:
4040:
4003:
3975:
3916:
3862:
3818:property of the functional calculus, the operator
3787:
3757:
3661:
3641:
3615:
3585:
3489:
3469:
3413:
3107:
2930:
2910:
2883:
2861:
2788:
2379:
2357:
2207:
2175:
2155:
2128:
2108:
1957:
1857:
1768:
1734:
1498:
1033:
991:
766:
639:
609:
578:
554:
499:
475:
323:
260:
148:
119:
99:
8464:
8330:
8225:
8168:
7650:
7621:
7570:
7552:
7511:
4374:For a particle in a region of constant potential
4198:
2537:
2515:
2026:
2004:
877:
862:
381:
366:
261:{\displaystyle {\hat {H}}={\hat {T}}+{\hat {V}},}
170:The Hamiltonian of a system represents the total
66:, who developed a revolutionary reformulation of
12256:
11150:
11063:
11061:
8759:with the Hamiltonian. To see this, suppose that
8569:Casting all of these into the Hamiltonian gives
8366:so the corresponding kinetic energy operator is
3932:); this gives rise to the physical principle of
10974:taking the place of the classical Hamiltonian.
10731:{\displaystyle \pi _{n}(t)=i\hbar a_{n}^{*}(t)}
324:{\displaystyle {\hat {V}}=V=V(\mathbf {r} ,t),}
11067:
10115:where the last step was obtained by expanding
9667:The instantaneous state of the system at time
8504:and the potential energy, which is due to the
8107:in an electromagnetic field, described by the
11999:
11207:
11121:
11058:
7725:, positioned in one place, the potential is:
7487:, positioned in one place, the potential is:
4605:
11130:. Manchester Physics Series (2nd ed.).
10961:
10955:
10828:
10822:
10761:
10755:
10640:If we define "conjugate momentum" variables
10563:
10557:
10458:
10452:
10418:
10391:
10385:
10358:
10304:
10298:
10100:
10073:
10019:
9979:
9952:
9937:
9866:
9843:
9797:
9754:
9628:
9609:
9486:
9433:
9399:
9347:
9326:
9286:
9012:
8981:
8925:
8898:
8868:
8841:
8805:
8774:
8063:Charged particle in an electromagnetic field
5037:where the three-dimensional position vector
4035:
4022:
3911:
3896:
3470:{\displaystyle \left|\psi (t)\right\rangle }
1547:
1532:
9025:must represent distinct states. Therefore,
2116:is the kinetic energy operator of particle
555:{\displaystyle {\hat {p}}=-i\hbar \nabla ,}
12006:
11992:
11214:
11200:
8910:
4797:{\displaystyle \omega ^{2}={\frac {k}{m}}}
4369:
178:
11151:Bransden, B. H.; Joachain, C. J. (1983).
8184:, which includes a contribution from the
7537:the dipole moment itself is the operator
6733:is the electrostatic potential of charge
1703:
1624:
1479:
1379:
1179:
12013:
6464:charges, the potential energy of charge
2798:However, complications can arise in the
11076:(2nd ed.). John Wiley & Sons.
9552:
7844:
7796:
7744:
7689:
6726:{\displaystyle \phi (\mathbf {r} _{i})}
3427:
12257:
11122:Grant, I. S.; Phillips, W. R. (2008).
11096:
10921:s as the generalized coordinates, the
9160:It is straightforward to show that if
8315:. The quantization prescription reads
8304:{\displaystyle m{\dot {\mathbf {r} }}}
6491:due to all other charges is (see also
11987:
11221:
11195:
4004:{\displaystyle \left|a\right\rangle }
1034:{\displaystyle \Psi (\mathbf {r} ,t)}
511:of the particle, the dot denotes the
8966:is nontrivial, at least one pair of
7696:{\displaystyle {\boldsymbol {\mu }}}
7412:Electric dipole in an electric field
1047:
8177:{\displaystyle \mathbf {\hat {p}} }
7677:Magnetic dipole in a magnetic field
4931:For three dimensions, this becomes
3477:is the state of the system at time
13:
10873:
10858:
10833:
10819:
10803:
10788:
10766:
10752:
10619:
10599:
10568:
10554:
10521:
10501:
10463:
10449:
10309:
10295:
9971:
9968:
9965:
9277:
9273:
8623:
8560:{\displaystyle {\hat {V}}=q\phi .}
8348:
7441:constituting charges of magnitude
7267:
7054:
6220:operators (components), about the
5696:
5686:
5594:
5584:
5492:
5482:
5347:
5337:
5315:
5305:
5283:
5273:
4981:
4868:
4858:
4560:
4469:
4459:
4348:
4270:
4260:
3939:
3325:
3208:
2850:
2837:
2702:
2331:
2321:
2294:
2284:
2257:
2247:
2227:
2191:
2144:
2092:
1669:
1661:
1578:
1570:
1439:
1431:
1311:
1303:
1286:
1278:
1122:
1114:
1011:
953:
778:Hamiltonian in classical mechanics
749:
738:
717:
706:
685:
674:
659:
628:
604:
573:
546:
461:
14:
12306:
11173:
11068:Resnick, R.; Eisberg, R. (1985).
10967:{\displaystyle \langle H\rangle }
10701:
10593:
10495:
9465:
9341:
8620:
8345:
7982:vector, whose components are the
6284:Electrostatic (Coulomb) potential
3863:{\displaystyle U=e^{-iHt/\hbar }}
3855:
3725:
3537:
1756:The formalism can be extended to
1751:
543:
204:energies of a system in the form
11968:
11967:
11179:
10948:s as the conjugate momenta, and
10740:then the above equations become
8633:
8477:
8461:
8424:
8407:
8327:
8291:
8258:
8241:
8222:
8192:
8165:
8141:
8046:
8038:
7964:
7885:
7804:
7752:
7711:
7660:
7647:
7631:
7618:
7567:
7549:
7521:
7508:
7473:
7427:
7372:
7357:
7180:
7165:
6933:
6918:
6782:{\displaystyle \mathbf {r} _{i}}
6769:
6710:
6665:
6650:
6561:
6420:
5141:
5128:
5120:
5045:
4594:This applies to the elementary "
4203:
3599:. It takes the same form as the
3086:
3050:
3020:
2990:
2763:
2742:
2727:
2614:
2593:
2578:
2534:
2512:
2023:
2001:
1933:
1912:
1897:
1018:
972:
904:
874:
859:
378:
363:
305:
11186:Hamiltonian (quantum mechanics)
10815:
7932:" (not to be confused with the
5059:using Cartesian coordinates is
4150:Expressions for the Hamiltonian
3809:holomorphic functional calculus
2208:{\displaystyle \nabla _{n}^{2}}
183:
159:
62:The Hamiltonian is named after
12209:Hanbury Brown and Twiss effect
11917:Relativistic quantum mechanics
11153:Physics of Atoms and Molecules
11144:
11115:
11099:Quanta: A Handbook of Concepts
11090:
10725:
10719:
10692:
10686:
10411:
10403:
10378:
10370:
10267:
10261:
10223:
10217:
10182:
10176:
10156:in terms of the basis states.
10093:
10085:
10016:
10010:
10003:
9995:
9991:
9985:
9949:
9943:
9907:
9901:
9863:
9857:
9850:
9837:
9831:
9790:
9786:
9780:
9751:
9745:
9738:
9621:
9502:
9498:
9492:
9449:
9445:
9439:
9396:
9390:
9383:
9379:
9367:
9363:
9359:
9353:
9323:
9317:
9310:
9302:
9298:
9292:
9241:
9229:
9145:
9132:
9076:be the Hermitian generator of
9005:
8974:
8928:
8918:
8911:
8901:
8891:
8884:
8861:
8834:
8798:
8767:
8584:
8539:
8381:
8208:canonical commutation relation
8001:
7783:
7603:
7383:
7351:
7191:
7159:
6982:
6944:
6912:
6720:
6705:
6676:
6644:
6571:
6556:
6425:
6415:
6209:{\displaystyle {\hat {J}}_{z}}
6194:
6171:{\displaystyle {\hat {J}}_{y}}
6156:
6133:{\displaystyle {\hat {J}}_{x}}
6118:
5976:
5917:
5858:
5807:
5768:
5227:
5147:
5136:
5084:
5066:
4946:
4821:
4525:
4422:
4313:
4223:
4199:General forms for one particle
3908:
3902:
3744:
3738:
3692:
3686:
3603:, which is one of the reasons
3572:
3566:
3523:
3517:
3459:
3453:
3395:
3136:
3102:
3081:
3066:
3045:
3036:
3015:
3006:
2985:
2779:
2722:
2630:
2573:
2468:
2447:
2406:
2215:is the Laplacian for particle
1979:
1949:
1892:
1880:
1849:
1828:
1791:
1729:
1723:
1693:
1688:
1682:
1675:
1650:
1644:
1614:
1609:
1603:
1596:
1592:
1586:
1544:
1538:
1266:
1260:
1247:
1241:
1028:
1014:
982:
968:
914:
900:
837:
822:
803:
528:
410:
348:
315:
301:
283:
249:
234:
219:
140:
91:
57:formulations of quantum theory
1:
11895:Quantum statistical mechanics
11672:Quantum differential calculus
11594:Delayed-choice quantum eraser
11377:Symmetry in quantum mechanics
11051:
10542:Similarly, one can show that
8733:occurs whenever a nontrivial
8682:. A wave propagating in the
7681:For a magnetic dipole moment
6354:which are frequently used in
3669:is independent of time, then
2163:is the gradient for particle
78:, it is typically denoted by
10273:{\displaystyle a_{n}^{*}(t)}
8787:is an energy eigenket. Then
8199:{\displaystyle \mathbf {A} }
8148:{\displaystyle \mathbf {A} }
7971:{\displaystyle \mathbf {S} }
7718:{\displaystyle \mathbf {B} }
7480:{\displaystyle \mathbf {E} }
7434:{\displaystyle \mathbf {d} }
5052:{\displaystyle \mathbf {r} }
647:. In three dimensions using
149:{\displaystyle {\check {H}}}
7:
11697:Quantum stochastic calculus
11687:Quantum measurement problem
11609:Mach–Zehnder interferometer
11101:. Oxford University Press.
10977:
9018:{\displaystyle U|a\rangle }
8811:{\displaystyle U|a\rangle }
4762:of the oscillator satisfy:
3926:one parameter unitary group
2156:{\displaystyle \nabla _{n}}
640:{\displaystyle \nabla ^{2}}
10:
12313:
9560:'s equations in classical
8987:{\displaystyle |a\rangle }
8780:{\displaystyle |a\rangle }
4612:simple harmonic oscillator
4606:Simple harmonic oscillator
4298:and in higher dimensions:
3431:
163:
100:{\displaystyle {\hat {H}}}
12240:Creation and annihilation
12227:
12196:
12180:
12154:
12128:
12102:
12095:
12070:
12054:
12028:
12021:
11963:
11925:
11877:
11757:Quantum complexity theory
11735:Quantum cellular automata
11710:
11642:
11576:
11489:
11453:
11440:Path integral formulation
11407:
11272:
11229:
8067:For a particle with mass
4134:, the energy is always a
4041:{\displaystyle \{E_{a}\}}
12188:Transition dipole moment
11824:Quantum machine learning
11804:Quantum key distribution
11794:Quantum image processing
11784:Quantum error correction
11634:Wheeler's delayed choice
11045:Lieb–Thirring inequality
10990:Two-state quantum system
10941:{\displaystyle \pi _{n}}
10660:{\displaystyle \pi _{n}}
10229:{\displaystyle a_{n}(t)}
10195:actually corresponds to
10188:{\displaystyle a_{n}(t)}
9913:{\displaystyle a_{n}(t)}
7956:is the electron charge,
6290:Coulomb potential energy
6101:moment of inertia tensor
4048:, solving the equation:
3917:{\displaystyle \{U(t)\}}
3601:Hamilton–Jacobi equation
651:the Laplace operator is
12290:Computational chemistry
12078:Anti-symmetric operator
12071:Operators for operators
11740:Quantum finite automata
6963:so the Hamiltonian is:
5792:), the Hamiltonian is:
5090:{\displaystyle (x,y,z)}
4806:so the Hamiltonian is:
4712:{\displaystyle \omega }
4400:{\displaystyle V=V_{0}}
4370:Constant-potential well
4144:spectrum of an operator
2893:mass polarization terms
610:{\displaystyle \nabla }
579:{\displaystyle \nabla }
179:Schrödinger Hamiltonian
12295:William Rowan Hamilton
11844:Quantum neural network
11184:Quotations related to
11097:Atkins, P. W. (1974).
11015:Conservation of energy
10968:
10942:
10915:
10886:
10732:
10661:
10632:
10534:
10435:Schrödinger's equation
10425:
10274:
10230:
10189:
10150:
10107:
9914:
9876:
9804:
9722:
9681:
9656:
9594:
9539:
9513:
9412:
9254:
9214:
9194:
9174:
9152:
9090:
9070:
9039:
9019:
8988:
8960:
8938:
8812:
8781:
8750:
8716:
8696:
8664:
8561:
8518:
8496:
8358:
8305:
8269:
8206:field and fulfils the
8200:
8178:
8149:
8124:
8101:
8081:
8054:
7972:
7950:
7922:
7893:
7812:
7760:
7719:
7697:
7668:
7580:
7529:
7481:
7455:
7435:
7418:electric dipole moment
7403:
7228:
7116:
7035:
6955:
6869:
6803:
6783:
6754:
6727:
6687:
6485:
6458:
6436:
6340:
6313:
6292:for two point charges
6274:
6254:
6234:
6210:
6172:
6134:
6089:
6088:{\displaystyle I_{zz}}
6059:
6058:{\displaystyle I_{yy}}
6029:
6028:{\displaystyle I_{xx}}
5997:
5760:
5200:
5091:
5053:
5029:
4923:
4798:
4756:
4736:
4713:
4688:
4586:
4502:
4401:
4361:
4290:
4189:
4169:
4124:
4102:
4042:
4005:
3977:
3956:in the following way:
3946:more general formalism
3918:
3864:
3789:
3759:
3663:
3643:
3617:
3587:
3491:
3471:
3415:
3387:
3286:
3245:
3189:
3109:
2981:
2932:
2912:
2885:
2863:
2790:
2683:
2504:
2439:
2381:
2359:
2209:
2177:
2157:
2130:
2110:
1959:
1859:
1820:
1770:
1736:
1500:
1035:
993:
768:
641:
611:
580:
556:
501:
477:
325:
262:
150:
121:
101:
64:William Rowan Hamilton
12285:Theoretical chemistry
12265:Hamiltonian mechanics
11869:Quantum teleportation
11397:Wave–particle duality
10985:Hamiltonian mechanics
10969:
10943:
10916:
10914:{\displaystyle a_{n}}
10887:
10733:
10662:
10633:
10535:
10426:
10275:
10231:
10190:
10151:
10108:
9915:
9877:
9805:
9723:
9682:
9657:
9595:
9562:Hamiltonian mechanics
9540:
9514:
9413:
9255:
9215:
9195:
9175:
9153:
9091:
9071:
9040:
9020:
8989:
8961:
8939:
8813:
8782:
8751:
8717:
8697:
8665:
8562:
8519:
8517:{\displaystyle \phi }
8497:
8359:
8306:
8270:
8210:, must be quantized;
8201:
8179:
8150:
8125:
8123:{\displaystyle \phi }
8102:
8082:
8055:
7973:
7951:
7923:
7921:{\displaystyle g_{s}}
7894:
7813:
7761:
7720:
7698:
7669:
7581:
7530:
7482:
7456:
7436:
7404:
7208:
7096:
7015:
6956:
6849:
6804:
6784:
6755:
6753:{\displaystyle q_{j}}
6728:
6688:
6486:
6484:{\displaystyle q_{j}}
6459:
6437:
6341:
6339:{\displaystyle q_{2}}
6314:
6312:{\displaystyle q_{1}}
6275:
6255:
6235:
6211:
6173:
6135:
6090:
6060:
6030:
5998:
5761:
5201:
5092:
5054:
5030:
4924:
4799:
4757:
4737:
4714:
4689:
4587:
4503:
4402:
4362:
4291:
4190:
4170:
4125:
4103:
4043:
4006:
3978:
3919:
3865:
3790:
3760:
3664:
3644:
3618:
3595:This equation is the
3588:
3492:
3472:
3416:
3367:
3266:
3225:
3169:
3110:
2961:
2933:
2913:
2886:
2864:
2791:
2663:
2484:
2419:
2382:
2360:
2210:
2178:
2158:
2131:
2111:
1960:
1860:
1800:
1771:
1737:
1501:
1036:
994:
769:
649:Cartesian coordinates
642:
612:
581:
557:
502:
478:
326:
263:
196:corresponding to the
151:
122:
102:
72:Hamiltonian mechanics
12015:Operators in physics
11900:Quantum field theory
11829:Quantum metamaterial
11774:Quantum cryptography
11504:Consistent histories
10952:
10925:
10898:
10746:
10673:
10644:
10548:
10443:
10289:
10243:
10204:
10163:
10119:
9934:
9888:
9818:
9734:
9691:
9671:
9606:
9568:
9553:Hamilton's equations
9529:
9430:
9268:
9226:
9204:
9184:
9164:
9102:
9080:
9060:
9029:
8998:
8970:
8950:
8824:
8791:
8763:
8740:
8706:
8686:
8575:
8530:
8508:
8372:
8321:
8281:
8216:
8188:
8159:
8137:
8114:
8091:
8071:
7992:
7960:
7940:
7905:
7839:
7774:
7731:
7707:
7685:
7594:
7543:
7493:
7469:
7445:
7423:
6969:
6815:
6793:
6764:
6737:
6699:
6501:
6468:
6448:
6364:
6323:
6296:
6264:
6244:
6224:
6184:
6146:
6108:
6069:
6039:
6009:
5798:
5214:
5103:
5063:
5041:
4937:
4812:
4768:
4746:
4726:
4703:
4620:
4516:
4510:in three dimensions
4413:
4378:
4304:
4214:
4179:
4159:
4114:
4054:
4019:
3987:
3967:
3893:
3824:
3779:
3675:
3653:
3627:
3607:
3597:Schrödinger equation
3503:
3481:
3442:
3434:Schrödinger equation
3428:Schrödinger equation
3123:
2952:
2922:
2902:
2875:
2808:
2393:
2371:
2223:
2187:
2167:
2140:
2120:
1969:
1871:
1782:
1760:
1514:
1061:
1008:
790:
782:Schrödinger equation
655:
624:
601:
570:
519:
491:
339:
274:
210:
131:
111:
82:
11885:Quantum fluctuation
11854:Quantum programming
11814:Quantum logic gates
11799:Quantum information
11779:Quantum electronics
11254:Classical mechanics
10718:
10616:
10485:
10331:
10260:
10062:
9925:system as a whole.
8524:field, is given by
7465:(time-independent)
7463:electrostatic field
7280:
7067:
6280:axes respectively.
5992:
5933:
5874:
5097:, its magnitude is
3801:functional calculus
3797:unbounded operators
3642:{\displaystyle t=0}
3338:
3221:
2715:
2348:
2311:
2274:
2240:
2204:
2105:
1673:
1582:
1443:
1315:
1290:
1126:
617:with itself is the
190:classical mechanics
68:Newtonian mechanics
12214:Quantum correlator
11938:in popular culture
11720:Quantum algorithms
11568:Von Neumann–Wigner
11548:Objective collapse
11259:Old quantum theory
10995:Operator (physics)
10964:
10938:
10911:
10882:
10728:
10704:
10657:
10628:
10602:
10530:
10466:
10421:
10347:
10312:
10282:partial derivative
10270:
10246:
10226:
10185:
10146:
10103:
10043:
10042:
9910:
9872:
9800:
9769:
9718:
9677:
9652:
9590:
9535:
9525:of the observable
9509:
9408:
9253:{\displaystyle =0}
9250:
9210:
9190:
9170:
9148:
9086:
9066:
9035:
9015:
8984:
8956:
8934:
8808:
8777:
8746:
8729:It turns out that
8712:
8692:
8660:
8557:
8514:
8492:
8354:
8301:
8265:
8196:
8174:
8145:
8120:
8097:
8077:
8050:
7968:
7946:
7934:gyromagnetic ratio
7918:
7889:
7808:
7756:
7715:
7693:
7664:
7576:
7525:
7477:
7451:
7431:
7399:
7397:
7324:
7266:
7132:
7053:
6951:
6885:
6799:
6779:
6750:
6723:
6683:
6617:
6542:
6481:
6454:
6432:
6336:
6309:
6270:
6250:
6230:
6206:
6168:
6130:
6085:
6055:
6025:
5993:
5969:
5910:
5851:
5779:degrees of freedom
5756:
5754:
5196:
5087:
5049:
5025:
4919:
4794:
4752:
4732:
4709:
4684:
4582:
4498:
4397:
4357:
4286:
4185:
4165:
4132:Hermitian operator
4120:
4098:
4038:
4001:
3973:
3914:
3860:
3785:
3755:
3659:
3639:
3613:
3583:
3487:
3467:
3411:
3409:
3324:
3207:
3105:
2928:
2908:
2881:
2859:
2786:
2784:
2701:
2377:
2355:
2334:
2297:
2260:
2226:
2205:
2190:
2173:
2153:
2126:
2106:
2091:
1955:
1855:
1766:
1746:divergence theorem
1732:
1653:
1562:
1496:
1494:
1423:
1295:
1227:
1106:
1031:
989:
987:
764:
637:
607:
576:
552:
497:
487:operator in which
473:
321:
258:
146:
117:
97:
49:energy eigenvalues
27:of a system is an
12280:Quantum chemistry
12275:Quantum mechanics
12252:
12251:
12248:
12247:
12235:Casimir invariant
12091:
12090:
11981:
11980:
11955:Quantum mysticism
11933:Schrödinger's cat
11864:Quantum simulator
11834:Quantum metrology
11762:Quantum computing
11725:Quantum amplifier
11702:Quantum spacetime
11667:Quantum cosmology
11657:Quantum chemistry
11372:Scattering theory
11320:Zero-point energy
11315:Degenerate levels
11223:Quantum mechanics
11137:978-0-471-92712-9
11025:Many-body problem
10880:
10847:
10810:
10780:
10626:
10582:
10528:
10487:
10338:
10333:
10238:complex conjugate
10025:
9976:
9884:The coefficients
9760:
9680:{\displaystyle t}
9538:{\displaystyle G}
9484:
9476:
9345:
9284:
9213:{\displaystyle G}
9193:{\displaystyle H}
9173:{\displaystyle U}
9089:{\displaystyle U}
9069:{\displaystyle G}
9047:rotation operator
9038:{\displaystyle H}
8959:{\displaystyle U}
8749:{\displaystyle U}
8715:{\displaystyle y}
8695:{\displaystyle x}
8606:
8587:
8542:
8467:
8450:
8431:
8414:
8398:
8384:
8333:
8298:
8248:
8228:
8171:
8100:{\displaystyle q}
8080:{\displaystyle m}
8035:
8004:
7949:{\displaystyle e}
7882:
7786:
7653:
7624:
7606:
7573:
7555:
7514:
7454:{\displaystyle q}
7388:
7309:
7307:
7264:
7196:
7117:
7094:
7051:
7013:
6985:
6949:
6870:
6847:
6802:{\displaystyle j}
6681:
6602:
6600:
6527:
6525:
6457:{\displaystyle N}
6430:
6273:{\displaystyle z}
6253:{\displaystyle y}
6233:{\displaystyle x}
6197:
6159:
6121:
6097:moment of inertia
5979:
5967:
5920:
5908:
5861:
5849:
5810:
5735:
5710:
5681:
5633:
5608:
5579:
5531:
5506:
5477:
5391:
5361:
5329:
5297:
5263:
5230:
5013:
4978:
4949:
4907:
4882:
4853:
4824:
4792:
4755:{\displaystyle m}
4735:{\displaystyle k}
4698:angular frequency
4672:
4637:
4596:particle in a box
4557:
4528:
4483:
4454:
4425:
4345:
4316:
4284:
4255:
4226:
4188:{\displaystyle q}
4175:, and charges by
4168:{\displaystyle m}
4123:{\displaystyle H}
4013:orthonormal basis
3976:{\displaystyle H}
3788:{\displaystyle H}
3662:{\displaystyle H}
3616:{\displaystyle H}
3556:
3548:
3490:{\displaystyle t}
3398:
3322:
3205:
3167:
3139:
2931:{\displaystyle V}
2911:{\displaystyle N}
2884:{\displaystyle M}
2834:
2800:many-body problem
2699:
2661:
2565:
2540:
2518:
2471:
2450:
2409:
2380:{\displaystyle N}
2350:
2313:
2276:
2176:{\displaystyle n}
2129:{\displaystyle n}
2089:
2054:
2029:
2007:
1982:
1883:
1852:
1831:
1794:
1769:{\displaystyle N}
1720:
1641:
1467:
1421:
1367:
1338:
1219:
1173:
1104:
1048:Expectation value
950:
892:
880:
865:
840:
825:
806:
762:
730:
698:
564:momentum operator
531:
500:{\displaystyle m}
458:
430:
413:
396:
384:
369:
351:
286:
252:
237:
222:
143:
120:{\displaystyle H}
94:
21:quantum mechanics
12302:
12228:Particle physics
12181:Electromagnetism
12155:Angular momentum
12100:
12099:
12026:
12025:
12008:
12001:
11994:
11985:
11984:
11971:
11970:
11682:Quantum geometry
11677:Quantum dynamics
11534:Superdeterminism
11430:Matrix mechanics
11285:Bra–ket notation
11216:
11209:
11202:
11193:
11192:
11183:
11167:
11166:
11148:
11142:
11141:
11129:
11126:Electromagnetism
11119:
11113:
11112:
11094:
11088:
11087:
11075:
11065:
11020:Potential theory
11000:Bra–ket notation
10973:
10971:
10970:
10965:
10947:
10945:
10944:
10939:
10937:
10936:
10920:
10918:
10917:
10912:
10910:
10909:
10891:
10889:
10888:
10883:
10881:
10879:
10871:
10870:
10869:
10856:
10848:
10846:
10845:
10844:
10831:
10817:
10811:
10809:
10801:
10800:
10799:
10786:
10781:
10779:
10778:
10777:
10764:
10750:
10737:
10735:
10734:
10729:
10717:
10712:
10685:
10684:
10666:
10664:
10663:
10658:
10656:
10655:
10637:
10635:
10634:
10629:
10627:
10625:
10617:
10615:
10610:
10597:
10583:
10581:
10580:
10579:
10566:
10552:
10539:
10537:
10536:
10531:
10529:
10527:
10519:
10518:
10517:
10516:
10499:
10488:
10486:
10484:
10479:
10478:
10461:
10447:
10430:
10428:
10427:
10422:
10414:
10406:
10401:
10381:
10373:
10368:
10357:
10356:
10346:
10334:
10332:
10330:
10325:
10324:
10307:
10293:
10279:
10277:
10276:
10271:
10259:
10254:
10235:
10233:
10232:
10227:
10216:
10215:
10194:
10192:
10191:
10186:
10175:
10174:
10155:
10153:
10152:
10147:
10145:
10141:
10140:
10112:
10110:
10109:
10104:
10096:
10088:
10083:
10072:
10071:
10061:
10056:
10055:
10041:
10040:
10006:
9998:
9978:
9977:
9975:
9974:
9962:
9957:
9919:
9917:
9916:
9911:
9900:
9899:
9881:
9879:
9878:
9873:
9853:
9830:
9829:
9809:
9807:
9806:
9801:
9793:
9779:
9778:
9768:
9741:
9727:
9725:
9724:
9719:
9717:
9713:
9712:
9686:
9684:
9683:
9678:
9661:
9659:
9658:
9653:
9651:
9650:
9649:
9624:
9619:
9599:
9597:
9596:
9591:
9589:
9585:
9547:angular momentum
9544:
9542:
9541:
9536:
9518:
9516:
9515:
9510:
9505:
9485:
9483:
9474:
9469:
9452:
9417:
9415:
9414:
9409:
9386:
9366:
9346:
9344:
9333:
9313:
9305:
9285:
9283:
9272:
9259:
9257:
9256:
9251:
9219:
9217:
9216:
9211:
9199:
9197:
9196:
9191:
9179:
9177:
9176:
9171:
9157:
9155:
9154:
9149:
9144:
9143:
9095:
9093:
9092:
9087:
9075:
9073:
9072:
9067:
9056:observable. Let
9044:
9042:
9041:
9036:
9024:
9022:
9021:
9016:
9008:
8993:
8991:
8990:
8985:
8977:
8965:
8963:
8962:
8957:
8943:
8941:
8940:
8935:
8921:
8894:
8883:
8882:
8864:
8859:
8858:
8837:
8817:
8815:
8814:
8809:
8801:
8786:
8784:
8783:
8778:
8770:
8755:
8753:
8752:
8747:
8735:unitary operator
8721:
8719:
8718:
8713:
8701:
8699:
8698:
8693:
8669:
8667:
8666:
8661:
8647:
8646:
8641:
8637:
8636:
8607:
8605:
8594:
8589:
8588:
8580:
8566:
8564:
8563:
8558:
8544:
8543:
8535:
8523:
8521:
8520:
8515:
8501:
8499:
8498:
8493:
8491:
8490:
8485:
8481:
8480:
8469:
8468:
8460:
8451:
8449:
8438:
8433:
8432:
8427:
8422:
8416:
8415:
8410:
8405:
8399:
8391:
8386:
8385:
8377:
8363:
8361:
8360:
8355:
8335:
8334:
8326:
8313:kinetic momentum
8310:
8308:
8307:
8302:
8300:
8299:
8294:
8289:
8274:
8272:
8271:
8266:
8261:
8250:
8249:
8244:
8239:
8230:
8229:
8221:
8205:
8203:
8202:
8197:
8195:
8183:
8181:
8180:
8175:
8173:
8172:
8164:
8154:
8152:
8151:
8146:
8144:
8132:vector potential
8129:
8127:
8126:
8121:
8109:scalar potential
8106:
8104:
8103:
8098:
8086:
8084:
8083:
8078:
8059:
8057:
8056:
8051:
8049:
8041:
8036:
8034:
8026:
8022:
8021:
8011:
8006:
8005:
7997:
7977:
7975:
7974:
7969:
7967:
7955:
7953:
7952:
7947:
7927:
7925:
7924:
7919:
7917:
7916:
7898:
7896:
7895:
7890:
7888:
7883:
7881:
7873:
7869:
7868:
7858:
7853:
7852:
7847:
7831:
7830:
7826:
7817:
7815:
7814:
7809:
7807:
7799:
7788:
7787:
7779:
7765:
7763:
7762:
7757:
7755:
7747:
7724:
7722:
7721:
7716:
7714:
7702:
7700:
7699:
7694:
7692:
7673:
7671:
7670:
7665:
7663:
7655:
7654:
7646:
7634:
7626:
7625:
7617:
7608:
7607:
7599:
7585:
7583:
7582:
7577:
7575:
7574:
7566:
7557:
7556:
7548:
7534:
7532:
7531:
7526:
7524:
7516:
7515:
7507:
7486:
7484:
7483:
7478:
7476:
7461:, in a uniform,
7460:
7458:
7457:
7452:
7440:
7438:
7437:
7432:
7430:
7408:
7406:
7405:
7400:
7398:
7394:
7390:
7389:
7387:
7386:
7381:
7380:
7375:
7366:
7365:
7360:
7354:
7348:
7347:
7346:
7337:
7336:
7326:
7323:
7308:
7306:
7305:
7304:
7285:
7279:
7274:
7265:
7263:
7262:
7261:
7248:
7247:
7238:
7227:
7222:
7201:
7197:
7195:
7194:
7189:
7188:
7183:
7174:
7173:
7168:
7162:
7156:
7155:
7154:
7145:
7144:
7134:
7131:
7115:
7110:
7095:
7093:
7092:
7091:
7072:
7066:
7061:
7052:
7050:
7049:
7037:
7034:
7029:
7014:
7009:
7008:
6999:
6987:
6986:
6978:
6960:
6958:
6957:
6952:
6950:
6948:
6947:
6942:
6941:
6936:
6927:
6926:
6921:
6915:
6909:
6908:
6907:
6898:
6897:
6887:
6884:
6868:
6863:
6848:
6846:
6845:
6844:
6825:
6808:
6806:
6805:
6800:
6788:
6786:
6785:
6780:
6778:
6777:
6772:
6759:
6757:
6756:
6751:
6749:
6748:
6732:
6730:
6729:
6724:
6719:
6718:
6713:
6692:
6690:
6689:
6684:
6682:
6680:
6679:
6674:
6673:
6668:
6659:
6658:
6653:
6647:
6641:
6640:
6639:
6630:
6629:
6619:
6616:
6601:
6599:
6598:
6597:
6578:
6570:
6569:
6564:
6552:
6551:
6541:
6526:
6518:
6513:
6512:
6490:
6488:
6487:
6482:
6480:
6479:
6463:
6461:
6460:
6455:
6441:
6439:
6438:
6433:
6431:
6429:
6428:
6423:
6418:
6413:
6412:
6396:
6395:
6394:
6385:
6384:
6374:
6356:electromagnetism
6345:
6343:
6342:
6337:
6335:
6334:
6318:
6316:
6315:
6310:
6308:
6307:
6279:
6277:
6276:
6271:
6259:
6257:
6256:
6251:
6239:
6237:
6236:
6231:
6218:angular momentum
6215:
6213:
6212:
6207:
6205:
6204:
6199:
6198:
6190:
6179:
6177:
6175:
6174:
6169:
6167:
6166:
6161:
6160:
6152:
6141:
6139:
6137:
6136:
6131:
6129:
6128:
6123:
6122:
6114:
6094:
6092:
6091:
6086:
6084:
6083:
6064:
6062:
6061:
6056:
6054:
6053:
6034:
6032:
6031:
6026:
6024:
6023:
6002:
6000:
5999:
5994:
5991:
5986:
5981:
5980:
5972:
5968:
5966:
5965:
5964:
5948:
5947:
5938:
5932:
5927:
5922:
5921:
5913:
5909:
5907:
5906:
5905:
5889:
5888:
5879:
5873:
5868:
5863:
5862:
5854:
5850:
5848:
5847:
5846:
5830:
5829:
5820:
5812:
5811:
5803:
5765:
5763:
5762:
5757:
5755:
5751:
5747:
5746:
5745:
5736:
5731:
5730:
5729:
5716:
5711:
5709:
5708:
5707:
5694:
5693:
5684:
5682:
5680:
5672:
5671:
5662:
5649:
5645:
5644:
5643:
5634:
5629:
5628:
5627:
5614:
5609:
5607:
5606:
5605:
5592:
5591:
5582:
5580:
5578:
5570:
5569:
5560:
5547:
5543:
5542:
5541:
5532:
5527:
5526:
5525:
5512:
5507:
5505:
5504:
5503:
5490:
5489:
5480:
5478:
5476:
5468:
5467:
5458:
5442:
5438:
5434:
5433:
5432:
5420:
5419:
5407:
5406:
5392:
5387:
5386:
5385:
5372:
5367:
5363:
5362:
5360:
5359:
5358:
5345:
5344:
5335:
5330:
5328:
5327:
5326:
5313:
5312:
5303:
5298:
5296:
5295:
5294:
5281:
5280:
5271:
5264:
5262:
5254:
5253:
5244:
5232:
5231:
5223:
5205:
5203:
5202:
5197:
5195:
5194:
5182:
5181:
5169:
5168:
5156:
5155:
5150:
5144:
5139:
5131:
5123:
5115:
5114:
5096:
5094:
5093:
5088:
5058:
5056:
5055:
5050:
5048:
5034:
5032:
5031:
5026:
5024:
5023:
5014:
5009:
5008:
5007:
4994:
4989:
4988:
4979:
4977:
4969:
4968:
4959:
4951:
4950:
4942:
4928:
4926:
4925:
4920:
4918:
4917:
4908:
4903:
4902:
4901:
4888:
4883:
4881:
4880:
4879:
4866:
4865:
4856:
4854:
4852:
4844:
4843:
4834:
4826:
4825:
4817:
4803:
4801:
4800:
4795:
4793:
4785:
4780:
4779:
4761:
4759:
4758:
4753:
4741:
4739:
4738:
4733:
4718:
4716:
4715:
4710:
4693:
4691:
4690:
4685:
4683:
4682:
4673:
4668:
4667:
4666:
4653:
4648:
4647:
4638:
4630:
4591:
4589:
4588:
4583:
4581:
4580:
4568:
4567:
4558:
4556:
4548:
4547:
4538:
4530:
4529:
4521:
4507:
4505:
4504:
4499:
4497:
4496:
4484:
4482:
4481:
4480:
4467:
4466:
4457:
4455:
4453:
4445:
4444:
4435:
4427:
4426:
4418:
4406:
4404:
4403:
4398:
4396:
4395:
4366:
4364:
4363:
4358:
4356:
4355:
4346:
4344:
4336:
4335:
4326:
4318:
4317:
4309:
4295:
4293:
4292:
4287:
4285:
4283:
4282:
4281:
4268:
4267:
4258:
4256:
4254:
4246:
4245:
4236:
4228:
4227:
4219:
4194:
4192:
4191:
4186:
4174:
4172:
4171:
4166:
4129:
4127:
4126:
4121:
4107:
4105:
4104:
4099:
4094:
4083:
4082:
4070:
4047:
4045:
4044:
4039:
4034:
4033:
4010:
4008:
4007:
4002:
4000:
3982:
3980:
3979:
3974:
3944:However, in the
3934:detailed balance
3923:
3921:
3920:
3915:
3874:unitary operator
3869:
3867:
3866:
3861:
3859:
3858:
3854:
3794:
3792:
3791:
3786:
3764:
3762:
3761:
3756:
3751:
3747:
3729:
3728:
3724:
3699:
3695:
3668:
3666:
3665:
3660:
3648:
3646:
3645:
3640:
3622:
3620:
3619:
3614:
3592:
3590:
3589:
3584:
3579:
3575:
3557:
3555:
3546:
3541:
3530:
3526:
3496:
3494:
3493:
3488:
3476:
3474:
3473:
3468:
3466:
3462:
3420:
3418:
3417:
3412:
3410:
3406:
3405:
3400:
3399:
3391:
3386:
3381:
3360:
3356:
3352:
3351:
3350:
3337:
3332:
3323:
3321:
3320:
3319:
3306:
3305:
3296:
3285:
3280:
3259:
3255:
3254:
3244:
3239:
3220:
3215:
3206:
3204:
3203:
3191:
3188:
3183:
3168:
3163:
3162:
3153:
3141:
3140:
3132:
3114:
3112:
3111:
3106:
3095:
3094:
3089:
3059:
3058:
3053:
3029:
3028:
3023:
2999:
2998:
2993:
2980:
2975:
2937:
2935:
2934:
2929:
2917:
2915:
2914:
2909:
2890:
2888:
2887:
2882:
2868:
2866:
2865:
2860:
2858:
2857:
2845:
2844:
2835:
2833:
2825:
2824:
2815:
2795:
2793:
2792:
2787:
2785:
2772:
2771:
2766:
2751:
2750:
2745:
2736:
2735:
2730:
2714:
2709:
2700:
2698:
2697:
2685:
2682:
2677:
2662:
2657:
2656:
2647:
2636:
2623:
2622:
2617:
2602:
2601:
2596:
2587:
2586:
2581:
2566:
2564:
2563:
2562:
2549:
2548:
2547:
2542:
2541:
2533:
2526:
2525:
2520:
2519:
2511:
2506:
2503:
2498:
2477:
2473:
2472:
2464:
2458:
2457:
2452:
2451:
2443:
2438:
2433:
2411:
2410:
2402:
2387:-particle case:
2386:
2384:
2383:
2378:
2364:
2362:
2361:
2356:
2351:
2349:
2347:
2342:
2329:
2328:
2319:
2314:
2312:
2310:
2305:
2292:
2291:
2282:
2277:
2275:
2273:
2268:
2255:
2254:
2245:
2239:
2234:
2218:
2214:
2212:
2211:
2206:
2203:
2198:
2182:
2180:
2179:
2174:
2162:
2160:
2159:
2154:
2152:
2151:
2135:
2133:
2132:
2127:
2115:
2113:
2112:
2107:
2104:
2099:
2090:
2088:
2087:
2086:
2073:
2072:
2063:
2055:
2053:
2052:
2051:
2038:
2037:
2036:
2031:
2030:
2022:
2015:
2014:
2009:
2008:
2000:
1995:
1990:
1989:
1984:
1983:
1975:
1964:
1962:
1961:
1956:
1942:
1941:
1936:
1921:
1920:
1915:
1906:
1905:
1900:
1885:
1884:
1876:
1864:
1862:
1861:
1856:
1854:
1853:
1845:
1839:
1838:
1833:
1832:
1824:
1819:
1814:
1796:
1795:
1787:
1775:
1773:
1772:
1767:
1741:
1739:
1738:
1733:
1722:
1721:
1718:
1702:
1701:
1696:
1678:
1672:
1664:
1643:
1642:
1639:
1623:
1622:
1617:
1599:
1581:
1573:
1505:
1503:
1502:
1497:
1495:
1478:
1477:
1472:
1468:
1466:
1458:
1450:
1442:
1434:
1422:
1420:
1412:
1411:
1402:
1394:
1390:
1386:
1378:
1377:
1372:
1368:
1366:
1358:
1350:
1343:
1339:
1337:
1329:
1321:
1314:
1306:
1291:
1289:
1281:
1273:
1269:
1259:
1258:
1240:
1220:
1218:
1210:
1209:
1200:
1189:
1178:
1174:
1172:
1171:
1170:
1157:
1153:
1152:
1142:
1136:
1135:
1125:
1117:
1105:
1103:
1095:
1094:
1085:
1040:
1038:
1037:
1032:
1021:
998:
996:
995:
990:
988:
975:
961:
960:
951:
949:
941:
940:
931:
920:
907:
893:
891:
883:
882:
881:
873:
867:
866:
858:
854:
846:
842:
841:
833:
827:
826:
818:
808:
807:
799:
773:
771:
770:
765:
763:
761:
760:
755:
746:
745:
736:
731:
729:
728:
723:
714:
713:
704:
699:
697:
696:
691:
682:
681:
672:
667:
666:
646:
644:
643:
638:
636:
635:
616:
614:
613:
608:
585:
583:
582:
577:
561:
559:
558:
553:
533:
532:
524:
515:of vectors, and
506:
504:
503:
498:
482:
480:
479:
474:
469:
468:
459:
457:
449:
448:
439:
431:
429:
421:
420:
415:
414:
406:
402:
397:
395:
387:
386:
385:
377:
371:
370:
362:
358:
353:
352:
344:
333:potential energy
330:
328:
327:
322:
308:
288:
287:
279:
267:
265:
264:
259:
254:
253:
245:
239:
238:
230:
224:
223:
215:
188:By analogy with
155:
153:
152:
147:
145:
144:
136:
126:
124:
123:
118:
106:
104:
103:
98:
96:
95:
87:
37:potential energy
12312:
12311:
12305:
12304:
12303:
12301:
12300:
12299:
12270:Operator theory
12255:
12254:
12253:
12244:
12223:
12192:
12176:
12150:
12124:
12087:
12083:Ladder operator
12066:
12050:
12017:
12012:
11982:
11977:
11959:
11945:Wigner's friend
11921:
11912:Quantum gravity
11873:
11859:Quantum sensing
11839:Quantum network
11819:Quantum machine
11789:Quantum imaging
11752:Quantum circuit
11747:Quantum channel
11706:
11652:Quantum biology
11638:
11614:Elitzur–Vaidman
11589:Davisson–Germer
11572:
11524:Hidden-variable
11514:de Broglie–Bohm
11491:Interpretations
11485:
11449:
11403:
11290:Complementarity
11268:
11225:
11220:
11176:
11171:
11170:
11163:
11149:
11145:
11138:
11120:
11116:
11109:
11095:
11091:
11084:
11066:
11059:
11054:
11049:
10980:
10953:
10950:
10949:
10932:
10928:
10926:
10923:
10922:
10905:
10901:
10899:
10896:
10895:
10872:
10865:
10861:
10857:
10855:
10840:
10836:
10832:
10818:
10816:
10802:
10795:
10791:
10787:
10785:
10773:
10769:
10765:
10751:
10749:
10747:
10744:
10743:
10713:
10708:
10680:
10676:
10674:
10671:
10670:
10651:
10647:
10645:
10642:
10641:
10618:
10611:
10606:
10598:
10596:
10575:
10571:
10567:
10553:
10551:
10549:
10546:
10545:
10520:
10509:
10508:
10504:
10500:
10498:
10480:
10471:
10470:
10462:
10448:
10446:
10444:
10441:
10440:
10410:
10402:
10394:
10377:
10369:
10361:
10352:
10348:
10342:
10326:
10317:
10316:
10308:
10294:
10292:
10290:
10287:
10286:
10255:
10250:
10244:
10241:
10240:
10211:
10207:
10205:
10202:
10201:
10170:
10166:
10164:
10161:
10160:
10130:
10126:
10122:
10120:
10117:
10116:
10092:
10084:
10076:
10067:
10063:
10057:
10048:
10047:
10033:
10029:
10002:
9994:
9964:
9963:
9958:
9956:
9955:
9935:
9932:
9931:
9895:
9891:
9889:
9886:
9885:
9849:
9825:
9821:
9819:
9816:
9815:
9789:
9774:
9770:
9764:
9737:
9735:
9732:
9731:
9702:
9698:
9694:
9692:
9689:
9688:
9672:
9669:
9668:
9642:
9638:
9634:
9620:
9612:
9607:
9604:
9603:
9575:
9571:
9569:
9566:
9565:
9555:
9530:
9527:
9526:
9501:
9473:
9468:
9448:
9431:
9428:
9427:
9382:
9362:
9337:
9332:
9309:
9301:
9276:
9271:
9269:
9266:
9265:
9227:
9224:
9223:
9205:
9202:
9201:
9200:, then so does
9185:
9182:
9181:
9165:
9162:
9161:
9139:
9135:
9103:
9100:
9099:
9081:
9078:
9077:
9061:
9058:
9057:
9030:
9027:
9026:
9004:
8999:
8996:
8995:
8973:
8971:
8968:
8967:
8951:
8948:
8947:
8917:
8890:
8878:
8874:
8860:
8854:
8850:
8833:
8825:
8822:
8821:
8797:
8792:
8789:
8788:
8766:
8764:
8761:
8760:
8741:
8738:
8737:
8707:
8704:
8703:
8687:
8684:
8683:
8675:
8642:
8632:
8613:
8609:
8608:
8598:
8593:
8579:
8578:
8576:
8573:
8572:
8534:
8533:
8531:
8528:
8527:
8509:
8506:
8505:
8486:
8476:
8459:
8458:
8457:
8453:
8452:
8442:
8437:
8423:
8421:
8420:
8406:
8404:
8403:
8390:
8376:
8375:
8373:
8370:
8369:
8325:
8324:
8322:
8319:
8318:
8290:
8288:
8287:
8282:
8279:
8278:
8257:
8240:
8238:
8237:
8220:
8219:
8217:
8214:
8213:
8191:
8189:
8186:
8185:
8163:
8162:
8160:
8157:
8156:
8140:
8138:
8135:
8134:
8115:
8112:
8111:
8092:
8089:
8088:
8072:
8069:
8068:
8065:
8045:
8037:
8027:
8017:
8013:
8012:
8010:
7996:
7995:
7993:
7990:
7989:
7963:
7961:
7958:
7957:
7941:
7938:
7937:
7912:
7908:
7906:
7903:
7902:
7884:
7874:
7864:
7860:
7859:
7857:
7848:
7843:
7842:
7840:
7837:
7836:
7828:
7824:
7823:
7803:
7795:
7778:
7777:
7775:
7772:
7771:
7751:
7743:
7732:
7729:
7728:
7710:
7708:
7705:
7704:
7688:
7686:
7683:
7682:
7679:
7659:
7645:
7644:
7630:
7616:
7615:
7598:
7597:
7595:
7592:
7591:
7565:
7564:
7547:
7546:
7544:
7541:
7540:
7520:
7506:
7505:
7494:
7491:
7490:
7472:
7470:
7467:
7466:
7446:
7443:
7442:
7426:
7424:
7421:
7420:
7414:
7396:
7395:
7382:
7376:
7371:
7370:
7361:
7356:
7355:
7350:
7349:
7342:
7338:
7332:
7328:
7327:
7325:
7313:
7300:
7296:
7289:
7284:
7275:
7270:
7257:
7253:
7249:
7243:
7239:
7237:
7233:
7229:
7223:
7212:
7199:
7198:
7190:
7184:
7179:
7178:
7169:
7164:
7163:
7158:
7157:
7150:
7146:
7140:
7136:
7135:
7133:
7121:
7111:
7100:
7087:
7083:
7076:
7071:
7062:
7057:
7045:
7041:
7036:
7030:
7019:
7004:
7000:
6998:
6988:
6977:
6976:
6972:
6970:
6967:
6966:
6943:
6937:
6932:
6931:
6922:
6917:
6916:
6911:
6910:
6903:
6899:
6893:
6889:
6888:
6886:
6874:
6864:
6853:
6840:
6836:
6829:
6824:
6816:
6813:
6812:
6794:
6791:
6790:
6773:
6768:
6767:
6765:
6762:
6761:
6744:
6740:
6738:
6735:
6734:
6714:
6709:
6708:
6700:
6697:
6696:
6675:
6669:
6664:
6663:
6654:
6649:
6648:
6643:
6642:
6635:
6631:
6625:
6621:
6620:
6618:
6606:
6593:
6589:
6582:
6577:
6565:
6560:
6559:
6547:
6543:
6531:
6517:
6508:
6504:
6502:
6499:
6498:
6475:
6471:
6469:
6466:
6465:
6449:
6446:
6445:
6424:
6419:
6414:
6408:
6404:
6397:
6390:
6386:
6380:
6376:
6375:
6373:
6365:
6362:
6361:
6330:
6326:
6324:
6321:
6320:
6303:
6299:
6297:
6294:
6293:
6286:
6265:
6262:
6261:
6245:
6242:
6241:
6225:
6222:
6221:
6200:
6189:
6188:
6187:
6185:
6182:
6181:
6162:
6151:
6150:
6149:
6147:
6144:
6143:
6142:
6124:
6113:
6112:
6111:
6109:
6106:
6105:
6104:
6076:
6072:
6070:
6067:
6066:
6046:
6042:
6040:
6037:
6036:
6016:
6012:
6010:
6007:
6006:
5987:
5982:
5971:
5970:
5957:
5953:
5949:
5943:
5939:
5937:
5928:
5923:
5912:
5911:
5898:
5894:
5890:
5884:
5880:
5878:
5869:
5864:
5853:
5852:
5839:
5835:
5831:
5825:
5821:
5819:
5802:
5801:
5799:
5796:
5795:
5771:
5753:
5752:
5741:
5737:
5725:
5721:
5717:
5715:
5703:
5699:
5695:
5689:
5685:
5683:
5673:
5667:
5663:
5661:
5657:
5653:
5639:
5635:
5623:
5619:
5615:
5613:
5601:
5597:
5593:
5587:
5583:
5581:
5571:
5565:
5561:
5559:
5555:
5551:
5537:
5533:
5521:
5517:
5513:
5511:
5499:
5495:
5491:
5485:
5481:
5479:
5469:
5463:
5459:
5457:
5453:
5449:
5440:
5439:
5428:
5424:
5415:
5411:
5402:
5398:
5397:
5393:
5381:
5377:
5373:
5371:
5354:
5350:
5346:
5340:
5336:
5334:
5322:
5318:
5314:
5308:
5304:
5302:
5290:
5286:
5282:
5276:
5272:
5270:
5269:
5265:
5255:
5249:
5245:
5243:
5233:
5222:
5221:
5217:
5215:
5212:
5211:
5190:
5186:
5177:
5173:
5164:
5160:
5151:
5146:
5145:
5140:
5135:
5127:
5119:
5110:
5106:
5104:
5101:
5100:
5064:
5061:
5060:
5044:
5042:
5039:
5038:
5019:
5015:
5003:
4999:
4995:
4993:
4984:
4980:
4970:
4964:
4960:
4958:
4941:
4940:
4938:
4935:
4934:
4913:
4909:
4897:
4893:
4889:
4887:
4875:
4871:
4867:
4861:
4857:
4855:
4845:
4839:
4835:
4833:
4816:
4815:
4813:
4810:
4809:
4784:
4775:
4771:
4769:
4766:
4765:
4747:
4744:
4743:
4727:
4724:
4723:
4721:spring constant
4704:
4701:
4700:
4678:
4674:
4662:
4658:
4654:
4652:
4643:
4639:
4629:
4621:
4618:
4617:
4608:
4600:step potentials
4598:" problem, and
4576:
4572:
4563:
4559:
4549:
4543:
4539:
4537:
4520:
4519:
4517:
4514:
4513:
4492:
4488:
4476:
4472:
4468:
4462:
4458:
4456:
4446:
4440:
4436:
4434:
4417:
4416:
4414:
4411:
4410:
4391:
4387:
4379:
4376:
4375:
4372:
4351:
4347:
4337:
4331:
4327:
4325:
4308:
4307:
4305:
4302:
4301:
4277:
4273:
4269:
4263:
4259:
4257:
4247:
4241:
4237:
4235:
4218:
4217:
4215:
4212:
4211:
4206:
4201:
4180:
4177:
4176:
4160:
4157:
4156:
4152:
4115:
4112:
4111:
4084:
4078:
4074:
4060:
4055:
4052:
4051:
4029:
4025:
4020:
4017:
4016:
3990:
3988:
3985:
3984:
3968:
3965:
3964:
3959:The eigenkets (
3942:
3940:Dirac formalism
3894:
3891:
3890:
3850:
3837:
3833:
3825:
3822:
3821:
3780:
3777:
3776:
3734:
3730:
3720:
3707:
3703:
3682:
3678:
3676:
3673:
3672:
3654:
3651:
3650:
3628:
3625:
3624:
3608:
3605:
3604:
3562:
3558:
3545:
3540:
3513:
3509:
3504:
3501:
3500:
3482:
3479:
3478:
3449:
3445:
3443:
3440:
3439:
3436:
3430:
3408:
3407:
3401:
3390:
3389:
3388:
3382:
3371:
3358:
3357:
3346:
3342:
3333:
3328:
3315:
3311:
3307:
3301:
3297:
3295:
3291:
3287:
3281:
3270:
3257:
3256:
3250:
3246:
3240:
3229:
3216:
3211:
3199:
3195:
3190:
3184:
3173:
3158:
3154:
3152:
3142:
3131:
3130:
3126:
3124:
3121:
3120:
3090:
3085:
3084:
3054:
3049:
3048:
3024:
3019:
3018:
2994:
2989:
2988:
2976:
2965:
2953:
2950:
2949:
2923:
2920:
2919:
2903:
2900:
2899:
2876:
2873:
2872:
2853:
2849:
2840:
2836:
2826:
2820:
2816:
2814:
2809:
2806:
2805:
2783:
2782:
2767:
2762:
2761:
2746:
2741:
2740:
2731:
2726:
2725:
2710:
2705:
2693:
2689:
2684:
2678:
2667:
2652:
2648:
2646:
2634:
2633:
2618:
2613:
2612:
2597:
2592:
2591:
2582:
2577:
2576:
2558:
2554:
2550:
2543:
2532:
2531:
2530:
2521:
2510:
2509:
2508:
2507:
2505:
2499:
2488:
2475:
2474:
2463:
2462:
2453:
2442:
2441:
2440:
2434:
2423:
2412:
2401:
2400:
2396:
2394:
2391:
2390:
2372:
2369:
2368:
2343:
2338:
2330:
2324:
2320:
2318:
2306:
2301:
2293:
2287:
2283:
2281:
2269:
2264:
2256:
2250:
2246:
2244:
2235:
2230:
2224:
2221:
2220:
2216:
2199:
2194:
2188:
2185:
2184:
2168:
2165:
2164:
2147:
2143:
2141:
2138:
2137:
2121:
2118:
2117:
2100:
2095:
2082:
2078:
2074:
2068:
2064:
2062:
2047:
2043:
2039:
2032:
2021:
2020:
2019:
2010:
1999:
1998:
1997:
1996:
1994:
1985:
1974:
1973:
1972:
1970:
1967:
1966:
1937:
1932:
1931:
1916:
1911:
1910:
1901:
1896:
1895:
1875:
1874:
1872:
1869:
1868:
1844:
1843:
1834:
1823:
1822:
1821:
1815:
1804:
1786:
1785:
1783:
1780:
1779:
1761:
1758:
1757:
1754:
1717:
1713:
1697:
1692:
1691:
1674:
1665:
1657:
1638:
1634:
1618:
1613:
1612:
1595:
1574:
1566:
1515:
1512:
1511:
1493:
1492:
1473:
1459:
1451:
1449:
1445:
1444:
1435:
1427:
1413:
1407:
1403:
1401:
1392:
1391:
1373:
1359:
1351:
1349:
1345:
1344:
1330:
1322:
1320:
1316:
1307:
1299:
1282:
1274:
1254:
1250:
1233:
1232:
1228:
1226:
1225:
1221:
1211:
1205:
1201:
1199:
1187:
1186:
1166:
1162:
1158:
1148:
1144:
1143:
1141:
1137:
1131:
1127:
1118:
1110:
1096:
1090:
1086:
1084:
1074:
1064:
1062:
1059:
1058:
1050:
1017:
1009:
1006:
1005:
986:
985:
971:
956:
952:
942:
936:
932:
930:
918:
917:
903:
884:
872:
871:
857:
856:
855:
853:
844:
843:
832:
831:
817:
816:
809:
798:
797:
793:
791:
788:
787:
756:
748:
747:
741:
737:
735:
724:
716:
715:
709:
705:
703:
692:
684:
683:
677:
673:
671:
662:
658:
656:
653:
652:
631:
627:
625:
622:
621:
602:
599:
598:
571:
568:
567:
523:
522:
520:
517:
516:
492:
489:
488:
464:
460:
450:
444:
440:
438:
422:
416:
405:
404:
403:
401:
388:
376:
375:
361:
360:
359:
357:
343:
342:
340:
337:
336:
304:
278:
277:
275:
272:
271:
244:
243:
229:
228:
214:
213:
211:
208:
207:
186:
181:
168:
162:
135:
134:
132:
129:
128:
112:
109:
108:
86:
85:
83:
80:
79:
76:vector notation
45:energy spectrum
43:, the system's
17:
12:
11:
5:
12310:
12309:
12298:
12297:
12292:
12287:
12282:
12277:
12272:
12267:
12250:
12249:
12246:
12245:
12243:
12242:
12237:
12231:
12229:
12225:
12224:
12222:
12221:
12216:
12211:
12206:
12200:
12198:
12194:
12193:
12191:
12190:
12184:
12182:
12178:
12177:
12175:
12174:
12169:
12164:
12158:
12156:
12152:
12151:
12149:
12148:
12146:Kinetic energy
12143:
12138:
12132:
12130:
12126:
12125:
12123:
12122:
12117:
12112:
12106:
12104:
12097:
12093:
12092:
12089:
12088:
12086:
12085:
12080:
12074:
12072:
12068:
12067:
12065:
12064:
12058:
12056:
12052:
12051:
12049:
12048:
12043:
12038:
12032:
12030:
12029:Space and time
12023:
12019:
12018:
12011:
12010:
12003:
11996:
11988:
11979:
11978:
11976:
11975:
11964:
11961:
11960:
11958:
11957:
11952:
11947:
11942:
11941:
11940:
11929:
11927:
11923:
11922:
11920:
11919:
11914:
11909:
11908:
11907:
11897:
11892:
11890:Casimir effect
11887:
11881:
11879:
11875:
11874:
11872:
11871:
11866:
11861:
11856:
11851:
11849:Quantum optics
11846:
11841:
11836:
11831:
11826:
11821:
11816:
11811:
11806:
11801:
11796:
11791:
11786:
11781:
11776:
11771:
11770:
11769:
11759:
11754:
11749:
11744:
11743:
11742:
11732:
11727:
11722:
11716:
11714:
11708:
11707:
11705:
11704:
11699:
11694:
11689:
11684:
11679:
11674:
11669:
11664:
11659:
11654:
11648:
11646:
11640:
11639:
11637:
11636:
11631:
11626:
11624:Quantum eraser
11621:
11616:
11611:
11606:
11601:
11596:
11591:
11586:
11580:
11578:
11574:
11573:
11571:
11570:
11565:
11560:
11555:
11550:
11545:
11540:
11539:
11538:
11537:
11536:
11521:
11516:
11511:
11506:
11501:
11495:
11493:
11487:
11486:
11484:
11483:
11478:
11473:
11468:
11463:
11457:
11455:
11451:
11450:
11448:
11447:
11442:
11437:
11432:
11427:
11422:
11417:
11411:
11409:
11405:
11404:
11402:
11401:
11400:
11399:
11394:
11384:
11379:
11374:
11369:
11364:
11359:
11354:
11349:
11344:
11339:
11334:
11329:
11324:
11323:
11322:
11317:
11312:
11307:
11297:
11295:Density matrix
11292:
11287:
11282:
11276:
11274:
11270:
11269:
11267:
11266:
11261:
11256:
11251:
11250:
11249:
11239:
11233:
11231:
11227:
11226:
11219:
11218:
11211:
11204:
11196:
11190:
11189:
11175:
11174:External links
11172:
11169:
11168:
11161:
11143:
11136:
11114:
11107:
11089:
11082:
11056:
11055:
11053:
11050:
11048:
11047:
11042:
11040:Magnetic field
11037:
11035:Electric field
11032:
11030:Electrostatics
11027:
11022:
11017:
11012:
11010:Linear algebra
11007:
11002:
10997:
10992:
10987:
10981:
10979:
10976:
10963:
10960:
10957:
10935:
10931:
10908:
10904:
10878:
10875:
10868:
10864:
10860:
10854:
10851:
10843:
10839:
10835:
10830:
10827:
10824:
10821:
10814:
10808:
10805:
10798:
10794:
10790:
10784:
10776:
10772:
10768:
10763:
10760:
10757:
10754:
10727:
10724:
10721:
10716:
10711:
10707:
10703:
10700:
10697:
10694:
10691:
10688:
10683:
10679:
10654:
10650:
10624:
10621:
10614:
10609:
10605:
10601:
10595:
10592:
10589:
10586:
10578:
10574:
10570:
10565:
10562:
10559:
10556:
10526:
10523:
10515:
10512:
10507:
10503:
10497:
10494:
10491:
10483:
10477:
10474:
10469:
10465:
10460:
10457:
10454:
10451:
10420:
10417:
10413:
10409:
10405:
10400:
10397:
10393:
10390:
10387:
10384:
10380:
10376:
10372:
10367:
10364:
10360:
10355:
10351:
10345:
10341:
10337:
10329:
10323:
10320:
10315:
10311:
10306:
10303:
10300:
10297:
10269:
10266:
10263:
10258:
10253:
10249:
10225:
10222:
10219:
10214:
10210:
10184:
10181:
10178:
10173:
10169:
10144:
10139:
10136:
10133:
10129:
10125:
10102:
10099:
10095:
10091:
10087:
10082:
10079:
10075:
10070:
10066:
10060:
10054:
10051:
10046:
10039:
10036:
10032:
10028:
10024:
10021:
10018:
10015:
10012:
10009:
10005:
10001:
9997:
9993:
9990:
9987:
9984:
9981:
9973:
9970:
9967:
9961:
9954:
9951:
9948:
9945:
9942:
9939:
9909:
9906:
9903:
9898:
9894:
9871:
9868:
9865:
9862:
9859:
9856:
9852:
9848:
9845:
9842:
9839:
9836:
9833:
9828:
9824:
9799:
9796:
9792:
9788:
9785:
9782:
9777:
9773:
9767:
9763:
9759:
9756:
9753:
9750:
9747:
9744:
9740:
9716:
9711:
9708:
9705:
9701:
9697:
9676:
9648:
9645:
9641:
9637:
9633:
9630:
9627:
9623:
9618:
9615:
9611:
9588:
9584:
9581:
9578:
9574:
9554:
9551:
9534:
9523:expected value
9508:
9504:
9500:
9497:
9494:
9491:
9488:
9482:
9479:
9472:
9467:
9464:
9461:
9458:
9455:
9451:
9447:
9444:
9441:
9438:
9435:
9407:
9404:
9401:
9398:
9395:
9392:
9389:
9385:
9381:
9378:
9375:
9372:
9369:
9365:
9361:
9358:
9355:
9352:
9349:
9343:
9340:
9336:
9331:
9328:
9325:
9322:
9319:
9316:
9312:
9308:
9304:
9300:
9297:
9294:
9291:
9288:
9282:
9279:
9275:
9249:
9246:
9243:
9240:
9237:
9234:
9231:
9209:
9189:
9180:commutes with
9169:
9147:
9142:
9138:
9134:
9131:
9128:
9125:
9122:
9119:
9116:
9113:
9110:
9107:
9085:
9065:
9034:
9014:
9011:
9007:
9003:
8983:
8980:
8976:
8955:
8933:
8930:
8927:
8924:
8920:
8916:
8913:
8909:
8906:
8903:
8900:
8897:
8893:
8889:
8886:
8881:
8877:
8873:
8870:
8867:
8863:
8857:
8853:
8849:
8846:
8843:
8840:
8836:
8832:
8829:
8807:
8804:
8800:
8796:
8776:
8773:
8769:
8745:
8711:
8691:
8674:
8671:
8659:
8656:
8653:
8650:
8645:
8640:
8635:
8631:
8628:
8625:
8622:
8619:
8616:
8612:
8604:
8601:
8597:
8592:
8586:
8583:
8556:
8553:
8550:
8547:
8541:
8538:
8513:
8489:
8484:
8479:
8475:
8472:
8466:
8463:
8456:
8448:
8445:
8441:
8436:
8430:
8426:
8419:
8413:
8409:
8402:
8397:
8394:
8389:
8383:
8380:
8353:
8350:
8347:
8344:
8341:
8338:
8332:
8329:
8297:
8293:
8286:
8264:
8260:
8256:
8253:
8247:
8243:
8236:
8233:
8227:
8224:
8194:
8170:
8167:
8143:
8119:
8096:
8076:
8064:
8061:
8048:
8044:
8040:
8033:
8030:
8025:
8020:
8016:
8009:
8003:
8000:
7984:Pauli matrices
7966:
7945:
7915:
7911:
7887:
7880:
7877:
7872:
7867:
7863:
7856:
7851:
7846:
7806:
7802:
7798:
7794:
7791:
7785:
7782:
7754:
7750:
7746:
7742:
7739:
7736:
7713:
7691:
7678:
7675:
7662:
7658:
7652:
7649:
7643:
7640:
7637:
7633:
7629:
7623:
7620:
7614:
7611:
7605:
7602:
7572:
7569:
7563:
7560:
7554:
7551:
7523:
7519:
7513:
7510:
7504:
7501:
7498:
7475:
7450:
7429:
7413:
7410:
7393:
7385:
7379:
7374:
7369:
7364:
7359:
7353:
7345:
7341:
7335:
7331:
7322:
7319:
7316:
7312:
7303:
7299:
7295:
7292:
7288:
7283:
7278:
7273:
7269:
7260:
7256:
7252:
7246:
7242:
7236:
7232:
7226:
7221:
7218:
7215:
7211:
7207:
7204:
7202:
7200:
7193:
7187:
7182:
7177:
7172:
7167:
7161:
7153:
7149:
7143:
7139:
7130:
7127:
7124:
7120:
7114:
7109:
7106:
7103:
7099:
7090:
7086:
7082:
7079:
7075:
7070:
7065:
7060:
7056:
7048:
7044:
7040:
7033:
7028:
7025:
7022:
7018:
7012:
7007:
7003:
6997:
6994:
6991:
6989:
6984:
6981:
6975:
6974:
6946:
6940:
6935:
6930:
6925:
6920:
6914:
6906:
6902:
6896:
6892:
6883:
6880:
6877:
6873:
6867:
6862:
6859:
6856:
6852:
6843:
6839:
6835:
6832:
6828:
6823:
6820:
6798:
6776:
6771:
6747:
6743:
6722:
6717:
6712:
6707:
6704:
6678:
6672:
6667:
6662:
6657:
6652:
6646:
6638:
6634:
6628:
6624:
6615:
6612:
6609:
6605:
6596:
6592:
6588:
6585:
6581:
6576:
6573:
6568:
6563:
6558:
6555:
6550:
6546:
6540:
6537:
6534:
6530:
6524:
6521:
6516:
6511:
6507:
6478:
6474:
6453:
6427:
6422:
6417:
6411:
6407:
6403:
6400:
6393:
6389:
6383:
6379:
6372:
6369:
6352:Gaussian units
6333:
6329:
6306:
6302:
6285:
6282:
6269:
6249:
6229:
6216:are the total
6203:
6196:
6193:
6165:
6158:
6155:
6127:
6120:
6117:
6082:
6079:
6075:
6052:
6049:
6045:
6022:
6019:
6015:
5990:
5985:
5978:
5975:
5963:
5960:
5956:
5952:
5946:
5942:
5936:
5931:
5926:
5919:
5916:
5904:
5901:
5897:
5893:
5887:
5883:
5877:
5872:
5867:
5860:
5857:
5845:
5842:
5838:
5834:
5828:
5824:
5818:
5815:
5809:
5806:
5790:chemical bonds
5770:
5767:
5750:
5744:
5740:
5734:
5728:
5724:
5720:
5714:
5706:
5702:
5698:
5692:
5688:
5679:
5676:
5670:
5666:
5660:
5656:
5652:
5648:
5642:
5638:
5632:
5626:
5622:
5618:
5612:
5604:
5600:
5596:
5590:
5586:
5577:
5574:
5568:
5564:
5558:
5554:
5550:
5546:
5540:
5536:
5530:
5524:
5520:
5516:
5510:
5502:
5498:
5494:
5488:
5484:
5475:
5472:
5466:
5462:
5456:
5452:
5448:
5445:
5443:
5441:
5437:
5431:
5427:
5423:
5418:
5414:
5410:
5405:
5401:
5396:
5390:
5384:
5380:
5376:
5370:
5366:
5357:
5353:
5349:
5343:
5339:
5333:
5325:
5321:
5317:
5311:
5307:
5301:
5293:
5289:
5285:
5279:
5275:
5268:
5261:
5258:
5252:
5248:
5242:
5239:
5236:
5234:
5229:
5226:
5220:
5219:
5193:
5189:
5185:
5180:
5176:
5172:
5167:
5163:
5159:
5154:
5149:
5143:
5138:
5134:
5130:
5126:
5122:
5118:
5113:
5109:
5086:
5083:
5080:
5077:
5074:
5071:
5068:
5047:
5022:
5018:
5012:
5006:
5002:
4998:
4992:
4987:
4983:
4976:
4973:
4967:
4963:
4957:
4954:
4948:
4945:
4916:
4912:
4906:
4900:
4896:
4892:
4886:
4878:
4874:
4870:
4864:
4860:
4851:
4848:
4842:
4838:
4832:
4829:
4823:
4820:
4791:
4788:
4783:
4778:
4774:
4751:
4731:
4708:
4681:
4677:
4671:
4665:
4661:
4657:
4651:
4646:
4642:
4636:
4633:
4628:
4625:
4607:
4604:
4579:
4575:
4571:
4566:
4562:
4555:
4552:
4546:
4542:
4536:
4533:
4527:
4524:
4495:
4491:
4487:
4479:
4475:
4471:
4465:
4461:
4452:
4449:
4443:
4439:
4433:
4430:
4424:
4421:
4394:
4390:
4386:
4383:
4371:
4368:
4354:
4350:
4343:
4340:
4334:
4330:
4324:
4321:
4315:
4312:
4280:
4276:
4272:
4266:
4262:
4253:
4250:
4244:
4240:
4234:
4231:
4225:
4222:
4205:
4202:
4200:
4197:
4184:
4164:
4151:
4148:
4119:
4097:
4093:
4090:
4087:
4081:
4077:
4073:
4069:
4066:
4063:
4059:
4037:
4032:
4028:
4024:
3999:
3996:
3993:
3972:
3941:
3938:
3913:
3910:
3907:
3904:
3901:
3898:
3879:time evolution
3857:
3853:
3849:
3846:
3843:
3840:
3836:
3832:
3829:
3807:, or just the
3784:
3754:
3750:
3746:
3743:
3740:
3737:
3733:
3727:
3723:
3719:
3716:
3713:
3710:
3706:
3702:
3698:
3694:
3691:
3688:
3685:
3681:
3658:
3638:
3635:
3632:
3612:
3582:
3578:
3574:
3571:
3568:
3565:
3561:
3554:
3551:
3544:
3539:
3536:
3533:
3529:
3525:
3522:
3519:
3516:
3512:
3508:
3486:
3465:
3461:
3458:
3455:
3452:
3448:
3432:Main article:
3429:
3426:
3404:
3397:
3394:
3385:
3380:
3377:
3374:
3370:
3366:
3363:
3361:
3359:
3355:
3349:
3345:
3341:
3336:
3331:
3327:
3318:
3314:
3310:
3304:
3300:
3294:
3290:
3284:
3279:
3276:
3273:
3269:
3265:
3262:
3260:
3258:
3253:
3249:
3243:
3238:
3235:
3232:
3228:
3224:
3219:
3214:
3210:
3202:
3198:
3194:
3187:
3182:
3179:
3176:
3172:
3166:
3161:
3157:
3151:
3148:
3145:
3143:
3138:
3135:
3129:
3128:
3104:
3101:
3098:
3093:
3088:
3083:
3080:
3077:
3074:
3071:
3068:
3065:
3062:
3057:
3052:
3047:
3044:
3041:
3038:
3035:
3032:
3027:
3022:
3017:
3014:
3011:
3008:
3005:
3002:
2997:
2992:
2987:
2984:
2979:
2974:
2971:
2968:
2964:
2960:
2957:
2927:
2907:
2880:
2856:
2852:
2848:
2843:
2839:
2832:
2829:
2823:
2819:
2813:
2781:
2778:
2775:
2770:
2765:
2760:
2757:
2754:
2749:
2744:
2739:
2734:
2729:
2724:
2721:
2718:
2713:
2708:
2704:
2696:
2692:
2688:
2681:
2676:
2673:
2670:
2666:
2660:
2655:
2651:
2645:
2642:
2639:
2637:
2635:
2632:
2629:
2626:
2621:
2616:
2611:
2608:
2605:
2600:
2595:
2590:
2585:
2580:
2575:
2572:
2569:
2561:
2557:
2553:
2546:
2539:
2536:
2529:
2524:
2517:
2514:
2502:
2497:
2494:
2491:
2487:
2483:
2480:
2478:
2476:
2470:
2467:
2461:
2456:
2449:
2446:
2437:
2432:
2429:
2426:
2422:
2418:
2415:
2413:
2408:
2405:
2399:
2398:
2376:
2354:
2346:
2341:
2337:
2333:
2327:
2323:
2317:
2309:
2304:
2300:
2296:
2290:
2286:
2280:
2272:
2267:
2263:
2259:
2253:
2249:
2243:
2238:
2233:
2229:
2202:
2197:
2193:
2172:
2150:
2146:
2125:
2103:
2098:
2094:
2085:
2081:
2077:
2071:
2067:
2061:
2058:
2050:
2046:
2042:
2035:
2028:
2025:
2018:
2013:
2006:
2003:
1993:
1988:
1981:
1978:
1954:
1951:
1948:
1945:
1940:
1935:
1930:
1927:
1924:
1919:
1914:
1909:
1904:
1899:
1894:
1891:
1888:
1882:
1879:
1851:
1848:
1842:
1837:
1830:
1827:
1818:
1813:
1810:
1807:
1803:
1799:
1793:
1790:
1765:
1753:
1752:Many particles
1750:
1731:
1728:
1725:
1716:
1712:
1709:
1706:
1700:
1695:
1690:
1687:
1684:
1681:
1677:
1671:
1668:
1663:
1660:
1656:
1652:
1649:
1646:
1637:
1633:
1630:
1627:
1621:
1616:
1611:
1608:
1605:
1602:
1598:
1594:
1591:
1588:
1585:
1580:
1577:
1572:
1569:
1565:
1561:
1558:
1555:
1552:
1549:
1546:
1543:
1540:
1537:
1534:
1531:
1528:
1525:
1522:
1519:
1491:
1488:
1485:
1482:
1476:
1471:
1465:
1462:
1457:
1454:
1448:
1441:
1438:
1433:
1430:
1426:
1419:
1416:
1410:
1406:
1400:
1397:
1395:
1393:
1389:
1385:
1382:
1376:
1371:
1365:
1362:
1357:
1354:
1348:
1342:
1336:
1333:
1328:
1325:
1319:
1313:
1310:
1305:
1302:
1298:
1294:
1288:
1285:
1280:
1277:
1272:
1268:
1265:
1262:
1257:
1253:
1249:
1246:
1243:
1239:
1236:
1231:
1224:
1217:
1214:
1208:
1204:
1198:
1195:
1192:
1190:
1188:
1185:
1182:
1177:
1169:
1165:
1161:
1156:
1151:
1147:
1140:
1134:
1130:
1124:
1121:
1116:
1113:
1109:
1102:
1099:
1093:
1089:
1083:
1080:
1077:
1075:
1073:
1070:
1067:
1066:
1049:
1046:
1030:
1027:
1024:
1020:
1016:
1013:
984:
981:
978:
974:
970:
967:
964:
959:
955:
948:
945:
939:
935:
929:
926:
923:
921:
919:
916:
913:
910:
906:
902:
899:
896:
890:
887:
879:
876:
870:
864:
861:
852:
849:
847:
845:
839:
836:
830:
824:
821:
815:
812:
810:
805:
802:
796:
795:
759:
754:
751:
744:
740:
734:
727:
722:
719:
712:
708:
702:
695:
690:
687:
680:
676:
670:
665:
661:
634:
630:
606:
575:
551:
548:
545:
542:
539:
536:
530:
527:
496:
485:kinetic energy
472:
467:
463:
456:
453:
447:
443:
437:
434:
428:
425:
419:
412:
409:
400:
394:
391:
383:
380:
374:
368:
365:
356:
350:
347:
320:
317:
314:
311:
307:
303:
300:
297:
294:
291:
285:
282:
257:
251:
248:
242:
236:
233:
227:
221:
218:
185:
182:
180:
177:
164:Main article:
161:
158:
142:
139:
116:
93:
90:
53:time-evolution
47:or its set of
33:kinetic energy
15:
9:
6:
4:
3:
2:
12308:
12307:
12296:
12293:
12291:
12288:
12286:
12283:
12281:
12278:
12276:
12273:
12271:
12268:
12266:
12263:
12262:
12260:
12241:
12238:
12236:
12233:
12232:
12230:
12226:
12220:
12217:
12215:
12212:
12210:
12207:
12205:
12202:
12201:
12199:
12195:
12189:
12186:
12185:
12183:
12179:
12173:
12170:
12168:
12165:
12163:
12160:
12159:
12157:
12153:
12147:
12144:
12142:
12139:
12137:
12134:
12133:
12131:
12127:
12121:
12118:
12116:
12113:
12111:
12108:
12107:
12105:
12101:
12098:
12094:
12084:
12081:
12079:
12076:
12075:
12073:
12069:
12063:
12060:
12059:
12057:
12053:
12047:
12044:
12042:
12039:
12037:
12036:d'Alembertian
12034:
12033:
12031:
12027:
12024:
12020:
12016:
12009:
12004:
12002:
11997:
11995:
11990:
11989:
11986:
11974:
11966:
11965:
11962:
11956:
11953:
11951:
11948:
11946:
11943:
11939:
11936:
11935:
11934:
11931:
11930:
11928:
11924:
11918:
11915:
11913:
11910:
11906:
11903:
11902:
11901:
11898:
11896:
11893:
11891:
11888:
11886:
11883:
11882:
11880:
11876:
11870:
11867:
11865:
11862:
11860:
11857:
11855:
11852:
11850:
11847:
11845:
11842:
11840:
11837:
11835:
11832:
11830:
11827:
11825:
11822:
11820:
11817:
11815:
11812:
11810:
11809:Quantum logic
11807:
11805:
11802:
11800:
11797:
11795:
11792:
11790:
11787:
11785:
11782:
11780:
11777:
11775:
11772:
11768:
11765:
11764:
11763:
11760:
11758:
11755:
11753:
11750:
11748:
11745:
11741:
11738:
11737:
11736:
11733:
11731:
11728:
11726:
11723:
11721:
11718:
11717:
11715:
11713:
11709:
11703:
11700:
11698:
11695:
11693:
11690:
11688:
11685:
11683:
11680:
11678:
11675:
11673:
11670:
11668:
11665:
11663:
11662:Quantum chaos
11660:
11658:
11655:
11653:
11650:
11649:
11647:
11645:
11641:
11635:
11632:
11630:
11629:Stern–Gerlach
11627:
11625:
11622:
11620:
11617:
11615:
11612:
11610:
11607:
11605:
11602:
11600:
11597:
11595:
11592:
11590:
11587:
11585:
11582:
11581:
11579:
11575:
11569:
11566:
11564:
11563:Transactional
11561:
11559:
11556:
11554:
11553:Quantum logic
11551:
11549:
11546:
11544:
11541:
11535:
11532:
11531:
11530:
11527:
11526:
11525:
11522:
11520:
11517:
11515:
11512:
11510:
11507:
11505:
11502:
11500:
11497:
11496:
11494:
11492:
11488:
11482:
11479:
11477:
11474:
11472:
11469:
11467:
11464:
11462:
11459:
11458:
11456:
11452:
11446:
11443:
11441:
11438:
11436:
11433:
11431:
11428:
11426:
11423:
11421:
11418:
11416:
11413:
11412:
11410:
11406:
11398:
11395:
11393:
11390:
11389:
11388:
11387:Wave function
11385:
11383:
11380:
11378:
11375:
11373:
11370:
11368:
11365:
11363:
11362:Superposition
11360:
11358:
11357:Quantum state
11355:
11353:
11350:
11348:
11345:
11343:
11340:
11338:
11335:
11333:
11330:
11328:
11325:
11321:
11318:
11316:
11313:
11311:
11310:Excited state
11308:
11306:
11303:
11302:
11301:
11298:
11296:
11293:
11291:
11288:
11286:
11283:
11281:
11278:
11277:
11275:
11271:
11265:
11262:
11260:
11257:
11255:
11252:
11248:
11245:
11244:
11243:
11240:
11238:
11235:
11234:
11232:
11228:
11224:
11217:
11212:
11210:
11205:
11203:
11198:
11197:
11194:
11187:
11182:
11178:
11177:
11164:
11162:0-582-44401-2
11158:
11154:
11147:
11139:
11133:
11128:
11127:
11118:
11110:
11108:0-19-855493-1
11104:
11100:
11093:
11085:
11083:0-471-87373-X
11079:
11074:
11073:
11064:
11062:
11057:
11046:
11043:
11041:
11038:
11036:
11033:
11031:
11028:
11026:
11023:
11021:
11018:
11016:
11013:
11011:
11008:
11006:
11005:Quantum state
11003:
11001:
10998:
10996:
10993:
10991:
10988:
10986:
10983:
10982:
10975:
10958:
10933:
10929:
10906:
10902:
10892:
10876:
10866:
10862:
10852:
10849:
10841:
10837:
10825:
10812:
10806:
10796:
10792:
10782:
10774:
10770:
10758:
10741:
10738:
10722:
10714:
10709:
10705:
10698:
10695:
10689:
10681:
10677:
10668:
10652:
10648:
10638:
10622:
10612:
10607:
10603:
10590:
10587:
10584:
10576:
10572:
10560:
10543:
10540:
10524:
10513:
10510:
10505:
10492:
10489:
10481:
10475:
10472:
10467:
10455:
10438:
10436:
10431:
10415:
10407:
10398:
10395:
10388:
10382:
10374:
10365:
10362:
10353:
10349:
10343:
10339:
10335:
10327:
10321:
10318:
10313:
10301:
10284:
10283:
10264:
10256:
10251:
10247:
10239:
10220:
10212:
10208:
10198:
10179:
10171:
10167:
10157:
10142:
10137:
10134:
10131:
10127:
10123:
10113:
10097:
10089:
10080:
10077:
10068:
10064:
10058:
10052:
10049:
10044:
10037:
10034:
10030:
10026:
10022:
10013:
10007:
9999:
9988:
9982:
9959:
9946:
9940:
9929:
9926:
9923:
9904:
9896:
9892:
9882:
9869:
9860:
9854:
9846:
9840:
9834:
9826:
9822:
9813:
9810:
9794:
9783:
9775:
9771:
9765:
9761:
9757:
9748:
9742:
9729:
9714:
9709:
9706:
9703:
9699:
9695:
9674:
9665:
9662:
9646:
9643:
9639:
9635:
9631:
9625:
9616:
9613:
9601:
9586:
9582:
9579:
9576:
9572:
9563:
9559:
9550:
9548:
9532:
9524:
9519:
9506:
9495:
9489:
9480:
9477:
9470:
9462:
9459:
9456:
9453:
9442:
9436:
9425:
9423:
9418:
9405:
9402:
9393:
9387:
9376:
9373:
9370:
9356:
9350:
9338:
9334:
9329:
9320:
9314:
9306:
9295:
9289:
9280:
9263:
9260:
9247:
9244:
9238:
9235:
9232:
9221:
9207:
9187:
9167:
9158:
9140:
9136:
9129:
9126:
9123:
9120:
9117:
9114:
9111:
9108:
9105:
9097:
9083:
9063:
9055:
9050:
9048:
9032:
9009:
9001:
8978:
8953:
8944:
8931:
8922:
8914:
8907:
8904:
8895:
8887:
8879:
8875:
8871:
8865:
8855:
8851:
8847:
8844:
8838:
8830:
8827:
8819:
8802:
8794:
8771:
8758:
8743:
8736:
8732:
8727:
8725:
8709:
8689:
8681:
8670:
8657:
8654:
8651:
8648:
8643:
8638:
8629:
8626:
8617:
8614:
8610:
8602:
8599:
8595:
8590:
8581:
8570:
8567:
8554:
8551:
8548:
8545:
8536:
8525:
8511:
8502:
8487:
8482:
8473:
8470:
8454:
8446:
8443:
8439:
8434:
8428:
8417:
8411:
8400:
8395:
8392:
8387:
8378:
8367:
8364:
8351:
8342:
8339:
8336:
8316:
8314:
8295:
8284:
8275:
8262:
8254:
8251:
8245:
8234:
8231:
8211:
8209:
8133:
8117:
8110:
8094:
8074:
8060:
8042:
8031:
8028:
8023:
8018:
8014:
8007:
7998:
7987:
7985:
7981:
7980:spin operator
7943:
7935:
7931:
7928:is the "spin
7913:
7909:
7899:
7878:
7875:
7870:
7865:
7861:
7854:
7849:
7834:
7832:
7818:
7800:
7792:
7789:
7780:
7769:
7766:
7748:
7740:
7737:
7734:
7726:
7674:
7656:
7641:
7638:
7635:
7627:
7612:
7609:
7600:
7589:
7586:
7561:
7558:
7538:
7535:
7517:
7502:
7499:
7496:
7488:
7464:
7448:
7419:
7409:
7391:
7377:
7367:
7362:
7343:
7339:
7333:
7329:
7320:
7317:
7314:
7310:
7301:
7297:
7293:
7290:
7286:
7281:
7276:
7271:
7258:
7254:
7250:
7244:
7240:
7234:
7230:
7224:
7219:
7216:
7213:
7209:
7205:
7203:
7185:
7175:
7170:
7151:
7147:
7141:
7137:
7128:
7125:
7122:
7118:
7112:
7107:
7104:
7101:
7097:
7088:
7084:
7080:
7077:
7073:
7068:
7063:
7058:
7046:
7042:
7038:
7031:
7026:
7023:
7020:
7016:
7010:
7005:
7001:
6995:
6992:
6990:
6979:
6964:
6961:
6938:
6928:
6923:
6904:
6900:
6894:
6890:
6881:
6878:
6875:
6871:
6865:
6860:
6857:
6854:
6850:
6841:
6837:
6833:
6830:
6826:
6821:
6818:
6810:
6796:
6774:
6745:
6741:
6715:
6702:
6693:
6670:
6660:
6655:
6636:
6632:
6626:
6622:
6613:
6610:
6607:
6603:
6594:
6590:
6586:
6583:
6579:
6574:
6566:
6553:
6548:
6544:
6538:
6535:
6532:
6528:
6522:
6519:
6514:
6509:
6505:
6496:
6494:
6476:
6472:
6451:
6442:
6409:
6405:
6401:
6398:
6391:
6387:
6381:
6377:
6370:
6367:
6359:
6357:
6353:
6350:—rather than
6349:
6331:
6327:
6304:
6300:
6291:
6281:
6267:
6247:
6227:
6219:
6201:
6191:
6163:
6153:
6125:
6115:
6102:
6098:
6080:
6077:
6073:
6050:
6047:
6043:
6020:
6017:
6013:
6003:
5988:
5983:
5973:
5961:
5958:
5954:
5950:
5944:
5940:
5934:
5929:
5924:
5914:
5902:
5899:
5895:
5891:
5885:
5881:
5875:
5870:
5865:
5855:
5843:
5840:
5836:
5832:
5826:
5822:
5816:
5813:
5804:
5793:
5791:
5788:
5784:
5781:, say due to
5780:
5776:
5766:
5748:
5742:
5738:
5732:
5726:
5722:
5718:
5712:
5704:
5700:
5690:
5677:
5674:
5668:
5664:
5658:
5654:
5650:
5646:
5640:
5636:
5630:
5624:
5620:
5616:
5610:
5602:
5598:
5588:
5575:
5572:
5566:
5562:
5556:
5552:
5548:
5544:
5538:
5534:
5528:
5522:
5518:
5514:
5508:
5500:
5496:
5486:
5473:
5470:
5464:
5460:
5454:
5450:
5446:
5444:
5435:
5429:
5425:
5421:
5416:
5412:
5408:
5403:
5399:
5394:
5388:
5382:
5378:
5374:
5368:
5364:
5355:
5351:
5341:
5331:
5323:
5319:
5309:
5299:
5291:
5287:
5277:
5266:
5259:
5256:
5250:
5246:
5240:
5237:
5235:
5224:
5209:
5206:
5191:
5187:
5183:
5178:
5174:
5170:
5165:
5161:
5157:
5152:
5132:
5124:
5116:
5111:
5107:
5098:
5081:
5078:
5075:
5072:
5069:
5035:
5020:
5016:
5010:
5004:
5000:
4996:
4990:
4985:
4974:
4971:
4965:
4961:
4955:
4952:
4943:
4932:
4929:
4914:
4910:
4904:
4898:
4894:
4890:
4884:
4876:
4872:
4862:
4849:
4846:
4840:
4836:
4830:
4827:
4818:
4807:
4804:
4789:
4786:
4781:
4776:
4772:
4763:
4749:
4729:
4722:
4706:
4699:
4694:
4679:
4675:
4669:
4663:
4659:
4655:
4649:
4644:
4640:
4634:
4631:
4626:
4623:
4615:
4613:
4603:
4601:
4597:
4592:
4577:
4573:
4569:
4564:
4553:
4550:
4544:
4540:
4534:
4531:
4522:
4511:
4508:
4493:
4489:
4485:
4477:
4473:
4463:
4450:
4447:
4441:
4437:
4431:
4428:
4419:
4408:
4392:
4388:
4384:
4381:
4367:
4352:
4341:
4338:
4332:
4328:
4322:
4319:
4310:
4299:
4296:
4278:
4274:
4264:
4251:
4248:
4242:
4238:
4232:
4229:
4220:
4209:
4204:Free particle
4196:
4182:
4162:
4147:
4145:
4139:
4137:
4133:
4117:
4108:
4095:
4091:
4088:
4085:
4079:
4075:
4071:
4067:
4064:
4061:
4057:
4049:
4030:
4026:
4014:
4011:, provide an
3997:
3994:
3991:
3970:
3962:
3957:
3955:
3954:Hilbert space
3951:
3947:
3937:
3935:
3931:
3928:(more than a
3927:
3905:
3899:
3888:
3887:
3882:
3880:
3875:
3870:
3851:
3847:
3844:
3841:
3838:
3834:
3830:
3827:
3819:
3817:
3812:
3810:
3806:
3802:
3798:
3782:
3774:
3770:
3765:
3752:
3748:
3741:
3735:
3731:
3721:
3717:
3714:
3711:
3708:
3704:
3700:
3696:
3689:
3683:
3679:
3670:
3656:
3636:
3633:
3630:
3610:
3602:
3598:
3593:
3580:
3576:
3569:
3563:
3559:
3552:
3549:
3542:
3534:
3531:
3527:
3520:
3514:
3510:
3506:
3498:
3484:
3463:
3456:
3450:
3446:
3435:
3425:
3421:
3402:
3392:
3383:
3378:
3375:
3372:
3368:
3364:
3362:
3353:
3347:
3343:
3339:
3334:
3329:
3316:
3312:
3308:
3302:
3298:
3292:
3288:
3282:
3277:
3274:
3271:
3267:
3263:
3261:
3251:
3247:
3241:
3236:
3233:
3230:
3226:
3222:
3217:
3212:
3200:
3196:
3192:
3185:
3180:
3177:
3174:
3170:
3164:
3159:
3155:
3149:
3146:
3144:
3133:
3118:
3115:
3099:
3096:
3091:
3078:
3075:
3072:
3069:
3063:
3060:
3055:
3042:
3039:
3033:
3030:
3025:
3012:
3009:
3003:
3000:
2995:
2982:
2977:
2972:
2969:
2966:
2962:
2958:
2955:
2947:
2944:
2941:
2925:
2905:
2896:
2894:
2878:
2869:
2854:
2846:
2841:
2830:
2827:
2821:
2817:
2811:
2803:
2801:
2796:
2776:
2773:
2768:
2758:
2755:
2752:
2747:
2737:
2732:
2719:
2716:
2711:
2706:
2694:
2690:
2686:
2679:
2674:
2671:
2668:
2664:
2658:
2653:
2649:
2643:
2640:
2638:
2627:
2624:
2619:
2609:
2606:
2603:
2598:
2588:
2583:
2570:
2567:
2559:
2555:
2551:
2544:
2527:
2522:
2500:
2495:
2492:
2489:
2485:
2481:
2479:
2465:
2459:
2454:
2444:
2435:
2430:
2427:
2424:
2420:
2416:
2414:
2403:
2388:
2374:
2365:
2352:
2344:
2339:
2335:
2325:
2315:
2307:
2302:
2298:
2288:
2278:
2270:
2265:
2261:
2251:
2241:
2236:
2231:
2200:
2195:
2170:
2148:
2123:
2101:
2096:
2083:
2079:
2075:
2069:
2065:
2059:
2056:
2048:
2044:
2040:
2033:
2016:
2011:
1991:
1986:
1976:
1952:
1946:
1943:
1938:
1928:
1925:
1922:
1917:
1907:
1902:
1889:
1886:
1877:
1865:
1846:
1840:
1835:
1825:
1816:
1811:
1808:
1805:
1801:
1797:
1788:
1777:
1763:
1749:
1747:
1742:
1726:
1714:
1710:
1707:
1704:
1698:
1685:
1679:
1666:
1658:
1654:
1647:
1635:
1631:
1628:
1625:
1619:
1606:
1600:
1589:
1583:
1575:
1567:
1563:
1559:
1556:
1553:
1550:
1541:
1535:
1529:
1526:
1523:
1520:
1517:
1509:
1506:
1489:
1486:
1483:
1480:
1474:
1469:
1463:
1460:
1455:
1452:
1446:
1436:
1428:
1424:
1417:
1414:
1408:
1404:
1398:
1396:
1387:
1383:
1380:
1374:
1369:
1363:
1360:
1355:
1352:
1346:
1340:
1334:
1331:
1326:
1323:
1317:
1308:
1300:
1296:
1292:
1283:
1275:
1270:
1263:
1255:
1251:
1244:
1237:
1234:
1229:
1222:
1215:
1212:
1206:
1202:
1196:
1193:
1191:
1183:
1180:
1175:
1167:
1163:
1159:
1154:
1149:
1145:
1138:
1132:
1128:
1119:
1111:
1107:
1100:
1097:
1091:
1087:
1081:
1078:
1076:
1071:
1068:
1056:
1053:
1045:
1042:
1025:
1022:
1004:
1003:wave function
999:
979:
976:
965:
962:
957:
946:
943:
937:
933:
927:
924:
922:
911:
908:
897:
894:
888:
885:
868:
850:
848:
834:
828:
819:
813:
811:
800:
785:
783:
779:
774:
757:
752:
742:
732:
725:
720:
710:
700:
693:
688:
678:
668:
663:
650:
632:
620:
596:
592:
589:
565:
549:
540:
537:
534:
525:
514:
510:
494:
486:
470:
465:
454:
451:
445:
441:
435:
432:
426:
423:
417:
407:
398:
392:
389:
372:
354:
345:
335:operator and
334:
318:
312:
309:
298:
295:
292:
289:
280:
268:
255:
246:
240:
231:
225:
216:
205:
203:
199:
195:
191:
176:
173:
167:
157:
137:
114:
88:
77:
73:
69:
65:
60:
58:
54:
50:
46:
42:
38:
34:
30:
26:
22:
12204:Displacement
12140:
12136:Total energy
11692:Quantum mind
11604:Franck–Hertz
11466:Klein–Gordon
11415:Formulations
11408:Formulations
11337:Interference
11331:
11327:Entanglement
11305:Ground state
11300:Energy level
11273:Fundamentals
11237:Introduction
11188:at Wikiquote
11152:
11146:
11125:
11117:
11098:
11092:
11071:
10893:
10742:
10739:
10669:
10639:
10544:
10541:
10439:
10433:By applying
10432:
10285:
10196:
10158:
10114:
9930:
9927:
9883:
9814:
9811:
9730:
9666:
9663:
9602:
9556:
9520:
9426:
9419:
9264:
9261:
9222:
9159:
9098:
9051:
8945:
8820:
8728:
8723:
8676:
8571:
8568:
8526:
8503:
8368:
8365:
8317:
8276:
8212:
8066:
7988:
7900:
7835:
7819:
7770:
7767:
7727:
7680:
7590:
7587:
7539:
7536:
7489:
7415:
6965:
6962:
6811:
6694:
6497:
6443:
6360:
6287:
6004:
5794:
5772:
5210:
5207:
5099:
5036:
4933:
4930:
4808:
4805:
4764:
4719:, effective
4695:
4616:
4609:
4593:
4512:
4509:
4409:
4373:
4300:
4297:
4210:
4207:
4153:
4140:
4109:
4050:
3961:eigenvectors
3958:
3943:
3884:
3877:
3876:. It is the
3871:
3820:
3816:homomorphism
3813:
3773:power series
3766:
3671:
3594:
3499:
3437:
3422:
3119:
3116:
2948:
2945:
2939:
2897:
2892:
2870:
2804:
2797:
2389:
2366:
1866:
1778:
1755:
1743:
1510:
1507:
1057:
1054:
1051:
1043:
1000:
786:
775:
269:
206:
187:
184:One particle
169:
160:Introduction
61:
48:
44:
24:
18:
12141:Hamiltonian
12103:Fundamental
11950:EPR paradox
11730:Quantum bus
11599:Double-slit
11577:Experiments
11543:Many-worlds
11481:Schrödinger
11445:Phase space
11435:Schrödinger
11425:Interaction
11382:Uncertainty
11352:Nonlocality
11347:Measurement
11342:Decoherence
11332:Hamiltonian
11155:. Longman.
9262:Therefore,
8087:and charge
5775:rigid rotor
5769:Rigid rotor
4742:, and mass
4136:real number
3769:exponential
1776:particles:
595:dot product
513:dot product
70:, known as
25:Hamiltonian
12259:Categories
12062:C-symmetry
11878:Extensions
11712:Technology
11558:Relational
11509:Copenhagen
11420:Heisenberg
11367:Tunnelling
11230:Background
11052:References
9521:Thus, the
8731:degeneracy
8724:degenerate
8680:wavelength
4696:where the
3983:, denoted
3886:propagator
3805:continuous
12055:Particles
11584:Bell test
11454:Equations
11280:Born rule
10962:⟩
10956:⟨
10930:π
10874:∂
10863:π
10859:∂
10853:−
10834:∂
10829:⟩
10823:⟨
10820:∂
10804:∂
10789:∂
10771:π
10767:∂
10762:⟩
10756:⟨
10753:∂
10715:∗
10702:ℏ
10678:π
10649:π
10620:∂
10613:∗
10600:∂
10594:ℏ
10588:−
10569:∂
10564:⟩
10558:⟨
10555:∂
10522:∂
10502:∂
10496:ℏ
10482:∗
10464:∂
10459:⟩
10453:⟨
10450:∂
10419:⟩
10416:ψ
10392:⟨
10386:⟩
10359:⟨
10340:∑
10328:∗
10310:∂
10305:⟩
10299:⟨
10296:∂
10257:∗
10128:ψ
10101:⟩
10074:⟨
10059:∗
10027:∑
10020:⟩
10008:ψ
9983:ψ
9980:⟨
9953:⟩
9938:⟨
9867:⟩
9855:ψ
9844:⟨
9798:⟩
9762:∑
9755:⟩
9743:ψ
9700:ψ
9636:δ
9629:⟩
9610:⟨
9490:ψ
9487:⟨
9466:ℏ
9460:−
9437:ψ
9434:⟨
9400:⟩
9388:ψ
9351:ψ
9348:⟨
9342:ℏ
9327:⟩
9315:ψ
9290:ψ
9287:⟨
9278:∂
9274:∂
9137:ε
9121:ε
9115:−
9054:conserved
9013:⟩
8982:⟩
8926:⟩
8899:⟩
8869:⟩
8842:⟩
8806:⟩
8775:⟩
8655:ϕ
8627:−
8624:∇
8621:ℏ
8615:−
8585:^
8552:ϕ
8540:^
8512:ϕ
8471:−
8465:^
8429:˙
8418:⋅
8412:˙
8382:^
8349:∇
8346:ℏ
8340:−
8331:^
8296:˙
8246:˙
8226:^
8169:^
8118:ϕ
8043:⋅
8002:^
7845:μ
7801:⋅
7797:μ
7793:−
7784:^
7749:⋅
7745:μ
7741:−
7690:μ
7657:⋅
7651:^
7639:−
7628:⋅
7622:^
7613:−
7604:^
7571:^
7553:^
7518:⋅
7512:^
7503:−
7368:−
7318:≠
7311:∑
7298:ε
7294:π
7268:∇
7241:ℏ
7235:−
7210:∑
7176:−
7126:≠
7119:∑
7098:∑
7085:ε
7081:π
7055:∇
7017:∑
7002:ℏ
6996:−
6983:^
6929:−
6879:≠
6872:∑
6851:∑
6838:ε
6834:π
6703:ϕ
6661:−
6611:≠
6604:∑
6591:ε
6587:π
6554:ϕ
6536:≠
6529:∑
6406:ε
6402:π
6195:^
6157:^
6119:^
5977:^
5941:ℏ
5935:−
5918:^
5882:ℏ
5876:−
5859:^
5823:ℏ
5817:−
5808:^
5723:ω
5697:∂
5687:∂
5665:ℏ
5659:−
5621:ω
5595:∂
5585:∂
5563:ℏ
5557:−
5519:ω
5493:∂
5483:∂
5461:ℏ
5455:−
5379:ω
5348:∂
5338:∂
5316:∂
5306:∂
5284:∂
5274:∂
5247:ℏ
5241:−
5228:^
5125:⋅
5001:ω
4982:∇
4962:ℏ
4956:−
4947:^
4895:ω
4869:∂
4859:∂
4837:ℏ
4831:−
4822:^
4773:ω
4707:ω
4660:ω
4561:∇
4541:ℏ
4535:−
4526:^
4470:∂
4460:∂
4438:ℏ
4432:−
4423:^
4349:∇
4329:ℏ
4323:−
4314:^
4271:∂
4261:∂
4239:ℏ
4233:−
4224:^
3930:semigroup
3856:ℏ
3839:−
3814:By the *-
3736:ψ
3726:ℏ
3709:−
3684:ψ
3564:ψ
3538:ℏ
3515:ψ
3451:ψ
3396:^
3369:∑
3326:∇
3299:ℏ
3293:−
3268:∑
3227:∑
3209:∇
3171:∑
3156:ℏ
3150:−
3137:^
3073:⋯
2963:∑
2851:∇
2847:⋅
2838:∇
2818:ℏ
2812:−
2756:…
2703:∇
2665:∑
2650:ℏ
2644:−
2607:…
2538:^
2528:⋅
2516:^
2486:∑
2469:^
2448:^
2421:∑
2407:^
2332:∂
2322:∂
2295:∂
2285:∂
2258:∂
2248:∂
2228:∇
2192:∇
2145:∇
2093:∇
2066:ℏ
2060:−
2027:^
2017:⋅
2005:^
1980:^
1926:…
1881:^
1850:^
1829:^
1802:∑
1792:^
1711:≥
1680:ψ
1670:∞
1662:∞
1659:−
1655:∫
1632:≥
1601:ψ
1579:∞
1571:∞
1568:−
1564:∫
1548:⟩
1533:⟨
1487:≥
1456:ψ
1440:∞
1432:∞
1429:−
1425:∫
1405:ℏ
1375:∗
1356:ψ
1327:ψ
1312:∞
1304:∞
1301:−
1297:∫
1293:−
1287:∞
1279:∞
1276:−
1256:∗
1252:ψ
1235:ψ
1203:ℏ
1197:−
1155:ψ
1133:∗
1129:ψ
1123:∞
1115:∞
1112:−
1108:∫
1088:ℏ
1082:−
1012:Ψ
954:∇
934:ℏ
928:−
878:^
869:⋅
863:^
838:^
823:^
804:^
750:∂
739:∂
718:∂
707:∂
686:∂
675:∂
660:∇
629:∇
619:Laplacian
605:∇
574:∇
547:∇
544:ℏ
538:−
529:^
462:∇
442:ℏ
436:−
411:^
382:^
373:⋅
367:^
349:^
284:^
250:^
235:^
220:^
202:potential
194:operators
141:ˇ
92:^
12120:Rotation
12115:Position
12110:Momentum
11973:Category
11767:Timeline
11519:Ensemble
11499:Bayesian
11392:Collapse
11264:Glossary
11247:Timeline
10978:See also
10514:′
10476:′
10399:′
10366:′
10322:′
10236:and its
10143:⟩
10081:′
10053:′
10038:′
9715:⟩
9647:′
9617:′
9583:⟩
9558:Hamilton
8757:commutes
7986:, hence
7930:g-factor
6348:SI units
6095:are the
4092:⟩
4068:⟩
3998:⟩
3881:operator
3749:⟩
3697:⟩
3577:⟩
3528:⟩
3464:⟩
1238:′
591:operator
566:where a
41:spectrum
29:operator
12219:Squeeze
12167:Orbital
12096:Quantum
12022:General
11926:Related
11905:History
11644:Science
11476:Rydberg
11242:History
9922:complex
8311:is the
7978:is the
7827:⁄
7416:For an
6103:), and
3924:form a
3497:, then
586:is the
562:is the
507:is the
483:is the
331:is the
198:kinetic
12197:Optics
12129:Energy
12041:Parity
11619:Popper
11159:
11134:
11105:
11080:
9812:where
9475:
8946:Since
8277:where
7901:where
7820:For a
6695:where
6260:, and
6065:, and
6005:where
5787:triple
5783:double
5773:For a
4610:For a
4110:Since
3547:
2871:where
2183:, and
1867:where
593:. The
270:where
172:energy
39:. Its
23:, the
12162:Total
11529:Local
11471:Pauli
11461:Dirac
10159:Each
7822:spin-
4130:is a
3963:) of
3950:Dirac
3872:is a
12172:Spin
12046:Time
11157:ISBN
11132:ISBN
11103:ISBN
11078:ISBN
9920:are
9422:dual
8994:and
8130:and
6319:and
6288:The
6180:and
3767:The
2898:For
509:mass
200:and
35:and
10667:by
10197:two
7936:),
6760:at
6495:):
6358:):
5785:or
3948:of
3883:or
3775:in
2940:not
2938:is
1719:min
1640:min
597:of
588:del
127:or
19:In
12261::
11060:^
9687:,
9549:.
9424:,
9406:0.
9220::
9096::
8726:.
6809::
6240:,
6035:,
4602:.
4195:.
4138:.
3936:.
2219::
2136:,
1748:.
784::
156:.
59:.
12007:e
12000:t
11993:v
11215:e
11208:t
11201:v
11165:.
11140:.
11111:.
11086:.
10959:H
10934:n
10907:n
10903:a
10877:t
10867:n
10850:=
10842:n
10838:a
10826:H
10813:,
10807:t
10797:n
10793:a
10783:=
10775:n
10759:H
10726:)
10723:t
10720:(
10710:n
10706:a
10699:i
10696:=
10693:)
10690:t
10687:(
10682:n
10653:n
10623:t
10608:n
10604:a
10591:i
10585:=
10577:n
10573:a
10561:H
10525:t
10511:n
10506:a
10493:i
10490:=
10473:n
10468:a
10456:H
10412:|
10408:H
10404:|
10396:n
10389:=
10383:n
10379:|
10375:H
10371:|
10363:n
10354:n
10350:a
10344:n
10336:=
10319:n
10314:a
10302:H
10268:)
10265:t
10262:(
10252:n
10248:a
10224:)
10221:t
10218:(
10213:n
10209:a
10183:)
10180:t
10177:(
10172:n
10168:a
10138:)
10135:t
10132:(
10124:|
10098:n
10094:|
10090:H
10086:|
10078:n
10069:n
10065:a
10050:n
10045:a
10035:n
10031:n
10023:=
10017:)
10014:t
10011:(
10004:|
10000:H
9996:|
9992:)
9989:t
9986:(
9972:f
9969:e
9966:d
9960:=
9950:)
9947:t
9944:(
9941:H
9908:)
9905:t
9902:(
9897:n
9893:a
9870:.
9864:)
9861:t
9858:(
9851:|
9847:n
9841:=
9838:)
9835:t
9832:(
9827:n
9823:a
9795:n
9791:|
9787:)
9784:t
9781:(
9776:n
9772:a
9766:n
9758:=
9752:)
9749:t
9746:(
9739:|
9710:)
9707:t
9704:(
9696:|
9675:t
9644:n
9640:n
9632:=
9626:n
9622:|
9614:n
9587:}
9580:n
9577:|
9573:{
9533:G
9507:.
9503:|
9499:)
9496:t
9493:(
9481:t
9478:d
9471:d
9463:i
9457:=
9454:H
9450:|
9446:)
9443:t
9440:(
9403:=
9397:)
9394:t
9391:(
9384:|
9380:]
9377:H
9374:,
9371:G
9368:[
9364:|
9360:)
9357:t
9354:(
9339:i
9335:1
9330:=
9324:)
9321:t
9318:(
9311:|
9307:G
9303:|
9299:)
9296:t
9293:(
9281:t
9248:0
9245:=
9242:]
9239:G
9236:,
9233:H
9230:[
9208:G
9188:H
9168:U
9146:)
9141:2
9133:(
9130:O
9127:+
9124:G
9118:i
9112:I
9109:=
9106:U
9084:U
9064:G
9033:H
9010:a
9006:|
9002:U
8979:a
8975:|
8954:U
8932:.
8929:)
8923:a
8919:|
8915:U
8912:(
8908:H
8905:=
8902:)
8896:a
8892:|
8888:U
8885:(
8880:a
8876:E
8872:=
8866:a
8862:|
8856:a
8852:E
8848:U
8845:=
8839:a
8835:|
8831:H
8828:U
8803:a
8799:|
8795:U
8772:a
8768:|
8744:U
8710:y
8690:x
8658:.
8652:q
8649:+
8644:2
8639:)
8634:A
8630:q
8618:i
8611:(
8603:m
8600:2
8596:1
8591:=
8582:H
8555:.
8549:q
8546:=
8537:V
8488:2
8483:)
8478:A
8474:q
8462:p
8455:(
8447:m
8444:2
8440:1
8435:=
8425:r
8408:r
8401:m
8396:2
8393:1
8388:=
8379:T
8352:,
8343:i
8337:=
8328:p
8292:r
8285:m
8263:,
8259:A
8255:q
8252:+
8242:r
8235:m
8232:=
8223:p
8193:A
8166:p
8142:A
8095:q
8075:m
8047:B
8039:S
8032:m
8029:2
8024:e
8019:s
8015:g
8008:=
7999:H
7965:S
7944:e
7914:s
7910:g
7886:S
7879:m
7876:2
7871:e
7866:s
7862:g
7855:=
7850:S
7829:2
7825:1
7805:B
7790:=
7781:H
7753:B
7738:=
7735:V
7712:B
7661:E
7648:r
7642:q
7636:=
7632:E
7619:d
7610:=
7601:H
7568:r
7562:q
7559:=
7550:d
7522:E
7509:d
7500:=
7497:V
7474:E
7449:q
7428:d
7392:)
7384:|
7378:j
7373:r
7363:i
7358:r
7352:|
7344:j
7340:q
7334:i
7330:q
7321:j
7315:i
7302:0
7291:8
7287:1
7282:+
7277:2
7272:j
7259:j
7255:m
7251:2
7245:2
7231:(
7225:N
7220:1
7217:=
7214:j
7206:=
7192:|
7186:j
7181:r
7171:i
7166:r
7160:|
7152:j
7148:q
7142:i
7138:q
7129:j
7123:i
7113:N
7108:1
7105:=
7102:j
7089:0
7078:8
7074:1
7069:+
7064:2
7059:j
7047:j
7043:m
7039:1
7032:N
7027:1
7024:=
7021:j
7011:2
7006:2
6993:=
6980:H
6945:|
6939:j
6934:r
6924:i
6919:r
6913:|
6905:j
6901:q
6895:i
6891:q
6882:j
6876:i
6866:N
6861:1
6858:=
6855:j
6842:0
6831:8
6827:1
6822:=
6819:V
6797:j
6775:i
6770:r
6746:j
6742:q
6721:)
6716:i
6711:r
6706:(
6677:|
6671:j
6666:r
6656:i
6651:r
6645:|
6637:j
6633:q
6627:i
6623:q
6614:j
6608:i
6595:0
6584:8
6580:1
6575:=
6572:)
6567:i
6562:r
6557:(
6549:i
6545:q
6539:j
6533:i
6523:2
6520:1
6515:=
6510:j
6506:V
6477:j
6473:q
6452:N
6426:|
6421:r
6416:|
6410:0
6399:4
6392:2
6388:q
6382:1
6378:q
6371:=
6368:V
6332:2
6328:q
6305:1
6301:q
6268:z
6248:y
6228:x
6202:z
6192:J
6178:,
6164:y
6154:J
6140:,
6126:x
6116:J
6081:z
6078:z
6074:I
6051:y
6048:y
6044:I
6021:x
6018:x
6014:I
5989:2
5984:z
5974:J
5962:z
5959:z
5955:I
5951:2
5945:2
5930:2
5925:y
5915:J
5903:y
5900:y
5896:I
5892:2
5886:2
5871:2
5866:x
5856:J
5844:x
5841:x
5837:I
5833:2
5827:2
5814:=
5805:H
5749:)
5743:2
5739:z
5733:2
5727:2
5719:m
5713:+
5705:2
5701:z
5691:2
5678:m
5675:2
5669:2
5655:(
5651:+
5647:)
5641:2
5637:y
5631:2
5625:2
5617:m
5611:+
5603:2
5599:y
5589:2
5576:m
5573:2
5567:2
5553:(
5549:+
5545:)
5539:2
5535:x
5529:2
5523:2
5515:m
5509:+
5501:2
5497:x
5487:2
5474:m
5471:2
5465:2
5451:(
5447:=
5436:)
5430:2
5426:z
5422:+
5417:2
5413:y
5409:+
5404:2
5400:x
5395:(
5389:2
5383:2
5375:m
5369:+
5365:)
5356:2
5352:z
5342:2
5332:+
5324:2
5320:y
5310:2
5300:+
5292:2
5288:x
5278:2
5267:(
5260:m
5257:2
5251:2
5238:=
5225:H
5192:2
5188:z
5184:+
5179:2
5175:y
5171:+
5166:2
5162:x
5158:=
5153:2
5148:|
5142:r
5137:|
5133:=
5129:r
5121:r
5117:=
5112:2
5108:r
5085:)
5082:z
5079:,
5076:y
5073:,
5070:x
5067:(
5046:r
5021:2
5017:r
5011:2
5005:2
4997:m
4991:+
4986:2
4975:m
4972:2
4966:2
4953:=
4944:H
4915:2
4911:x
4905:2
4899:2
4891:m
4885:+
4877:2
4873:x
4863:2
4850:m
4847:2
4841:2
4828:=
4819:H
4790:m
4787:k
4782:=
4777:2
4750:m
4730:k
4680:2
4676:x
4670:2
4664:2
4656:m
4650:=
4645:2
4641:x
4635:2
4632:k
4627:=
4624:V
4578:0
4574:V
4570:+
4565:2
4554:m
4551:2
4545:2
4532:=
4523:H
4494:0
4490:V
4486:+
4478:2
4474:x
4464:2
4451:m
4448:2
4442:2
4429:=
4420:H
4393:0
4389:V
4385:=
4382:V
4353:2
4342:m
4339:2
4333:2
4320:=
4311:H
4279:2
4275:x
4265:2
4252:m
4249:2
4243:2
4230:=
4221:H
4183:q
4163:m
4118:H
4096:.
4089:a
4086:|
4080:a
4076:E
4072:=
4065:a
4062:|
4058:H
4036:}
4031:a
4027:E
4023:{
3995:a
3992:|
3971:H
3912:}
3909:)
3906:t
3903:(
3900:U
3897:{
3852:/
3848:t
3845:H
3842:i
3835:e
3831:=
3828:U
3783:H
3753:.
3745:)
3742:0
3739:(
3732:|
3722:/
3718:t
3715:H
3712:i
3705:e
3701:=
3693:)
3690:t
3687:(
3680:|
3657:H
3637:0
3634:=
3631:t
3611:H
3581:.
3573:)
3570:t
3567:(
3560:|
3553:t
3550:d
3543:d
3535:i
3532:=
3524:)
3521:t
3518:(
3511:|
3507:H
3485:t
3460:)
3457:t
3454:(
3447:|
3403:i
3393:H
3384:N
3379:1
3376:=
3373:i
3365:=
3354:)
3348:i
3344:V
3340:+
3335:2
3330:i
3317:i
3313:m
3309:2
3303:2
3289:(
3283:N
3278:1
3275:=
3272:i
3264:=
3252:i
3248:V
3242:N
3237:1
3234:=
3231:i
3223:+
3218:2
3213:i
3201:i
3197:m
3193:1
3186:N
3181:1
3178:=
3175:i
3165:2
3160:2
3147:=
3134:H
3103:)
3100:t
3097:,
3092:N
3087:r
3082:(
3079:V
3076:+
3070:+
3067:)
3064:t
3061:,
3056:2
3051:r
3046:(
3043:V
3040:+
3037:)
3034:t
3031:,
3026:1
3021:r
3016:(
3013:V
3010:=
3007:)
3004:t
3001:,
2996:i
2991:r
2986:(
2983:V
2978:N
2973:1
2970:=
2967:i
2959:=
2956:V
2926:V
2906:N
2879:M
2855:j
2842:i
2831:M
2828:2
2822:2
2780:)
2777:t
2774:,
2769:N
2764:r
2759:,
2753:,
2748:2
2743:r
2738:,
2733:1
2728:r
2723:(
2720:V
2717:+
2712:2
2707:n
2695:n
2691:m
2687:1
2680:N
2675:1
2672:=
2669:n
2659:2
2654:2
2641:=
2631:)
2628:t
2625:,
2620:N
2615:r
2610:,
2604:,
2599:2
2594:r
2589:,
2584:1
2579:r
2574:(
2571:V
2568:+
2560:n
2556:m
2552:2
2545:n
2535:p
2523:n
2513:p
2501:N
2496:1
2493:=
2490:n
2482:=
2466:V
2460:+
2455:n
2445:T
2436:N
2431:1
2428:=
2425:n
2417:=
2404:H
2375:N
2353:,
2345:2
2340:n
2336:z
2326:2
2316:+
2308:2
2303:n
2299:y
2289:2
2279:+
2271:2
2266:n
2262:x
2252:2
2242:=
2237:2
2232:n
2217:n
2201:2
2196:n
2171:n
2149:n
2124:n
2102:2
2097:n
2084:n
2080:m
2076:2
2070:2
2057:=
2049:n
2045:m
2041:2
2034:n
2024:p
2012:n
2002:p
1992:=
1987:n
1977:T
1953:,
1950:)
1947:t
1944:,
1939:N
1934:r
1929:,
1923:,
1918:2
1913:r
1908:,
1903:1
1898:r
1893:(
1890:V
1887:=
1878:V
1847:V
1841:+
1836:n
1826:T
1817:N
1812:1
1809:=
1806:n
1798:=
1789:H
1764:N
1730:)
1727:x
1724:(
1715:V
1708:x
1705:d
1699:2
1694:|
1689:)
1686:x
1683:(
1676:|
1667:+
1651:)
1648:x
1645:(
1636:V
1629:x
1626:d
1620:2
1615:|
1610:)
1607:x
1604:(
1597:|
1593:)
1590:x
1587:(
1584:V
1576:+
1560:+
1557:E
1554:K
1551:=
1545:)
1542:x
1539:(
1536:V
1530:+
1527:E
1524:K
1521:=
1518:E
1490:0
1484:x
1481:d
1475:2
1470:|
1464:x
1461:d
1453:d
1447:|
1437:+
1418:m
1415:2
1409:2
1399:=
1388:)
1384:x
1381:d
1370:)
1364:x
1361:d
1353:d
1347:(
1341:)
1335:x
1332:d
1324:d
1318:(
1309:+
1284:+
1271:]
1267:)
1264:x
1261:(
1248:)
1245:x
1242:(
1230:[
1223:(
1216:m
1213:2
1207:2
1194:=
1184:x
1181:d
1176:)
1168:2
1164:x
1160:d
1150:2
1146:d
1139:(
1120:+
1101:m
1098:2
1092:2
1079:=
1072:E
1069:K
1029:)
1026:t
1023:,
1019:r
1015:(
983:)
980:t
977:,
973:r
969:(
966:V
963:+
958:2
947:m
944:2
938:2
925:=
915:)
912:t
909:,
905:r
901:(
898:V
895:+
889:m
886:2
875:p
860:p
851:=
835:V
829:+
820:T
814:=
801:H
758:2
753:z
743:2
733:+
726:2
721:y
711:2
701:+
694:2
689:x
679:2
669:=
664:2
633:2
550:,
541:i
535:=
526:p
495:m
471:,
466:2
455:m
452:2
446:2
433:=
427:m
424:2
418:2
408:p
399:=
393:m
390:2
379:p
364:p
355:=
346:T
319:,
316:)
313:t
310:,
306:r
302:(
299:V
296:=
293:V
290:=
281:V
256:,
247:V
241:+
232:T
226:=
217:H
138:H
115:H
89:H
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