906:. Qualitatively, it corresponds to a potential which is zero everywhere, except at a single point, where it takes an infinite value. This can be used to simulate situations where a particle is free to move in two regions of space with a barrier between the two regions. For example, an electron can move almost freely in a conducting material, but if two conducting surfaces are put close together, the interface between them acts as a barrier for the electron that can be approximated by a delta potential.
3130:
1646:
5655:
2006:
4128:
929:
2894:
1396:
3731:, that is, a barrier has the same probability of reflecting the particle as a well. This is a significant difference from classical mechanics, where the reflection probability would be 1 for the barrier (the particle simply bounces back), and 0 for the well (the particle passes through the well undisturbed).
4008:
The above model is one-dimensional while the space around us is three-dimensional. So, in fact, one should solve the Schrödinger equation in three dimensions. On the other hand, many systems only change along one coordinate direction and are translationally invariant along the others. The Schrödinger
2619:
3971:
4001:(STM) relies on this tunneling effect. In that case, the barrier is due to the air between the tip of the STM and the underlying object. The strength of the barrier is related to the separation being stronger the further apart the two are. For a more general model of this situation, see
3994:. Often, the surfaces of such materials are covered with oxide layers or are not ideal for other reasons. This thin, non-conducting layer may then be modeled by a local delta-function potential as above. Electrons may then tunnel from one material to the other giving rise to a current.
5005:
4771:
3725:
1820:
2714:
3002:
3495:
2633:
1797:
A second relation can be found by studying the derivative of the wavefunction. Normally, we could also impose differentiability at the origin, but this is not possible because of the delta potential. However, if we integrate the Schrödinger equation around
4624:
4272:
2406:
1098:
2399:
3116:
3737:
4800:
4629:
3506:
1641:{\displaystyle \psi (x)={\begin{cases}\psi _{\text{L}}(x)=A_{\text{r}}e^{ikx}+A_{\text{l}}e^{-ikx},&{\text{ if }}x<0,\\\psi _{\text{R}}(x)=B_{\text{r}}e^{ikx}+B_{\text{l}}e^{-ikx},&{\text{ if }}x>0,\end{cases}}}
2248:
1314:
106:
1793:
2901:
3314:
2131:
2001:{\displaystyle -{\frac {\hbar ^{2}}{2m}}\int _{-\varepsilon }^{+\varepsilon }\psi ''(x)\,dx+\int _{-\varepsilon }^{+\varepsilon }V(x)\psi (x)\,dx=E\int _{-\varepsilon }^{+\varepsilon }\psi (x)\,dx.}
2889:{\displaystyle \psi (x)={\begin{cases}\psi _{\text{L}}(x)=A_{\text{l}}e^{\kappa x},&{\text{ if }}x\leq 0,\\\psi _{\text{R}}(x)=B_{\text{r}}e^{-\kappa x},&{\text{ if }}x\geq 0.\end{cases}}}
5446:
3197:
4083:
1389:
5426:
5265:
4507:
1167:
4144:
970:
2257:
3018:
4498:
4419:
5120:
4387:
5184:
5038:
4466:
2690:
is imaginary, and the wave functions which were oscillating for positive energies in the calculation above are now exponentially increasing or decreasing functions of
5360:
1203:
is positive. The delta has been defined to occur at the origin for simplicity; a shift in the delta function's argument does not change any of the following results.
917:, which is obtained if one maintains the product of the width of the well and the potential constant while decreasing the well's width and increasing the potential.
5304:
3980:
The calculation presented above may at first seem unrealistic and hardly useful. However, it has proved to be a suitable model for a variety of real-life systems.
1231:
1670:
5402:
4791:
372:
3987:
materials. In the bulk of the materials, the motion of the electrons is quasi-free and can be described by the kinetic term in the above
Hamiltonian with an
2136:
873:
2614:{\displaystyle {\begin{cases}A_{r}+A_{l}-B_{r}-B_{l}&=0,\\-A_{r}+A_{l}+B_{r}-B_{l}&={\frac {2m\lambda }{ik\hbar ^{2}}}(A_{r}+A_{l}).\end{cases}}}
580:
124:
3966:{\displaystyle T=|t|^{2}=1-R={\cfrac {1}{1+{\cfrac {m^{2}\lambda ^{2}}{\hbar ^{4}k^{2}}}}}={\cfrac {1}{1+{\cfrac {m\lambda ^{2}}{2\hbar ^{2}E}}}}.}
36:
3503:
of the model, the amplitudes for incidence from the right are the same as those from the left. The result is that there is a non-zero probability
2636:
The graph of the bound state wavefunction solution to the delta function potential is continuous everywhere, but its derivative is not defined at
2020:
1393:
In general, due to the presence of the delta potential in the origin, the coefficients of the solution need not be the same in both half-spaces:
3008:
must be negative, that is, the bound state only exists for the well, and not for the barrier. The
Fourier transform of this wave function is a
5675:
5000:{\displaystyle d_{\pm }(\lambda )={\frac {1}{2}}q(\lambda +1)\pm {\frac {1}{2}}\left\{q^{2}(1+\lambda )^{2}-4\lambda q^{2}\left\right\}^{1/2},}
353:
4766:{\displaystyle {\begin{vmatrix}q-d&qe^{-dR}\\q\lambda e^{-dR}&q\lambda -d\end{vmatrix}}=0,\quad {\text{where }}E=-{\frac {d^{2}}{2}}.}
3720:{\displaystyle R=|r|^{2}={\cfrac {1}{1+{\cfrac {\hbar ^{4}k^{2}}{m^{2}\lambda ^{2}}}}}={\cfrac {1}{1+{\cfrac {2\hbar ^{2}E}{m\lambda ^{2}}}}}}
536:
5630:
3988:
459:
4432:(shown in brown in the diagram). Keeping in mind the relationship of this model with its three-dimensional molecular counterpart, we use
1350:
5189:
119:
5152:(shown in green in the diagram). They represent an approximation of the two lowest discrete energy states of the three-dimensional
920:
This article, for simplicity, only considers a one-dimensional potential well, but analysis could be expanded to more dimensions.
5695:
5690:
2997:{\displaystyle {\begin{cases}A_{\text{l}}=B_{\text{r}}={\sqrt {\kappa }},\\\kappa =-{\frac {m\lambda }{\hbar ^{2}}},\end{cases}}}
208:
3490:{\displaystyle t={\cfrac {1}{1-{\cfrac {m\lambda }{i\hbar ^{2}k}}}},\quad r={\cfrac {1}{{\cfrac {i\hbar ^{2}k}{m\lambda }}-1}}.}
315:
295:
5369:. For these specific parameters, there are many interesting properties that occur, one of which is the unusual effect that the
5186:
and are useful in its analysis. Analytical solutions for the energy eigenvalues for the case of symmetric charges are given by
866:
163:
1122:
285:
4005:. The delta function potential barrier is the limiting case of the model considered there for very high and narrow barriers.
555:
4131:
The symmetric and anti-symmetric wavefunctions for the double-well Dirac delta function model with "internuclear" distance
4002:
3998:
3151:
595:
333:
233:
5465:
4012:
348:
1317:
531:
526:
497:
129:
5548:
5436:
310:
300:
5567:
5407:
5050:
910:
565:
482:
4141:
The double-well Dirac delta function models a diatomic hydrogen molecule by the corresponding Schrödinger equation:
3253:. To find the amplitudes for reflection and transmission for incidence from the left, we put in the above equations
5685:
5659:
5441:
4277:
4093:
859:
511:
3220:
The quantum case can be studied in the following situation: a particle incident on the barrier from the left side
430:
516:
477:
405:
328:
188:
5370:
1667:
One obtains a relation between the coefficients by imposing that the wavefunction be continuous at the origin:
600:
5122:
The "+" case corresponds to a wave function symmetric about the midpoint (shown in red in the diagram), where
5680:
3984:
936:
769:
487:
445:
395:
343:
109:
5431:
774:
492:
400:
320:
290:
253:
5392:
5134:
4471:
4116:
4009:
equation may then be reduced to the case considered here by an Ansatz for the wave function of the type
2694:(see above). Requiring that the wave functions do not diverge at infinity eliminates half of the terms:
1321:
914:
243:
228:
4392:
155:
5138:. Correspondingly, the "â" case is the wave function that is anti-symmetric about the midpoint, where
4088:
Alternatively, it is possible to generalize the delta function to exist on the surface of some domain
5589:
5540:
5508:
4108:
1181:
903:
899:
804:
560:
472:
198:
2910:
2738:
2415:
1420:
5310:
charges, and for that matter the three-dimensional molecular problem, the solutions are given by a
4619:{\displaystyle \psi (x)=Ae^{-d\left|x+{\frac {R}{2}}\right|}+Be^{-d\left|x-{\frac {R}{2}}\right|}.}
3009:
659:
467:
385:
213:
193:
145:
5639:
For the 3-dimensional case look for the "delta shell potential"; further see K. Gottfried (1966),
5559:
5524:
5498:
5489:
Lange, Rutger-Jan (2012), "Potential theory, path integrals and the
Laplacian of the indicator",
5416:
5397:
5319:
5272:
5155:
1325:
380:
305:
238:
150:
5010:
4439:
4267:{\displaystyle -{\frac {\hbar ^{2}}{2m}}{\frac {d^{2}\psi (x)}{dx^{2}}}+V(x)\psi (x)=E\psi (x),}
1093:{\displaystyle -{\frac {\hbar ^{2}}{2m}}{\frac {d^{2}\psi (x)}{dx^{2}}}+V(x)\psi (x)=E\psi (x),}
5626:
5605:
5387:
2394:{\displaystyle -{\frac {\hbar ^{2}}{2m}}ik(-A_{r}+A_{l}+B_{r}-B_{l})+\lambda (A_{r}+A_{l})=0.}
887:
814:
789:
729:
724:
624:
590:
570:
168:
27:
5332:
3111:{\displaystyle E=-{\frac {\hbar ^{2}\kappa ^{2}}{2m}}=-{\frac {m\lambda ^{2}}{2\hbar ^{2}}}.}
5597:
5516:
954:
819:
809:
799:
699:
679:
664:
634:
502:
390:
5282:
5421:
1228:). In each of these parts the potential is zero, and the Schrödinger equation reduces to
1105:
844:
714:
694:
440:
280:
5593:
5512:
4776:
895:
779:
739:
719:
689:
669:
619:
585:
435:
425:
218:
4626:
Matching of the wavefunction at the Dirac delta-function peaks yields the determinant
2632:
5669:
5412:
5382:
4421:
is the "internuclear" distance with Dirac delta-function (negative) peaks located at
4100:
940:
839:
834:
764:
734:
704:
575:
521:
248:
223:
5563:
5528:
5580:
van Dijk, W.; Kiers, K. A. (1992). "Time delay in simple one-dimensional systems".
4433:
829:
824:
759:
744:
709:
203:
4500:
is a formally adjustable parameter. From the single-well case, we can infer the "
2403:
The boundary conditions thus give the following restrictions on the coefficients
5520:
2648:
1341:
794:
749:
684:
639:
4127:
784:
754:
674:
649:
644:
629:
5609:
4115:
Dirac Delta function model which represents a one-dimensional version of the
2017:, the right-hand side of this equation vanishes; the left-hand side becomes
1309:{\displaystyle {\frac {d^{2}\psi }{dx^{2}}}=-{\frac {2mE}{\hbar ^{2}}}\psi ;}
101:{\displaystyle i\hbar {\frac {d}{dt}}|\Psi \rangle ={\hat {H}}|\Psi \rangle }
275:
2898:
From the boundary conditions and normalization conditions, it follows that
1788:{\displaystyle \psi (0)=\psi _{L}(0)=\psi _{R}(0)=A_{r}+A_{l}=B_{r}+B_{l},}
5654:
3203:
For positive energies, the particle is free to move in either half-space:
3129:
5564:"The Calculation of Exchange Forces: General Results and Specific Models"
3500:
654:
4099:
The delta function model is actually a one-dimensional version of the
3727:
for the particle to be reflected. This does not depend on the sign of
928:
4501:
2243:{\displaystyle \int _{-\varepsilon }^{+\varepsilon }\psi ''(x)\,dx=.}
1113:
5601:
5279:. Note that the lowest energy corresponds to the symmetric solution
5503:
4126:
3128:
2631:
927:
5588:(6). American Association of Physics Teachers (AAPT): 520â527.
5040:. For the case of equal charges (symmetric homonuclear case),
4111:
The delta function model becomes particularly useful with the
5447:
List of quantum-mechanical systems with analytical solutions
2647:
In any one-dimensional attractive potential there will be a
2990:
2882:
2607:
2126:{\displaystyle -{\frac {\hbar ^{2}}{2m}}+\lambda \psi (0),}
1634:
5466:"quantum mechanics - Wave function with a delta potential"
5427:
Particle in a one-dimensional lattice (periodic potential)
3199:. Dashed: classical result. Solid line: quantum mechanics.
5362:. Thus, one has a non-trivial bound state solution with
3929:
3907:
3893:
3881:
3840:
3811:
3797:
3785:
3689:
3664:
3650:
3638:
3597:
3568:
3554:
3542:
3457:
3432:
3424:
3412:
3368:
3353:
3339:
3327:
3217:. It may be scattered at the delta-function potential.
4638:
3932:
3910:
3896:
3884:
3843:
3814:
3800:
3788:
3692:
3667:
3653:
3641:
3600:
3571:
3557:
3545:
3460:
3435:
3427:
3415:
3371:
3356:
3342:
3330:
5335:
5285:
5192:
5158:
5053:
5013:
4803:
4779:
4632:
4510:
4474:
4442:
4395:
4280:
4147:
4015:
3740:
3509:
3317:
3154:
3021:
2904:
2717:
2409:
2260:
2139:
2023:
1823:
1673:
1399:
1353:
1234:
1125:
973:
39:
3983:
One such example regards the interfaces between two
3192:{\displaystyle {\frac {m\lambda ^{2}}{2\hbar ^{2}}}}
3141:) probability of a delta potential well. The energy
4078:{\displaystyle \Psi (x,y,z)=\psi (x)\phi (y,z)\,\!}
5354:
5298:
5259:
5178:
5114:
5032:
4999:
4785:
4765:
4618:
4492:
4460:
4413:
4381:
4266:
4077:
3965:
3719:
3489:
3191:
3110:
2996:
2888:
2613:
2393:
2242:
2125:
2000:
1787:
1640:
1383:
1308:
1161:
1092:
100:
4074:
16:Model of an energy potential in quantum mechanics
1384:{\displaystyle k={\frac {\sqrt {2mE}}{\hbar }}.}
5625:(2nd ed.). Prentice Hall. pp. 68â78.
5260:{\displaystyle d_{\pm }=q+W(\pm qRe^{-qR})/R,}
1658:represents a wave traveling to the right, and
1648:where, in the case of positive energies (real
5403:Particle in a spherically symmetric potential
1214:The potential splits the space in two parts (
867:
8:
3307:even though we do not have any equations in
95:
69:
5325:One of the most interesting cases is when
3286:(no incoming particle from the right) and
874:
860:
18:
5502:
5340:
5334:
5320:Lambert W function § Generalizations
5290:
5284:
5246:
5231:
5197:
5191:
5169:
5164:
5159:
5157:
5093:
5085:
5058:
5052:
5024:
5012:
4984:
4980:
4950:
4939:
4918:
4899:
4877:
4857:
4826:
4808:
4802:
4778:
4749:
4743:
4729:
4685:
4659:
4633:
4631:
4596:
4578:
4551:
4533:
4509:
4473:
4441:
4394:
4356:
4321:
4279:
4207:
4180:
4173:
4157:
4151:
4146:
4073:
4014:
3941:
3933:
3919:
3911:
3904:
3897:
3885:
3878:
3859:
3849:
3844:
3830:
3820:
3815:
3808:
3801:
3789:
3782:
3761:
3756:
3747:
3739:
3701:
3693:
3676:
3668:
3661:
3654:
3642:
3635:
3616:
3606:
3601:
3587:
3577:
3572:
3565:
3558:
3546:
3539:
3530:
3525:
3516:
3508:
3461:
3444:
3436:
3429:
3428:
3416:
3409:
3380:
3372:
3357:
3350:
3343:
3331:
3324:
3316:
3180:
3165:
3155:
3153:
3096:
3081:
3071:
3048:
3038:
3031:
3020:
2976:
2962:
2939:
2930:
2917:
2905:
2903:
2865:
2848:
2838:
2816:
2791:
2777:
2767:
2745:
2733:
2716:
2592:
2579:
2563:
2539:
2528:
2515:
2502:
2489:
2461:
2448:
2435:
2422:
2410:
2408:
2376:
2363:
2341:
2328:
2315:
2302:
2270:
2264:
2259:
2178:
2152:
2144:
2138:
2081:
2056:
2033:
2027:
2022:
1988:
1967:
1959:
1942:
1909:
1901:
1887:
1861:
1853:
1833:
1827:
1822:
1776:
1763:
1750:
1737:
1715:
1693:
1672:
1614:
1594:
1584:
1565:
1555:
1533:
1508:
1488:
1478:
1459:
1449:
1427:
1415:
1398:
1360:
1352:
1292:
1275:
1260:
1242:
1235:
1233:
1124:
1033:
1006:
999:
983:
977:
972:
87:
76:
75:
61:
46:
38:
5641:Quantum Mechanics Volume I: Fundamentals
1162:{\displaystyle V(x)=\lambda \delta (x),}
5545:Dimensional Scaling in Chemical Physics
5543:, J.S. Avery, and O. Goscinski (eds.),
5457:
4443:
4154:
3938:
3846:
3673:
3574:
3441:
3377:
3177:
3093:
3035:
2973:
2560:
2267:
2030:
1830:
1373:
1289:
980:
43:
26:
5047:, and the pseudo-quadratic reduces to
3015:The energy of the bound state is then
4119:, as shown in the following section.
1119:The delta potential is the potential
7:
3734:The probability for transmission is
2651:. To find its energy, note that for
953:of a particle in one dimension in a
5558:T. C. Scott, J. F. Babb,
4493:{\displaystyle 0<\lambda <1}
4408:
4016:
406:Sum-over-histories (path integral)
92:
66:
22:Part of a series of articles about
14:
5623:Introduction to Quantum Mechanics
5329: †1, which results in
4414:{\displaystyle 0<R<\infty }
4107:method developed by the group of
5653:
5115:{\displaystyle d_{\pm }=q\left.}
1207:Solving the Schrödinger equation
898:mathematically described by the
4793:is found to be governed by the
4728:
3402:
2250:Substituting the definition of
5491:Journal of High Energy Physics
5243:
5215:
4962:
4956:
4896:
4883:
4851:
4839:
4820:
4814:
4520:
4514:
4290:
4284:
4258:
4252:
4240:
4234:
4228:
4222:
4195:
4189:
4070:
4058:
4052:
4046:
4037:
4019:
3757:
3748:
3526:
3517:
3299:(transmission), and solve for
2828:
2822:
2757:
2751:
2727:
2721:
2598:
2572:
2382:
2356:
2347:
2292:
2234:
2231:
2222:
2208:
2199:
2188:
2175:
2169:
2117:
2111:
2099:
2096:
2090:
2071:
2065:
2049:
1985:
1979:
1939:
1933:
1927:
1921:
1884:
1878:
1727:
1721:
1705:
1699:
1683:
1677:
1545:
1539:
1439:
1433:
1409:
1403:
1153:
1147:
1135:
1129:
1084:
1078:
1066:
1060:
1054:
1048:
1021:
1015:
909:The delta potential well is a
556:Relativistic quantum mechanics
88:
81:
62:
1:
5676:Quantum mechanical potentials
5643:, ch. III, sec. 15.
4382:{\displaystyle V(x)=-q\left,}
4003:Finite potential barrier (QM)
3999:scanning tunneling microscope
1347:is related to the energy by
596:Quantum statistical mechanics
5621:Griffiths, David J. (2005).
5570:, 99, pp. 2841â2854, (1993).
5179:{\displaystyle {\ce {H2^+}}}
2711:. The wave function is then
2254:into this expression yields
1318:linear differential equation
5582:American Journal of Physics
5408:Quantum harmonic oscillator
4274:where the potential is now
3004:from which it follows that
1664:one traveling to the left.
566:Quantum information science
5712:
5442:Laplacian of the indicator
5033:{\displaystyle d=d_{\pm }}
4461:{\displaystyle \hbar =m=1}
4094:Laplacian of the indicator
5373:is unity at zero energy.
4504:" for the solution to be
5562:and John D. Morgan III,
5371:transmission coefficient
5007:which has two solutions
601:Quantum machine learning
354:Wheeler's delayed-choice
5696:Exactly solvable models
5521:10.1007/JHEP11(2012)032
5437:HolsteinâHerring method
5355:{\displaystyle d_{-}=0}
3976:Remarks and application
1197:delta potential barrier
311:LeggettâGarg inequality
5470:Physics Stack Exchange
5432:Hydrogen molecular ion
5356:
5300:
5261:
5180:
5116:
5034:
5001:
4787:
4767:
4620:
4494:
4462:
4415:
4383:
4268:
4138:
4123:Double delta potential
4079:
3967:
3721:
3491:
3231:. It may be reflected
3200:
3193:
3112:
2998:
2890:
2644:
2615:
2395:
2244:
2127:
2002:
1789:
1642:
1385:
1324:, whose solutions are
1310:
1163:
1094:
932:
924:Single delta potential
102:
5393:Finite potential well
5357:
5301:
5299:{\displaystyle d_{+}}
5262:
5181:
5117:
5035:
5002:
4788:
4768:
4621:
4495:
4463:
4416:
4384:
4269:
4130:
4117:Hydrogen molecule ion
4080:
3968:
3722:
3492:
3263:(incoming particle),
3194:
3132:
3113:
2999:
2891:
2635:
2616:
2396:
2245:
2128:
2003:
1790:
1643:
1386:
1322:constant coefficients
1311:
1164:
1095:
935:The time-independent
931:
915:finite potential well
296:Elitzur–Vaidman
286:Davisson–Germer
103:
5691:Schrödinger equation
5662:at Wikimedia Commons
5547:, Springer, (1992).
5333:
5283:
5190:
5156:
5051:
5011:
4801:
4777:
4630:
4508:
4472:
4440:
4393:
4278:
4145:
4109:Dudley R. Herschbach
4013:
3738:
3507:
3315:
3152:
3019:
2902:
2715:
2407:
2258:
2137:
2021:
1821:
1671:
1397:
1351:
1232:
1189:delta potential well
1182:Dirac delta function
1123:
971:
937:Schrödinger equation
904:generalized function
900:Dirac delta function
561:Quantum field theory
473:Consistent histories
110:Schrödinger equation
37:
5594:1992AmJPh..60..520V
5513:2012JHEP...11..032L
5174:
4105:dimensional scaling
3997:The operation of a
3931:
3909:
3895:
3883:
3842:
3813:
3799:
3787:
3691:
3666:
3652:
3640:
3599:
3570:
3556:
3544:
3459:
3434:
3426:
3414:
3370:
3355:
3341:
3329:
3010:Lorentzian function
2160:
2089:
2064:
1975:
1917:
1869:
1805:, over an interval
1326:linear combinations
1195:is negative, and a
349:Stern–Gerlach
146:Classical mechanics
5417:hydrogen-like atom
5398:Particle in a ring
5352:
5296:
5257:
5176:
5160:
5112:
5030:
4997:
4783:
4763:
4713:
4616:
4490:
4458:
4411:
4379:
4264:
4139:
4075:
3963:
3956:
3951:
3926:
3890:
3871:
3866:
3837:
3794:
3717:
3713:
3708:
3686:
3647:
3628:
3623:
3594:
3551:
3499:Due to the mirror
3487:
3480:
3469:
3454:
3421:
3395:
3390:
3365:
3336:
3201:
3189:
3137:) and reflection (
3108:
2994:
2989:
2886:
2881:
2645:
2611:
2606:
2391:
2240:
2140:
2123:
2077:
2052:
1998:
1955:
1897:
1849:
1785:
1638:
1633:
1381:
1306:
1159:
1090:
933:
537:Von NeumannâWigner
517:Objective-collapse
316:Mach–Zehnder
306:Leggett inequality
301:Franck–Hertz
151:Old quantum theory
98:
5686:Scattering theory
5658:Media related to
5632:978-0-13-111892-8
5388:Particle in a box
5306:. In the case of
5163:
4865:
4834:
4786:{\displaystyle d}
4758:
4732:
4604:
4559:
4364:
4329:
4214:
4171:
4103:according to the
3958:
3953:
3930:
3908:
3894:
3882:
3873:
3868:
3841:
3812:
3798:
3786:
3715:
3710:
3690:
3665:
3651:
3639:
3630:
3625:
3598:
3569:
3555:
3543:
3482:
3471:
3458:
3433:
3425:
3413:
3397:
3392:
3369:
3354:
3340:
3328:
3311:. The result is
3187:
3103:
3063:
2982:
2944:
2933:
2920:
2868:
2841:
2819:
2794:
2770:
2748:
2570:
2284:
2047:
1847:
1617:
1587:
1558:
1536:
1511:
1481:
1452:
1430:
1376:
1372:
1298:
1267:
1116:of the particle.
1040:
997:
888:quantum mechanics
884:
883:
591:Scattering theory
571:Quantum computing
344:Schrödinger's cat
276:Bell's inequality
84:
59:
28:Quantum mechanics
5703:
5657:
5636:
5614:
5613:
5577:
5571:
5556:
5550:
5538:
5532:
5531:
5506:
5486:
5480:
5479:
5477:
5476:
5462:
5368:
5361:
5359:
5358:
5353:
5345:
5344:
5305:
5303:
5302:
5297:
5295:
5294:
5271:is the standard
5266:
5264:
5263:
5258:
5250:
5242:
5241:
5202:
5201:
5185:
5183:
5182:
5177:
5175:
5173:
5168:
5161:
5148:, and is called
5147:
5132:, and is called
5131:
5121:
5119:
5118:
5113:
5108:
5104:
5103:
5102:
5098:
5097:
5063:
5062:
5046:
5039:
5037:
5036:
5031:
5029:
5028:
5006:
5004:
5003:
4998:
4993:
4992:
4988:
4979:
4975:
4974:
4970:
4969:
4968:
4955:
4954:
4923:
4922:
4904:
4903:
4882:
4881:
4866:
4858:
4835:
4827:
4813:
4812:
4795:pseudo-quadratic
4792:
4790:
4789:
4784:
4772:
4770:
4769:
4764:
4759:
4754:
4753:
4744:
4733:
4730:
4718:
4717:
4696:
4695:
4670:
4669:
4625:
4623:
4622:
4617:
4612:
4611:
4610:
4606:
4605:
4597:
4567:
4566:
4565:
4561:
4560:
4552:
4499:
4497:
4496:
4491:
4467:
4465:
4464:
4459:
4431:
4420:
4418:
4417:
4412:
4388:
4386:
4385:
4380:
4375:
4371:
4370:
4366:
4365:
4357:
4335:
4331:
4330:
4322:
4273:
4271:
4270:
4265:
4215:
4213:
4212:
4211:
4198:
4185:
4184:
4174:
4172:
4170:
4162:
4161:
4152:
4137:
4084:
4082:
4081:
4076:
3993:
3972:
3970:
3969:
3964:
3959:
3957:
3955:
3954:
3952:
3950:
3946:
3945:
3927:
3925:
3924:
3923:
3905:
3891:
3889:
3879:
3874:
3872:
3870:
3869:
3867:
3865:
3864:
3863:
3854:
3853:
3838:
3836:
3835:
3834:
3825:
3824:
3809:
3795:
3793:
3783:
3766:
3765:
3760:
3751:
3730:
3726:
3724:
3723:
3718:
3716:
3714:
3712:
3711:
3709:
3707:
3706:
3705:
3687:
3685:
3681:
3680:
3662:
3648:
3646:
3636:
3631:
3629:
3627:
3626:
3624:
3622:
3621:
3620:
3611:
3610:
3595:
3593:
3592:
3591:
3582:
3581:
3566:
3552:
3550:
3540:
3535:
3534:
3529:
3520:
3496:
3494:
3493:
3488:
3483:
3481:
3479:
3472:
3470:
3468:
3455:
3453:
3449:
3448:
3430:
3422:
3420:
3410:
3398:
3396:
3394:
3393:
3391:
3389:
3385:
3384:
3366:
3364:
3351:
3337:
3335:
3325:
3310:
3306:
3302:
3298:
3285:
3275:
3262:
3252:
3241:
3230:
3216:
3209:
3198:
3196:
3195:
3190:
3188:
3186:
3185:
3184:
3171:
3170:
3169:
3156:
3147:
3117:
3115:
3114:
3109:
3104:
3102:
3101:
3100:
3087:
3086:
3085:
3072:
3064:
3062:
3054:
3053:
3052:
3043:
3042:
3032:
3007:
3003:
3001:
3000:
2995:
2993:
2992:
2983:
2981:
2980:
2971:
2963:
2945:
2940:
2935:
2934:
2931:
2922:
2921:
2918:
2895:
2893:
2892:
2887:
2885:
2884:
2869:
2866:
2859:
2858:
2843:
2842:
2839:
2821:
2820:
2817:
2795:
2792:
2785:
2784:
2772:
2771:
2768:
2750:
2749:
2746:
2710:
2689:
2680:
2679:
2657:
2642:
2620:
2618:
2617:
2612:
2610:
2609:
2597:
2596:
2584:
2583:
2571:
2569:
2568:
2567:
2551:
2540:
2533:
2532:
2520:
2519:
2507:
2506:
2494:
2493:
2466:
2465:
2453:
2452:
2440:
2439:
2427:
2426:
2400:
2398:
2397:
2392:
2381:
2380:
2368:
2367:
2346:
2345:
2333:
2332:
2320:
2319:
2307:
2306:
2285:
2283:
2275:
2274:
2265:
2253:
2249:
2247:
2246:
2241:
2221:
2198:
2168:
2159:
2151:
2132:
2130:
2129:
2124:
2085:
2060:
2048:
2046:
2038:
2037:
2028:
2016:
2010:In the limit as
2007:
2005:
2004:
1999:
1974:
1966:
1916:
1908:
1877:
1868:
1860:
1848:
1846:
1838:
1837:
1828:
1816:
1804:
1794:
1792:
1791:
1786:
1781:
1780:
1768:
1767:
1755:
1754:
1742:
1741:
1720:
1719:
1698:
1697:
1663:
1657:
1651:
1647:
1645:
1644:
1639:
1637:
1636:
1618:
1615:
1608:
1607:
1589:
1588:
1585:
1576:
1575:
1560:
1559:
1556:
1538:
1537:
1534:
1512:
1509:
1502:
1501:
1483:
1482:
1479:
1470:
1469:
1454:
1453:
1450:
1432:
1431:
1428:
1390:
1388:
1387:
1382:
1377:
1362:
1361:
1346:
1339:
1333:
1315:
1313:
1312:
1307:
1299:
1297:
1296:
1287:
1276:
1268:
1266:
1265:
1264:
1251:
1247:
1246:
1236:
1227:
1220:
1202:
1194:
1179:
1168:
1166:
1165:
1160:
1111:
1103:
1099:
1097:
1096:
1091:
1041:
1039:
1038:
1037:
1024:
1011:
1010:
1000:
998:
996:
988:
987:
978:
966:
952:
876:
869:
862:
503:Superdeterminism
156:Braâket notation
107:
105:
104:
99:
91:
86:
85:
77:
65:
60:
58:
47:
19:
5711:
5710:
5706:
5705:
5704:
5702:
5701:
5700:
5666:
5665:
5660:Delta potential
5650:
5633:
5620:
5617:
5602:10.1119/1.16866
5579:
5578:
5574:
5557:
5553:
5541:D.R. Herschbach
5539:
5535:
5488:
5487:
5483:
5474:
5472:
5464:
5463:
5459:
5455:
5422:Ring wave guide
5379:
5363:
5336:
5331:
5330:
5314:of the Lambert
5286:
5281:
5280:
5227:
5193:
5188:
5187:
5154:
5153:
5139:
5123:
5089:
5081:
5074:
5070:
5054:
5049:
5048:
5041:
5020:
5009:
5008:
4946:
4935:
4928:
4924:
4914:
4895:
4873:
4872:
4868:
4867:
4804:
4799:
4798:
4775:
4774:
4745:
4712:
4711:
4697:
4681:
4672:
4671:
4655:
4650:
4634:
4628:
4627:
4589:
4585:
4574:
4544:
4540:
4529:
4506:
4505:
4470:
4469:
4438:
4437:
4422:
4391:
4390:
4349:
4345:
4314:
4310:
4306:
4302:
4276:
4275:
4203:
4199:
4176:
4175:
4163:
4153:
4143:
4142:
4132:
4125:
4011:
4010:
3991:
3978:
3937:
3928:
3915:
3906:
3892:
3880:
3855:
3845:
3839:
3826:
3816:
3810:
3796:
3784:
3755:
3736:
3735:
3728:
3697:
3688:
3672:
3663:
3649:
3637:
3612:
3602:
3596:
3583:
3573:
3567:
3553:
3541:
3524:
3505:
3504:
3456:
3440:
3431:
3423:
3411:
3376:
3367:
3352:
3338:
3326:
3313:
3312:
3308:
3304:
3300:
3293:
3287:
3283:
3277:
3270:
3264:
3260:
3254:
3250:
3243:
3242:or transmitted
3239:
3232:
3228:
3221:
3211:
3204:
3176:
3172:
3161:
3157:
3150:
3149:
3148:is in units of
3142:
3127:
3092:
3088:
3077:
3073:
3055:
3044:
3034:
3033:
3017:
3016:
3005:
2988:
2987:
2972:
2964:
2950:
2949:
2926:
2913:
2906:
2900:
2899:
2880:
2879:
2863:
2844:
2834:
2812:
2809:
2808:
2789:
2773:
2763:
2741:
2734:
2713:
2712:
2708:
2701:
2695:
2669:
2667:
2659:
2652:
2637:
2630:
2605:
2604:
2588:
2575:
2559:
2552:
2541:
2534:
2524:
2511:
2498:
2485:
2479:
2478:
2467:
2457:
2444:
2431:
2418:
2411:
2405:
2404:
2372:
2359:
2337:
2324:
2311:
2298:
2276:
2266:
2256:
2255:
2251:
2214:
2191:
2161:
2135:
2134:
2039:
2029:
2019:
2018:
2011:
1870:
1839:
1829:
1819:
1818:
1806:
1799:
1772:
1759:
1746:
1733:
1711:
1689:
1669:
1668:
1659:
1653:
1649:
1632:
1631:
1612:
1590:
1580:
1561:
1551:
1529:
1526:
1525:
1506:
1484:
1474:
1455:
1445:
1423:
1416:
1395:
1394:
1349:
1348:
1344:
1335:
1329:
1288:
1277:
1256:
1252:
1238:
1237:
1230:
1229:
1222:
1215:
1209:
1200:
1192:
1187:It is called a
1170:
1121:
1120:
1109:
1106:Planck constant
1104:is the reduced
1101:
1029:
1025:
1002:
1001:
989:
979:
969:
968:
957:
943:
926:
892:delta potential
880:
851:
850:
849:
614:
606:
605:
551:
550:Advanced topics
543:
542:
541:
493:Hidden-variable
483:de BroglieâBohm
462:
460:Interpretations
452:
451:
450:
420:
412:
411:
410:
368:
360:
359:
358:
325:
281:CHSH inequality
270:
262:
261:
260:
189:Complementarity
183:
175:
174:
173:
141:
112:
51:
35:
34:
17:
12:
11:
5:
5709:
5707:
5699:
5698:
5693:
5688:
5683:
5681:Quantum models
5678:
5668:
5667:
5664:
5663:
5649:
5648:External links
5646:
5645:
5644:
5637:
5631:
5616:
5615:
5572:
5568:J. Chem. Phys.
5551:
5533:
5481:
5456:
5454:
5451:
5450:
5449:
5444:
5439:
5434:
5429:
5424:
5419:
5410:
5405:
5400:
5395:
5390:
5385:
5378:
5375:
5351:
5348:
5343:
5339:
5318:function (see
5312:generalization
5293:
5289:
5256:
5253:
5249:
5245:
5240:
5237:
5234:
5230:
5226:
5223:
5220:
5217:
5214:
5211:
5208:
5205:
5200:
5196:
5172:
5167:
5111:
5107:
5101:
5096:
5092:
5088:
5084:
5080:
5077:
5073:
5069:
5066:
5061:
5057:
5027:
5023:
5019:
5016:
4996:
4991:
4987:
4983:
4978:
4973:
4967:
4964:
4961:
4958:
4953:
4949:
4945:
4942:
4938:
4934:
4931:
4927:
4921:
4917:
4913:
4910:
4907:
4902:
4898:
4894:
4891:
4888:
4885:
4880:
4876:
4871:
4864:
4861:
4856:
4853:
4850:
4847:
4844:
4841:
4838:
4833:
4830:
4825:
4822:
4819:
4816:
4811:
4807:
4782:
4762:
4757:
4752:
4748:
4742:
4739:
4736:
4727:
4724:
4721:
4716:
4710:
4707:
4704:
4701:
4698:
4694:
4691:
4688:
4684:
4680:
4677:
4674:
4673:
4668:
4665:
4662:
4658:
4654:
4651:
4649:
4646:
4643:
4640:
4639:
4637:
4615:
4609:
4603:
4600:
4595:
4592:
4588:
4584:
4581:
4577:
4573:
4570:
4564:
4558:
4555:
4550:
4547:
4543:
4539:
4536:
4532:
4528:
4525:
4522:
4519:
4516:
4513:
4489:
4486:
4483:
4480:
4477:
4457:
4454:
4451:
4448:
4445:
4410:
4407:
4404:
4401:
4398:
4378:
4374:
4369:
4363:
4360:
4355:
4352:
4348:
4344:
4341:
4338:
4334:
4328:
4325:
4320:
4317:
4313:
4309:
4305:
4301:
4298:
4295:
4292:
4289:
4286:
4283:
4263:
4260:
4257:
4254:
4251:
4248:
4245:
4242:
4239:
4236:
4233:
4230:
4227:
4224:
4221:
4218:
4210:
4206:
4202:
4197:
4194:
4191:
4188:
4183:
4179:
4169:
4166:
4160:
4156:
4150:
4124:
4121:
4072:
4069:
4066:
4063:
4060:
4057:
4054:
4051:
4048:
4045:
4042:
4039:
4036:
4033:
4030:
4027:
4024:
4021:
4018:
3989:effective mass
3977:
3974:
3962:
3949:
3944:
3940:
3936:
3922:
3918:
3914:
3903:
3900:
3888:
3877:
3862:
3858:
3852:
3848:
3833:
3829:
3823:
3819:
3807:
3804:
3792:
3781:
3778:
3775:
3772:
3769:
3764:
3759:
3754:
3750:
3746:
3743:
3704:
3700:
3696:
3684:
3679:
3675:
3671:
3660:
3657:
3645:
3634:
3619:
3615:
3609:
3605:
3590:
3586:
3580:
3576:
3564:
3561:
3549:
3538:
3533:
3528:
3523:
3519:
3515:
3512:
3486:
3478:
3475:
3467:
3464:
3452:
3447:
3443:
3439:
3419:
3408:
3405:
3401:
3388:
3383:
3379:
3375:
3363:
3360:
3349:
3346:
3334:
3323:
3320:
3291:
3281:
3276:(reflection),
3268:
3258:
3248:
3237:
3226:
3183:
3179:
3175:
3168:
3164:
3160:
3133:Transmission (
3126:
3119:
3107:
3099:
3095:
3091:
3084:
3080:
3076:
3070:
3067:
3061:
3058:
3051:
3047:
3041:
3037:
3030:
3027:
3024:
2991:
2986:
2979:
2975:
2970:
2967:
2961:
2958:
2955:
2952:
2951:
2948:
2943:
2938:
2929:
2925:
2916:
2912:
2911:
2909:
2883:
2878:
2875:
2872:
2867: if
2864:
2862:
2857:
2854:
2851:
2847:
2837:
2833:
2830:
2827:
2824:
2815:
2811:
2810:
2807:
2804:
2801:
2798:
2793: if
2790:
2788:
2783:
2780:
2776:
2766:
2762:
2759:
2756:
2753:
2744:
2740:
2739:
2737:
2732:
2729:
2726:
2723:
2720:
2706:
2699:
2629:
2622:
2608:
2603:
2600:
2595:
2591:
2587:
2582:
2578:
2574:
2566:
2562:
2558:
2555:
2550:
2547:
2544:
2538:
2535:
2531:
2527:
2523:
2518:
2514:
2510:
2505:
2501:
2497:
2492:
2488:
2484:
2481:
2480:
2477:
2474:
2471:
2468:
2464:
2460:
2456:
2451:
2447:
2443:
2438:
2434:
2430:
2425:
2421:
2417:
2416:
2414:
2390:
2387:
2384:
2379:
2375:
2371:
2366:
2362:
2358:
2355:
2352:
2349:
2344:
2340:
2336:
2331:
2327:
2323:
2318:
2314:
2310:
2305:
2301:
2297:
2294:
2291:
2288:
2282:
2279:
2273:
2269:
2263:
2239:
2236:
2233:
2230:
2227:
2224:
2220:
2217:
2213:
2210:
2207:
2204:
2201:
2197:
2194:
2190:
2187:
2184:
2181:
2177:
2174:
2171:
2167:
2164:
2158:
2155:
2150:
2147:
2143:
2122:
2119:
2116:
2113:
2110:
2107:
2104:
2101:
2098:
2095:
2092:
2088:
2084:
2080:
2076:
2073:
2070:
2067:
2063:
2059:
2055:
2051:
2045:
2042:
2036:
2032:
2026:
1997:
1994:
1991:
1987:
1984:
1981:
1978:
1973:
1970:
1965:
1962:
1958:
1954:
1951:
1948:
1945:
1941:
1938:
1935:
1932:
1929:
1926:
1923:
1920:
1915:
1912:
1907:
1904:
1900:
1896:
1893:
1890:
1886:
1883:
1880:
1876:
1873:
1867:
1864:
1859:
1856:
1852:
1845:
1842:
1836:
1832:
1826:
1784:
1779:
1775:
1771:
1766:
1762:
1758:
1753:
1749:
1745:
1740:
1736:
1732:
1729:
1726:
1723:
1718:
1714:
1710:
1707:
1704:
1701:
1696:
1692:
1688:
1685:
1682:
1679:
1676:
1635:
1630:
1627:
1624:
1621:
1616: if
1613:
1611:
1606:
1603:
1600:
1597:
1593:
1583:
1579:
1574:
1571:
1568:
1564:
1554:
1550:
1547:
1544:
1541:
1532:
1528:
1527:
1524:
1521:
1518:
1515:
1510: if
1507:
1505:
1500:
1497:
1494:
1491:
1487:
1477:
1473:
1468:
1465:
1462:
1458:
1448:
1444:
1441:
1438:
1435:
1426:
1422:
1421:
1419:
1414:
1411:
1408:
1405:
1402:
1380:
1375:
1371:
1368:
1365:
1359:
1356:
1305:
1302:
1295:
1291:
1286:
1283:
1280:
1274:
1271:
1263:
1259:
1255:
1250:
1245:
1241:
1208:
1205:
1158:
1155:
1152:
1149:
1146:
1143:
1140:
1137:
1134:
1131:
1128:
1089:
1086:
1083:
1080:
1077:
1074:
1071:
1068:
1065:
1062:
1059:
1056:
1053:
1050:
1047:
1044:
1036:
1032:
1028:
1023:
1020:
1017:
1014:
1009:
1005:
995:
992:
986:
982:
976:
925:
922:
896:potential well
882:
881:
879:
878:
871:
864:
856:
853:
852:
848:
847:
842:
837:
832:
827:
822:
817:
812:
807:
802:
797:
792:
787:
782:
777:
772:
767:
762:
757:
752:
747:
742:
737:
732:
727:
722:
717:
712:
707:
702:
697:
692:
687:
682:
677:
672:
667:
662:
657:
652:
647:
642:
637:
632:
627:
622:
616:
615:
612:
611:
608:
607:
604:
603:
598:
593:
588:
586:Density matrix
583:
578:
573:
568:
563:
558:
552:
549:
548:
545:
544:
540:
539:
534:
529:
524:
519:
514:
509:
508:
507:
506:
505:
490:
485:
480:
475:
470:
464:
463:
458:
457:
454:
453:
449:
448:
443:
438:
433:
428:
422:
421:
418:
417:
414:
413:
409:
408:
403:
398:
393:
388:
383:
377:
376:
375:
369:
366:
365:
362:
361:
357:
356:
351:
346:
340:
339:
338:
337:
336:
334:Delayed-choice
329:Quantum eraser
324:
323:
318:
313:
308:
303:
298:
293:
288:
283:
278:
272:
271:
268:
267:
264:
263:
259:
258:
257:
256:
246:
241:
236:
231:
226:
221:
219:Quantum number
216:
211:
206:
201:
196:
191:
185:
184:
181:
180:
177:
176:
172:
171:
166:
160:
159:
158:
153:
148:
142:
139:
138:
135:
134:
133:
132:
127:
122:
114:
113:
108:
97:
94:
90:
83:
80:
74:
71:
68:
64:
57:
54:
50:
45:
42:
31:
30:
24:
23:
15:
13:
10:
9:
6:
4:
3:
2:
5708:
5697:
5694:
5692:
5689:
5687:
5684:
5682:
5679:
5677:
5674:
5673:
5671:
5661:
5656:
5652:
5651:
5647:
5642:
5638:
5634:
5628:
5624:
5619:
5618:
5611:
5607:
5603:
5599:
5595:
5591:
5587:
5583:
5576:
5573:
5569:
5565:
5561:
5555:
5552:
5549:
5546:
5542:
5537:
5534:
5530:
5526:
5522:
5518:
5514:
5510:
5505:
5500:
5496:
5492:
5485:
5482:
5471:
5467:
5461:
5458:
5452:
5448:
5445:
5443:
5440:
5438:
5435:
5433:
5430:
5428:
5425:
5423:
5420:
5418:
5414:
5413:Hydrogen atom
5411:
5409:
5406:
5404:
5401:
5399:
5396:
5394:
5391:
5389:
5386:
5384:
5383:Free particle
5381:
5380:
5376:
5374:
5372:
5366:
5349:
5346:
5341:
5337:
5328:
5323:
5321:
5317:
5313:
5309:
5291:
5287:
5278:
5276:
5270:
5254:
5251:
5247:
5238:
5235:
5232:
5228:
5224:
5221:
5218:
5212:
5209:
5206:
5203:
5198:
5194:
5170:
5165:
5151:
5146:
5142:
5137:
5136:
5130:
5126:
5109:
5105:
5099:
5094:
5090:
5086:
5082:
5078:
5075:
5071:
5067:
5064:
5059:
5055:
5044:
5025:
5021:
5017:
5014:
4994:
4989:
4985:
4981:
4976:
4971:
4965:
4959:
4951:
4947:
4943:
4940:
4936:
4932:
4929:
4925:
4919:
4915:
4911:
4908:
4905:
4900:
4892:
4889:
4886:
4878:
4874:
4869:
4862:
4859:
4854:
4848:
4845:
4842:
4836:
4831:
4828:
4823:
4817:
4809:
4805:
4796:
4780:
4760:
4755:
4750:
4746:
4740:
4737:
4734:
4725:
4722:
4719:
4714:
4708:
4705:
4702:
4699:
4692:
4689:
4686:
4682:
4678:
4675:
4666:
4663:
4660:
4656:
4652:
4647:
4644:
4641:
4635:
4613:
4607:
4601:
4598:
4593:
4590:
4586:
4582:
4579:
4575:
4571:
4568:
4562:
4556:
4553:
4548:
4545:
4541:
4537:
4534:
4530:
4526:
4523:
4517:
4511:
4503:
4487:
4484:
4481:
4478:
4475:
4455:
4452:
4449:
4446:
4435:
4429:
4425:
4405:
4402:
4399:
4396:
4376:
4372:
4367:
4361:
4358:
4353:
4350:
4346:
4342:
4339:
4336:
4332:
4326:
4323:
4318:
4315:
4311:
4307:
4303:
4299:
4296:
4293:
4287:
4281:
4261:
4255:
4249:
4246:
4243:
4237:
4231:
4225:
4219:
4216:
4208:
4204:
4200:
4192:
4186:
4181:
4177:
4167:
4164:
4158:
4148:
4135:
4129:
4122:
4120:
4118:
4114:
4110:
4106:
4102:
4101:Hydrogen atom
4097:
4095:
4091:
4086:
4067:
4064:
4061:
4055:
4049:
4043:
4040:
4034:
4031:
4028:
4025:
4022:
4006:
4004:
4000:
3995:
3990:
3986:
3981:
3975:
3973:
3960:
3947:
3942:
3934:
3920:
3916:
3912:
3901:
3898:
3886:
3875:
3860:
3856:
3850:
3831:
3827:
3821:
3817:
3805:
3802:
3790:
3779:
3776:
3773:
3770:
3767:
3762:
3752:
3744:
3741:
3732:
3702:
3698:
3694:
3682:
3677:
3669:
3658:
3655:
3643:
3632:
3617:
3613:
3607:
3603:
3588:
3584:
3578:
3562:
3559:
3547:
3536:
3531:
3521:
3513:
3510:
3502:
3497:
3484:
3476:
3473:
3465:
3462:
3450:
3445:
3437:
3417:
3406:
3403:
3399:
3386:
3381:
3373:
3361:
3358:
3347:
3344:
3332:
3321:
3318:
3297:
3290:
3280:
3274:
3267:
3257:
3247:
3236:
3225:
3218:
3214:
3207:
3181:
3173:
3166:
3162:
3158:
3145:
3140:
3136:
3131:
3124:
3120:
3118:
3105:
3097:
3089:
3082:
3078:
3074:
3068:
3065:
3059:
3056:
3049:
3045:
3039:
3028:
3025:
3022:
3013:
3011:
2984:
2977:
2968:
2965:
2959:
2956:
2953:
2946:
2941:
2936:
2927:
2923:
2914:
2907:
2896:
2876:
2873:
2870:
2860:
2855:
2852:
2849:
2845:
2835:
2831:
2825:
2813:
2805:
2802:
2799:
2796:
2786:
2781:
2778:
2774:
2764:
2760:
2754:
2742:
2735:
2730:
2724:
2718:
2705:
2698:
2693:
2688:
2684:
2677:
2673:
2666:
2662:
2655:
2650:
2640:
2634:
2627:
2624:Bound state (
2623:
2621:
2601:
2593:
2589:
2585:
2580:
2576:
2564:
2556:
2553:
2548:
2545:
2542:
2536:
2529:
2525:
2521:
2516:
2512:
2508:
2503:
2499:
2495:
2490:
2486:
2482:
2475:
2472:
2469:
2462:
2458:
2454:
2449:
2445:
2441:
2436:
2432:
2428:
2423:
2419:
2412:
2401:
2388:
2385:
2377:
2373:
2369:
2364:
2360:
2353:
2350:
2342:
2338:
2334:
2329:
2325:
2321:
2316:
2312:
2308:
2303:
2299:
2295:
2289:
2286:
2280:
2277:
2271:
2261:
2237:
2228:
2225:
2218:
2215:
2211:
2205:
2202:
2195:
2192:
2185:
2182:
2179:
2172:
2165:
2162:
2156:
2153:
2148:
2145:
2141:
2120:
2114:
2108:
2105:
2102:
2093:
2086:
2082:
2078:
2074:
2068:
2061:
2057:
2053:
2043:
2040:
2034:
2024:
2014:
2008:
1995:
1992:
1989:
1982:
1976:
1971:
1968:
1963:
1960:
1956:
1952:
1949:
1946:
1943:
1936:
1930:
1924:
1918:
1913:
1910:
1905:
1902:
1898:
1894:
1891:
1888:
1881:
1874:
1871:
1865:
1862:
1857:
1854:
1850:
1843:
1840:
1834:
1824:
1814:
1810:
1802:
1795:
1782:
1777:
1773:
1769:
1764:
1760:
1756:
1751:
1747:
1743:
1738:
1734:
1730:
1724:
1716:
1712:
1708:
1702:
1694:
1690:
1686:
1680:
1674:
1665:
1662:
1656:
1628:
1625:
1622:
1619:
1609:
1604:
1601:
1598:
1595:
1591:
1581:
1577:
1572:
1569:
1566:
1562:
1552:
1548:
1542:
1530:
1522:
1519:
1516:
1513:
1503:
1498:
1495:
1492:
1489:
1485:
1475:
1471:
1466:
1463:
1460:
1456:
1446:
1442:
1436:
1424:
1417:
1412:
1406:
1400:
1391:
1378:
1369:
1366:
1363:
1357:
1354:
1343:
1338:
1332:
1327:
1323:
1319:
1303:
1300:
1293:
1284:
1281:
1278:
1272:
1269:
1261:
1257:
1253:
1248:
1243:
1239:
1225:
1218:
1212:
1206:
1204:
1198:
1190:
1185:
1183:
1177:
1173:
1156:
1150:
1144:
1141:
1138:
1132:
1126:
1117:
1115:
1107:
1087:
1081:
1075:
1072:
1069:
1063:
1057:
1051:
1045:
1042:
1034:
1030:
1026:
1018:
1012:
1007:
1003:
993:
990:
984:
974:
964:
960:
956:
950:
946:
942:
941:wave function
938:
930:
923:
921:
918:
916:
912:
911:limiting case
907:
905:
901:
897:
893:
889:
877:
872:
870:
865:
863:
858:
857:
855:
854:
846:
843:
841:
838:
836:
833:
831:
828:
826:
823:
821:
818:
816:
813:
811:
808:
806:
803:
801:
798:
796:
793:
791:
788:
786:
783:
781:
778:
776:
773:
771:
768:
766:
763:
761:
758:
756:
753:
751:
748:
746:
743:
741:
738:
736:
733:
731:
728:
726:
723:
721:
718:
716:
713:
711:
708:
706:
703:
701:
698:
696:
693:
691:
688:
686:
683:
681:
678:
676:
673:
671:
668:
666:
663:
661:
658:
656:
653:
651:
648:
646:
643:
641:
638:
636:
633:
631:
628:
626:
623:
621:
618:
617:
610:
609:
602:
599:
597:
594:
592:
589:
587:
584:
582:
579:
577:
576:Quantum chaos
574:
572:
569:
567:
564:
562:
559:
557:
554:
553:
547:
546:
538:
535:
533:
532:Transactional
530:
528:
525:
523:
522:Quantum logic
520:
518:
515:
513:
510:
504:
501:
500:
499:
496:
495:
494:
491:
489:
486:
484:
481:
479:
476:
474:
471:
469:
466:
465:
461:
456:
455:
447:
444:
442:
439:
437:
434:
432:
429:
427:
424:
423:
416:
415:
407:
404:
402:
399:
397:
394:
392:
389:
387:
384:
382:
379:
378:
374:
371:
370:
364:
363:
355:
352:
350:
347:
345:
342:
341:
335:
332:
331:
330:
327:
326:
322:
319:
317:
314:
312:
309:
307:
304:
302:
299:
297:
294:
292:
289:
287:
284:
282:
279:
277:
274:
273:
266:
265:
255:
252:
251:
250:
249:Wave function
247:
245:
242:
240:
237:
235:
232:
230:
229:Superposition
227:
225:
222:
220:
217:
215:
212:
210:
207:
205:
202:
200:
197:
195:
192:
190:
187:
186:
179:
178:
170:
167:
165:
162:
161:
157:
154:
152:
149:
147:
144:
143:
137:
136:
131:
128:
126:
123:
121:
118:
117:
116:
115:
111:
78:
72:
55:
52:
48:
40:
33:
32:
29:
25:
21:
20:
5640:
5622:
5585:
5581:
5575:
5554:
5544:
5536:
5497:(11): 1â49,
5494:
5490:
5484:
5473:. Retrieved
5469:
5460:
5364:
5326:
5324:
5315:
5311:
5307:
5274:
5268:
5149:
5144:
5140:
5133:
5128:
5124:
5042:
4794:
4434:atomic units
4427:
4423:
4140:
4133:
4112:
4104:
4098:
4089:
4087:
4007:
3996:
3982:
3979:
3733:
3498:
3295:
3288:
3278:
3272:
3265:
3255:
3245:
3234:
3223:
3219:
3212:
3205:
3202:
3143:
3138:
3134:
3122:
3121:Scattering (
3014:
2897:
2703:
2696:
2691:
2686:
2682:
2675:
2671:
2664:
2660:
2653:
2646:
2638:
2625:
2402:
2012:
2009:
1812:
1808:
1800:
1796:
1666:
1660:
1654:
1392:
1340:, where the
1336:
1330:
1223:
1216:
1213:
1210:
1196:
1188:
1186:
1175:
1171:
1118:
962:
958:
948:
944:
934:
919:
908:
891:
885:
431:KleinâGordon
367:Formulations
204:Energy level
199:Entanglement
182:Fundamentals
169:Interference
120:Introduction
5560:A. Dalgarno
4731:where
4113:double-well
2649:bound state
1342:wave number
820:von Neumann
805:Schrödinger
581:EPR paradox
512:Many-worlds
446:Schrödinger
401:Schrödinger
396:Phase-space
386:Interaction
291:Double-slit
269:Experiments
244:Uncertainty
214:Nonlocality
209:Measurement
194:Decoherence
164:Hamiltonian
5670:Categories
5475:2021-03-29
5453:References
3985:conducting
2133:because
1316:this is a
815:Sommerfeld
730:Heisenberg
725:Gutzwiller
665:de Broglie
613:Scientists
527:Relational
478:Copenhagen
381:Heisenberg
239:Tunnelling
140:Background
5610:0002-9505
5504:1302.0864
5342:−
5233:−
5219:±
5199:±
5095:±
5087:−
5079:±
5060:±
5026:±
4960:λ
4952:±
4941:−
4933:−
4912:λ
4906:−
4893:λ
4855:±
4843:λ
4818:λ
4810:±
4797:equation
4741:−
4706:−
4703:λ
4687:−
4679:λ
4661:−
4645:−
4594:−
4580:−
4535:−
4512:ψ
4482:λ
4444:ℏ
4409:∞
4354:−
4343:δ
4340:λ
4308:δ
4297:−
4250:ψ
4232:ψ
4187:ψ
4155:ℏ
4149:−
4056:ϕ
4044:ψ
4017:Ψ
3939:ℏ
3917:λ
3847:ℏ
3828:λ
3774:−
3699:λ
3674:ℏ
3614:λ
3575:ℏ
3474:−
3466:λ
3442:ℏ
3378:ℏ
3362:λ
3348:−
3178:ℏ
3163:λ
3094:ℏ
3079:λ
3069:−
3046:κ
3036:ℏ
3029:−
2974:ℏ
2969:λ
2960:−
2954:κ
2942:κ
2874:≥
2853:κ
2850:−
2814:ψ
2800:≤
2779:κ
2743:ψ
2719:ψ
2561:ℏ
2549:λ
2522:−
2483:−
2455:−
2442:−
2354:λ
2335:−
2296:−
2268:ℏ
2262:−
2229:ε
2226:−
2216:ψ
2212:−
2206:ε
2193:ψ
2163:ψ
2157:ε
2149:ε
2146:−
2142:∫
2109:ψ
2106:λ
2079:ψ
2075:−
2054:ψ
2031:ℏ
2025:−
1977:ψ
1972:ε
1964:ε
1961:−
1957:∫
1931:ψ
1914:ε
1906:ε
1903:−
1899:∫
1872:ψ
1866:ε
1858:ε
1855:−
1851:∫
1831:ℏ
1825:−
1713:ψ
1691:ψ
1675:ψ
1596:−
1531:ψ
1490:−
1425:ψ
1401:ψ
1374:ℏ
1301:ψ
1290:ℏ
1273:−
1249:ψ
1145:δ
1142:λ
1076:ψ
1058:ψ
1013:ψ
981:ℏ
975:−
955:potential
845:Zeilinger
690:Ehrenfest
419:Equations
96:⟩
93:Ψ
82:^
70:⟩
67:Ψ
44:ℏ
5529:56188533
5377:See also
5277:function
5273:Lambert
5150:ungerade
4436:and set
3501:symmetry
2219:′
2196:′
2166:″
2087:′
2062:′
1875:″
1211:Source:
939:for the
770:Millikan
695:Einstein
680:Davisson
635:Blackett
620:Aharonov
488:Ensemble
468:Bayesian
373:Overview
254:Collapse
234:Symmetry
125:Glossary
5590:Bibcode
5509:Bibcode
5308:unequal
4468:. Here
3125:> 0)
2668:√
2628:< 0)
1180:is the
1112:is the
913:of the
810:Simmons
800:Rydberg
765:Moseley
745:Kramers
735:Hilbert
720:Glauber
715:Feynman
700:Everett
670:Compton
441:Rydberg
130:History
5629:
5608:
5527:
5267:where
5135:gerade
4773:Thus,
4502:ansatz
4389:where
3215:> 0
3208:< 0
3146:> 0
2656:< 0
1807:[â
1226:> 0
1219:< 0
1169:where
1114:energy
1108:, and
1100:where
840:Zeeman
835:Wigner
785:Planck
755:Landau
740:Jordan
391:Matrix
321:Popper
5525:S2CID
5499:arXiv
4092:(see
1815:]
1320:with
894:is a
795:Raman
780:Pauli
775:Onnes
710:Fermi
685:Debye
675:Dirac
640:Bloch
630:Bethe
498:Local
436:Pauli
426:Dirac
224:State
5627:ISBN
5606:ISSN
5495:2012
4485:<
4479:<
4406:<
4400:<
3303:and
1623:>
1517:<
1334:and
1221:and
1184:.
902:- a
890:the
830:Wien
825:Weyl
790:Rabi
760:Laue
750:Lamb
705:Fock
660:Bose
655:Born
650:Bohr
645:Bohm
625:Bell
5598:doi
5517:doi
5415:or
5367:= 0
5322:).
5143:= â
5045:= 1
4426:= ±
4136:= 2
4096:).
4085:.
3284:= 0
3261:= 1
3210:or
2709:= 0
2641:= 0
2015:â 0
1811:, +
1803:= 0
1652:),
1328:of
1199:if
1191:if
967:is
886:In
5672::
5604:.
5596:.
5586:60
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5468:.
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5110:.
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4863:2
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4852:)
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4806:d
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4447:=
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4333:)
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4316:x
4312:(
4304:[
4300:q
4294:=
4291:)
4288:x
4285:(
4282:V
4262:,
4259:)
4256:x
4253:(
4247:E
4244:=
4241:)
4238:x
4235:(
4229:)
4226:x
4223:(
4220:V
4217:+
4209:2
4205:x
4201:d
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4190:(
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4068:z
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4062:y
4059:(
4053:)
4050:x
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4041:=
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4023:x
4020:(
3992:m
3961:.
3948:E
3943:2
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3876:=
3861:2
3857:k
3851:4
3832:2
3822:2
3818:m
3806:+
3803:1
3791:1
3780:=
3777:R
3771:1
3768:=
3763:2
3758:|
3753:t
3749:|
3745:=
3742:T
3729:λ
3703:2
3695:m
3683:E
3678:2
3670:2
3659:+
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3644:1
3633:=
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3608:2
3604:m
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3548:1
3537:=
3532:2
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3522:r
3518:|
3514:=
3511:R
3485:.
3477:1
3463:m
3451:k
3446:2
3438:i
3418:1
3407:=
3404:r
3400:,
3387:k
3382:2
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3359:m
3345:1
3333:1
3322:=
3319:t
3309:t
3305:t
3301:r
3296:t
3292:r
3289:B
3282:l
3279:B
3273:r
3269:l
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3251:)
3249:r
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3244:(
3240:)
3238:l
3235:A
3233:(
3229:)
3227:r
3224:A
3222:(
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3174:2
3167:2
3159:m
3144:E
3139:R
3135:T
3123:E
3106:.
3098:2
3090:2
3083:2
3075:m
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3060:m
3057:2
3050:2
3040:2
3026:=
3023:E
3006:λ
2985:,
2978:2
2966:m
2957:=
2947:,
2937:=
2932:r
2928:B
2924:=
2919:l
2915:A
2908:{
2871:x
2861:,
2856:x
2846:e
2840:r
2836:B
2832:=
2829:)
2826:x
2823:(
2818:R
2806:,
2803:0
2797:x
2787:,
2782:x
2775:e
2769:l
2765:A
2761:=
2758:)
2755:x
2752:(
2747:L
2736:{
2731:=
2728:)
2725:x
2722:(
2707:l
2704:B
2700:r
2697:A
2692:x
2683:ħ
2681:/
2678:|
2676:E
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2672:m
2670:2
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2661:k
2654:E
2643:.
2639:x
2626:E
2602:.
2599:)
2594:l
2590:A
2586:+
2581:r
2577:A
2573:(
2565:2
2557:k
2554:i
2546:m
2543:2
2537:=
2530:l
2526:B
2517:r
2513:B
2509:+
2504:l
2500:A
2496:+
2491:r
2487:A
2476:,
2473:0
2470:=
2463:l
2459:B
2450:r
2446:B
2437:l
2433:A
2429:+
2424:r
2420:A
2413:{
2386:=
2383:)
2378:l
2374:A
2370:+
2365:r
2361:A
2357:(
2351:+
2348:)
2343:l
2339:B
2330:r
2326:B
2322:+
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2313:A
2309:+
2304:r
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2290:k
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2281:m
2278:2
2272:2
2252:Ï
2238:.
2235:]
2232:)
2223:(
2209:)
2203:+
2200:(
2189:[
2186:=
2183:x
2180:d
2176:)
2173:x
2170:(
2154:+
2121:,
2118:)
2115:0
2112:(
2103:+
2100:]
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2094:0
2091:(
2083:L
2072:)
2069:0
2066:(
2058:R
2050:[
2044:m
2041:2
2035:2
2013:Δ
1996:.
1993:x
1990:d
1986:)
1983:x
1980:(
1969:+
1953:E
1950:=
1947:x
1944:d
1940:)
1937:x
1934:(
1928:)
1925:x
1922:(
1919:V
1911:+
1895:+
1892:x
1889:d
1885:)
1882:x
1879:(
1863:+
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1841:2
1835:2
1813:Δ
1809:Δ
1801:x
1783:,
1778:l
1774:B
1770:+
1765:r
1761:B
1757:=
1752:l
1748:A
1744:+
1739:r
1735:A
1731:=
1728:)
1725:0
1722:(
1717:R
1709:=
1706:)
1703:0
1700:(
1695:L
1687:=
1684:)
1681:0
1678:(
1661:e
1655:e
1650:k
1629:,
1626:0
1620:x
1610:,
1605:x
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1599:i
1592:e
1586:l
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1578:+
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1557:r
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1549:=
1546:)
1543:x
1540:(
1535:R
1523:,
1520:0
1514:x
1504:,
1499:x
1496:k
1493:i
1486:e
1480:l
1476:A
1472:+
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1464:k
1461:i
1457:e
1451:r
1447:A
1443:=
1440:)
1437:x
1434:(
1429:L
1418:{
1413:=
1410:)
1407:x
1404:(
1379:.
1370:E
1367:m
1364:2
1358:=
1355:k
1345:k
1337:e
1331:e
1304:;
1294:2
1285:E
1282:m
1279:2
1270:=
1262:2
1258:x
1254:d
1244:2
1240:d
1224:x
1217:x
1201:λ
1193:λ
1178:)
1176:x
1174:(
1172:ÎŽ
1157:,
1154:)
1151:x
1148:(
1139:=
1136:)
1133:x
1130:(
1127:V
1110:E
1102:ħ
1088:,
1085:)
1082:x
1079:(
1073:E
1070:=
1067:)
1064:x
1061:(
1055:)
1052:x
1049:(
1046:V
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1035:2
1031:x
1027:d
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1019:x
1016:(
1008:2
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994:m
991:2
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963:x
961:(
959:V
951:)
949:x
947:(
945:Ï
875:e
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861:v
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79:H
73:=
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56:t
53:d
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41:i
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