917:. 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.
3141:
1657:
5666:
2017:
4139:
940:
2905:
1407:
3742:, 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).
4019:
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
2630:
3982:
4012:(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
4005:. 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.
5016:
4782:
3736:
1831:
2725:
3013:
3506:
2644:
1808:
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
4635:
4283:
2417:
1109:
2410:
3127:
3748:
4811:
4640:
3517:
1652:{\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}}}
2259:
1325:
117:
1804:
2912:
3325:
2142:
2012:{\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.}
2900:{\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}}}
5457:
3208:
4094:
1400:
5437:
5276:
4518:
1178:
4155:
981:
2268:
3029:
4509:
4430:
5131:
4398:
5195:
5049:
4477:
2701:
is imaginary, and the wave functions which were oscillating for positive energies in the calculation above are now exponentially increasing or decreasing functions of
5371:
1214:
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.
928:, 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.
5315:
3991:
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.
1242:
1681:
5413:
4802:
383:
3998:
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
2147:
884:
2625:{\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}}}
591:
135:
3977:{\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}}}}.}
47:
3514:
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
2647:
The graph of the bound state wavefunction solution to the delta function potential is continuous everywhere, but its derivative is not defined at
2031:
1404:
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:
3019:
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
5686:
5011:{\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},}
364:
4777:{\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}}.}
3731:{\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}}}}}}
547:
5641:
3999:
470:
4443:(shown in brown in the diagram). Keeping in mind the relationship of this model with its three-dimensional molecular counterpart, we use
1361:
5200:
130:
5163:(shown in green in the diagram). They represent an approximation of the two lowest discrete energy states of the three-dimensional
931:
This article, for simplicity, only considers a one-dimensional potential well, but analysis could be expanded to more dimensions.
5706:
5701:
3008:{\displaystyle {\begin{cases}A_{\text{l}}=B_{\text{r}}={\sqrt {\kappa }},\\\kappa =-{\frac {m\lambda }{\hbar ^{2}}},\end{cases}}}
219:
3501:{\displaystyle t={\cfrac {1}{1-{\cfrac {m\lambda }{i\hbar ^{2}k}}}},\quad r={\cfrac {1}{{\cfrac {i\hbar ^{2}k}{m\lambda }}-1}}.}
326:
306:
5380:. For these specific parameters, there are many interesting properties that occur, one of which is the unusual effect that the
5197:
and are useful in its analysis. Analytical solutions for the energy eigenvalues for the case of symmetric charges are given by
877:
174:
1133:
296:
4016:. The delta function potential barrier is the limiting case of the model considered there for very high and narrow barriers.
566:
4142:
The symmetric and anti-symmetric wavefunctions for the double-well Dirac delta function model with "internuclear" distance
4013:
4009:
3162:
606:
344:
244:
5476:
4023:
359:
1328:
542:
537:
508:
140:
5559:
5447:
321:
311:
5578:
5418:
5061:
921:
576:
493:
4152:
The double-well Dirac delta function models a diatomic hydrogen molecule by the corresponding Schrödinger equation:
3264:. To find the amplitudes for reflection and transmission for incidence from the left, we put in the above equations
5696:
5670:
5452:
4288:
4104:
870:
522:
3231:
The quantum case can be studied in the following situation: a particle incident on the barrier from the left side
441:
527:
488:
416:
339:
199:
5381:
1678:
One obtains a relation between the coefficients by imposing that the wavefunction be continuous at the origin:
611:
5133:
The "+" case corresponds to a wave function symmetric about the midpoint (shown in red in the diagram), where
5691:
3995:
947:
780:
498:
456:
406:
354:
120:
5442:
785:
503:
411:
331:
301:
264:
5403:
5145:
4482:
4127:
4020:
equation may then be reduced to the case considered here by an Ansatz for the wave function of the type
2705:(see above). Requiring that the wave functions do not diverge at infinity eliminates half of the terms:
1332:
925:
254:
239:
4403:
166:
5149:. Correspondingly, the "â" case is the wave function that is anti-symmetric about the midpoint, where
4099:
Alternatively, it is possible to generalize the delta function to exist on the surface of some domain
5600:
5551:
5519:
4119:
1192:
914:
910:
815:
571:
483:
209:
2921:
2749:
2426:
1431:
5321:
charges, and for that matter the three-dimensional molecular problem, the solutions are given by a
4630:{\displaystyle \psi (x)=Ae^{-d\left|x+{\frac {R}{2}}\right|}+Be^{-d\left|x-{\frac {R}{2}}\right|}.}
3020:
670:
478:
396:
224:
204:
156:
5650:
For the 3-dimensional case look for the "delta shell potential"; further see K. Gottfried (1966),
5570:
5535:
5509:
5500:
Lange, Rutger-Jan (2012), "Potential theory, path integrals and the
Laplacian of the indicator",
5427:
5408:
5330:
5283:
5166:
1336:
391:
316:
249:
161:
5021:
4450:
4278:{\displaystyle -{\frac {\hbar ^{2}}{2m}}{\frac {d^{2}\psi (x)}{dx^{2}}}+V(x)\psi (x)=E\psi (x),}
1104:{\displaystyle -{\frac {\hbar ^{2}}{2m}}{\frac {d^{2}\psi (x)}{dx^{2}}}+V(x)\psi (x)=E\psi (x),}
5637:
5616:
5398:
2405:{\displaystyle -{\frac {\hbar ^{2}}{2m}}ik(-A_{r}+A_{l}+B_{r}-B_{l})+\lambda (A_{r}+A_{l})=0.}
898:
825:
800:
740:
735:
635:
601:
581:
179:
38:
5343:
3122:{\displaystyle E=-{\frac {\hbar ^{2}\kappa ^{2}}{2m}}=-{\frac {m\lambda ^{2}}{2\hbar ^{2}}}.}
5608:
5527:
965:
830:
820:
810:
710:
690:
675:
645:
513:
401:
5293:
5432:
1239:). In each of these parts the potential is zero, and the Schrödinger equation reduces to
1116:
855:
725:
705:
451:
291:
5604:
5523:
17:
4787:
906:
790:
750:
730:
700:
680:
630:
596:
446:
436:
229:
4637:
Matching of the wavefunction at the Dirac delta-function peaks yields the determinant
2643:
5680:
5423:
5393:
4432:
is the "internuclear" distance with Dirac delta-function (negative) peaks located at
4111:
951:
850:
845:
775:
745:
715:
586:
532:
259:
234:
5574:
5539:
5591:
van Dijk, W.; Kiers, K. A. (1992). "Time delay in simple one-dimensional systems".
4444:
840:
835:
770:
755:
720:
214:
4511:
is a formally adjustable parameter. From the single-well case, we can infer the "
2414:
The boundary conditions thus give the following restrictions on the coefficients
5531:
2659:
1352:
805:
760:
695:
650:
4138:
795:
765:
685:
660:
655:
640:
5620:
4126:
Dirac Delta function model which represents a one-dimensional version of the
2028:, the right-hand side of this equation vanishes; the left-hand side becomes
1320:{\displaystyle {\frac {d^{2}\psi }{dx^{2}}}=-{\frac {2mE}{\hbar ^{2}}}\psi ;}
112:{\displaystyle i\hbar {\frac {d}{dt}}|\Psi \rangle ={\hat {H}}|\Psi \rangle }
286:
2909:
From the boundary conditions and normalization conditions, it follows that
1799:{\displaystyle \psi (0)=\psi _{L}(0)=\psi _{R}(0)=A_{r}+A_{l}=B_{r}+B_{l},}
5665:
3214:
For positive energies, the particle is free to move in either half-space:
3140:
5575:"The Calculation of Exchange Forces: General Results and Specific Models"
3511:
665:
4110:
The delta function model is actually a one-dimensional version of the
3738:
for the particle to be reflected. This does not depend on the sign of
939:
4512:
2254:{\displaystyle \int _{-\varepsilon }^{+\varepsilon }\psi ''(x)\,dx=.}
1124:
5612:
5290:. Note that the lowest energy corresponds to the symmetric solution
5514:
4137:
3139:
2642:
938:
5599:(6). American Association of Physics Teachers (AAPT): 520â527.
5051:. For the case of equal charges (symmetric homonuclear case),
4122:
The delta function model becomes particularly useful with the
5458:
List of quantum-mechanical systems with analytical solutions
2658:
In any one-dimensional attractive potential there will be a
3001:
2893:
2618:
2137:{\displaystyle -{\frac {\hbar ^{2}}{2m}}+\lambda \psi (0),}
1645:
5477:"quantum mechanics - Wave function with a delta potential"
5438:
Particle in a one-dimensional lattice (periodic potential)
3210:. Dashed: classical result. Solid line: quantum mechanics.
5373:. Thus, one has a non-trivial bound state solution with
3940:
3918:
3904:
3892:
3851:
3822:
3808:
3796:
3700:
3675:
3661:
3649:
3608:
3579:
3565:
3553:
3468:
3443:
3435:
3423:
3379:
3364:
3350:
3338:
3228:. It may be scattered at the delta-function potential.
4649:
3943:
3921:
3907:
3895:
3854:
3825:
3811:
3799:
3703:
3678:
3664:
3652:
3611:
3582:
3568:
3556:
3471:
3446:
3438:
3426:
3382:
3367:
3353:
3341:
5346:
5296:
5203:
5169:
5064:
5024:
4814:
4790:
4643:
4521:
4485:
4453:
4406:
4291:
4158:
4026:
3751:
3520:
3328:
3165:
3032:
2915:
2728:
2420:
2271:
2150:
2034:
1834:
1684:
1410:
1364:
1245:
1136:
984:
50:
3994:
One such example regards the interfaces between two
3203:{\displaystyle {\frac {m\lambda ^{2}}{2\hbar ^{2}}}}
3152:) probability of a delta potential well. The energy
4089:{\displaystyle \Psi (x,y,z)=\psi (x)\phi (y,z)\,\!}
5365:
5309:
5270:
5189:
5125:
5043:
5010:
4796:
4776:
4629:
4503:
4471:
4424:
4392:
4277:
4088:
3976:
3730:
3500:
3202:
3121:
3007:
2899:
2624:
2404:
2253:
2136:
2011:
1798:
1651:
1394:
1319:
1172:
1103:
111:
4085:
27:Model of an energy potential in quantum mechanics
1395:{\displaystyle k={\frac {\sqrt {2mE}}{\hbar }}.}
5636:(2nd ed.). Prentice Hall. pp. 68â78.
5271:{\displaystyle d_{\pm }=q+W(\pm qRe^{-qR})/R,}
1669:represents a wave traveling to the right, and
1659:where, in the case of positive energies (real
5414:Particle in a spherically symmetric potential
1225:The potential splits the space in two parts (
878:
8:
3318:even though we do not have any equations in
106:
80:
5336:One of the most interesting cases is when
3297:(no incoming particle from the right) and
885:
871:
29:
5513:
5351:
5345:
5331:Lambert W function § Generalizations
5301:
5295:
5257:
5242:
5208:
5202:
5180:
5175:
5170:
5168:
5104:
5096:
5069:
5063:
5035:
5023:
4995:
4991:
4961:
4950:
4929:
4910:
4888:
4868:
4837:
4819:
4813:
4789:
4760:
4754:
4740:
4696:
4670:
4644:
4642:
4607:
4589:
4562:
4544:
4520:
4484:
4452:
4405:
4367:
4332:
4290:
4218:
4191:
4184:
4168:
4162:
4157:
4084:
4025:
3952:
3944:
3930:
3922:
3915:
3908:
3896:
3889:
3870:
3860:
3855:
3841:
3831:
3826:
3819:
3812:
3800:
3793:
3772:
3767:
3758:
3750:
3712:
3704:
3687:
3679:
3672:
3665:
3653:
3646:
3627:
3617:
3612:
3598:
3588:
3583:
3576:
3569:
3557:
3550:
3541:
3536:
3527:
3519:
3472:
3455:
3447:
3440:
3439:
3427:
3420:
3391:
3383:
3368:
3361:
3354:
3342:
3335:
3327:
3191:
3176:
3166:
3164:
3107:
3092:
3082:
3059:
3049:
3042:
3031:
2987:
2973:
2950:
2941:
2928:
2916:
2914:
2876:
2859:
2849:
2827:
2802:
2788:
2778:
2756:
2744:
2727:
2603:
2590:
2574:
2550:
2539:
2526:
2513:
2500:
2472:
2459:
2446:
2433:
2421:
2419:
2387:
2374:
2352:
2339:
2326:
2313:
2281:
2275:
2270:
2189:
2163:
2155:
2149:
2092:
2067:
2044:
2038:
2033:
1999:
1978:
1970:
1953:
1920:
1912:
1898:
1872:
1864:
1844:
1838:
1833:
1787:
1774:
1761:
1748:
1726:
1704:
1683:
1625:
1605:
1595:
1576:
1566:
1544:
1519:
1499:
1489:
1470:
1460:
1438:
1426:
1409:
1371:
1363:
1303:
1286:
1271:
1253:
1246:
1244:
1135:
1044:
1017:
1010:
994:
988:
983:
98:
87:
86:
72:
57:
49:
5652:Quantum Mechanics Volume I: Fundamentals
1173:{\displaystyle V(x)=\lambda \delta (x),}
5556:Dimensional Scaling in Chemical Physics
5554:, J.S. Avery, and O. Goscinski (eds.),
5468:
4454:
4165:
3949:
3857:
3684:
3585:
3452:
3388:
3188:
3104:
3046:
2984:
2571:
2278:
2041:
1841:
1384:
1300:
991:
54:
37:
5058:, and the pseudo-quadratic reduces to
3026:The energy of the bound state is then
4130:, as shown in the following section.
1130:The delta potential is the potential
7:
3745:The probability for transmission is
2662:. To find its energy, note that for
964:of a particle in one dimension in a
5569:T. C. Scott, J. F. Babb,
4504:{\displaystyle 0<\lambda <1}
4419:
4027:
417:Sum-over-histories (path integral)
103:
77:
33:Part of a series of articles about
25:
5634:Introduction to Quantum Mechanics
5340: †1, which results in
4425:{\displaystyle 0<R<\infty }
4118:method developed by the group of
5664:
5126:{\displaystyle d_{\pm }=q\left.}
1218:Solving the Schrödinger equation
909:mathematically described by the
4804:is found to be governed by the
4739:
3413:
2261:Substituting the definition of
5502:Journal of High Energy Physics
5254:
5226:
4973:
4967:
4907:
4894:
4862:
4850:
4831:
4825:
4531:
4525:
4301:
4295:
4269:
4263:
4251:
4245:
4239:
4233:
4206:
4200:
4081:
4069:
4063:
4057:
4048:
4030:
3768:
3759:
3537:
3528:
3310:(transmission), and solve for
2839:
2833:
2768:
2762:
2738:
2732:
2609:
2583:
2393:
2367:
2358:
2303:
2245:
2242:
2233:
2219:
2210:
2199:
2186:
2180:
2128:
2122:
2110:
2107:
2101:
2082:
2076:
2060:
1996:
1990:
1950:
1944:
1938:
1932:
1895:
1889:
1738:
1732:
1716:
1710:
1694:
1688:
1556:
1550:
1450:
1444:
1420:
1414:
1164:
1158:
1146:
1140:
1095:
1089:
1077:
1071:
1065:
1059:
1032:
1026:
920:The delta potential well is a
567:Relativistic quantum mechanics
99:
92:
73:
1:
5687:Quantum mechanical potentials
5654:, ch. III, sec. 15.
4393:{\displaystyle V(x)=-q\left,}
4014:Finite potential barrier (QM)
4010:scanning tunneling microscope
1358:is related to the energy by
607:Quantum statistical mechanics
5632:Griffiths, David J. (2005).
5581:, 99, pp. 2841â2854, (1993).
5190:{\displaystyle {\ce {H2^+}}}
2722:. The wave function is then
2265:into this expression yields
1329:linear differential equation
5593:American Journal of Physics
5419:Quantum harmonic oscillator
4285:where the potential is now
3015:from which it follows that
1675:one traveling to the left.
577:Quantum information science
5723:
5453:Laplacian of the indicator
5044:{\displaystyle d=d_{\pm }}
4472:{\displaystyle \hbar =m=1}
4105:Laplacian of the indicator
5384:is unity at zero energy.
4515:" for the solution to be
5573:and John D. Morgan III,
5382:transmission coefficient
5018:which has two solutions
612:Quantum machine learning
365:Wheeler's delayed-choice
18:Delta function potential
5707:Exactly solvable models
5532:10.1007/JHEP11(2012)032
5448:HolsteinâHerring method
5366:{\displaystyle d_{-}=0}
3987:Remarks and application
1208:delta potential barrier
322:LeggettâGarg inequality
5481:Physics Stack Exchange
5443:Hydrogen molecular ion
5367:
5311:
5272:
5191:
5127:
5045:
5012:
4798:
4778:
4631:
4505:
4473:
4426:
4394:
4279:
4149:
4134:Double delta potential
4090:
3978:
3732:
3502:
3242:. It may be reflected
3211:
3204:
3123:
3009:
2901:
2655:
2626:
2406:
2255:
2138:
2013:
1800:
1653:
1396:
1335:, whose solutions are
1321:
1174:
1105:
943:
935:Single delta potential
113:
5404:Finite potential well
5368:
5312:
5310:{\displaystyle d_{+}}
5273:
5192:
5128:
5046:
5013:
4799:
4779:
4632:
4506:
4474:
4427:
4395:
4280:
4141:
4128:Hydrogen molecule ion
4091:
3979:
3733:
3503:
3274:(incoming particle),
3205:
3143:
3124:
3010:
2902:
2646:
2627:
2407:
2256:
2139:
2014:
1801:
1654:
1397:
1333:constant coefficients
1322:
1175:
1106:
946:The time-independent
942:
926:finite potential well
307:Elitzur–Vaidman
297:Davisson–Germer
114:
5702:Schrödinger equation
5673:at Wikimedia Commons
5558:, Springer, (1992).
5344:
5294:
5201:
5167:
5062:
5022:
4812:
4788:
4641:
4519:
4483:
4451:
4404:
4289:
4156:
4120:Dudley R. Herschbach
4024:
3749:
3518:
3326:
3163:
3030:
2913:
2726:
2418:
2269:
2148:
2032:
1832:
1682:
1408:
1362:
1243:
1200:delta potential well
1193:Dirac delta function
1134:
982:
948:Schrödinger equation
915:generalized function
911:Dirac delta function
572:Quantum field theory
484:Consistent histories
121:Schrödinger equation
48:
5605:1992AmJPh..60..520V
5524:2012JHEP...11..032L
5185:
4116:dimensional scaling
4008:The operation of a
3942:
3920:
3906:
3894:
3853:
3824:
3810:
3798:
3702:
3677:
3663:
3651:
3610:
3581:
3567:
3555:
3470:
3445:
3437:
3425:
3381:
3366:
3352:
3340:
3021:Lorentzian function
2171:
2100:
2075:
1986:
1928:
1880:
1816:, over an interval
1337:linear combinations
1206:is negative, and a
360:Stern–Gerlach
157:Classical mechanics
5428:hydrogen-like atom
5409:Particle in a ring
5363:
5307:
5268:
5187:
5171:
5123:
5041:
5008:
4794:
4774:
4724:
4627:
4501:
4469:
4422:
4390:
4275:
4150:
4086:
3974:
3967:
3962:
3937:
3901:
3882:
3877:
3848:
3805:
3728:
3724:
3719:
3697:
3658:
3639:
3634:
3605:
3562:
3510:Due to the mirror
3498:
3491:
3480:
3465:
3432:
3406:
3401:
3376:
3347:
3212:
3200:
3148:) and reflection (
3119:
3005:
3000:
2897:
2892:
2656:
2622:
2617:
2402:
2251:
2151:
2134:
2088:
2063:
2009:
1966:
1908:
1860:
1796:
1649:
1644:
1392:
1317:
1170:
1101:
944:
548:Von NeumannâWigner
528:Objective-collapse
327:Mach–Zehnder
317:Leggett inequality
312:Franck–Hertz
162:Old quantum theory
109:
5697:Scattering theory
5669:Media related to
5643:978-0-13-111892-8
5399:Particle in a box
5317:. In the case of
5174:
4876:
4845:
4797:{\displaystyle d}
4769:
4743:
4615:
4570:
4375:
4340:
4225:
4182:
4114:according to the
3969:
3964:
3941:
3919:
3905:
3893:
3884:
3879:
3852:
3823:
3809:
3797:
3726:
3721:
3701:
3676:
3662:
3650:
3641:
3636:
3609:
3580:
3566:
3554:
3493:
3482:
3469:
3444:
3436:
3424:
3408:
3403:
3380:
3365:
3351:
3339:
3322:. The result is
3198:
3114:
3074:
2993:
2955:
2944:
2931:
2879:
2852:
2830:
2805:
2781:
2759:
2581:
2295:
2058:
1858:
1628:
1598:
1569:
1547:
1522:
1492:
1463:
1441:
1387:
1383:
1309:
1278:
1127:of the particle.
1051:
1008:
899:quantum mechanics
895:
894:
602:Scattering theory
582:Quantum computing
355:Schrödinger's cat
287:Bell's inequality
95:
70:
39:Quantum mechanics
16:(Redirected from
5714:
5668:
5647:
5625:
5624:
5588:
5582:
5567:
5561:
5549:
5543:
5542:
5517:
5497:
5491:
5490:
5488:
5487:
5473:
5379:
5372:
5370:
5369:
5364:
5356:
5355:
5316:
5314:
5313:
5308:
5306:
5305:
5282:is the standard
5277:
5275:
5274:
5269:
5261:
5253:
5252:
5213:
5212:
5196:
5194:
5193:
5188:
5186:
5184:
5179:
5172:
5159:, and is called
5158:
5143:, and is called
5142:
5132:
5130:
5129:
5124:
5119:
5115:
5114:
5113:
5109:
5108:
5074:
5073:
5057:
5050:
5048:
5047:
5042:
5040:
5039:
5017:
5015:
5014:
5009:
5004:
5003:
4999:
4990:
4986:
4985:
4981:
4980:
4979:
4966:
4965:
4934:
4933:
4915:
4914:
4893:
4892:
4877:
4869:
4846:
4838:
4824:
4823:
4806:pseudo-quadratic
4803:
4801:
4800:
4795:
4783:
4781:
4780:
4775:
4770:
4765:
4764:
4755:
4744:
4741:
4729:
4728:
4707:
4706:
4681:
4680:
4636:
4634:
4633:
4628:
4623:
4622:
4621:
4617:
4616:
4608:
4578:
4577:
4576:
4572:
4571:
4563:
4510:
4508:
4507:
4502:
4478:
4476:
4475:
4470:
4442:
4431:
4429:
4428:
4423:
4399:
4397:
4396:
4391:
4386:
4382:
4381:
4377:
4376:
4368:
4346:
4342:
4341:
4333:
4284:
4282:
4281:
4276:
4226:
4224:
4223:
4222:
4209:
4196:
4195:
4185:
4183:
4181:
4173:
4172:
4163:
4148:
4095:
4093:
4092:
4087:
4004:
3983:
3981:
3980:
3975:
3970:
3968:
3966:
3965:
3963:
3961:
3957:
3956:
3938:
3936:
3935:
3934:
3916:
3902:
3900:
3890:
3885:
3883:
3881:
3880:
3878:
3876:
3875:
3874:
3865:
3864:
3849:
3847:
3846:
3845:
3836:
3835:
3820:
3806:
3804:
3794:
3777:
3776:
3771:
3762:
3741:
3737:
3735:
3734:
3729:
3727:
3725:
3723:
3722:
3720:
3718:
3717:
3716:
3698:
3696:
3692:
3691:
3673:
3659:
3657:
3647:
3642:
3640:
3638:
3637:
3635:
3633:
3632:
3631:
3622:
3621:
3606:
3604:
3603:
3602:
3593:
3592:
3577:
3563:
3561:
3551:
3546:
3545:
3540:
3531:
3507:
3505:
3504:
3499:
3494:
3492:
3490:
3483:
3481:
3479:
3466:
3464:
3460:
3459:
3441:
3433:
3431:
3421:
3409:
3407:
3405:
3404:
3402:
3400:
3396:
3395:
3377:
3375:
3362:
3348:
3346:
3336:
3321:
3317:
3313:
3309:
3296:
3286:
3273:
3263:
3252:
3241:
3227:
3220:
3209:
3207:
3206:
3201:
3199:
3197:
3196:
3195:
3182:
3181:
3180:
3167:
3158:
3128:
3126:
3125:
3120:
3115:
3113:
3112:
3111:
3098:
3097:
3096:
3083:
3075:
3073:
3065:
3064:
3063:
3054:
3053:
3043:
3018:
3014:
3012:
3011:
3006:
3004:
3003:
2994:
2992:
2991:
2982:
2974:
2956:
2951:
2946:
2945:
2942:
2933:
2932:
2929:
2906:
2904:
2903:
2898:
2896:
2895:
2880:
2877:
2870:
2869:
2854:
2853:
2850:
2832:
2831:
2828:
2806:
2803:
2796:
2795:
2783:
2782:
2779:
2761:
2760:
2757:
2721:
2700:
2691:
2690:
2668:
2653:
2631:
2629:
2628:
2623:
2621:
2620:
2608:
2607:
2595:
2594:
2582:
2580:
2579:
2578:
2562:
2551:
2544:
2543:
2531:
2530:
2518:
2517:
2505:
2504:
2477:
2476:
2464:
2463:
2451:
2450:
2438:
2437:
2411:
2409:
2408:
2403:
2392:
2391:
2379:
2378:
2357:
2356:
2344:
2343:
2331:
2330:
2318:
2317:
2296:
2294:
2286:
2285:
2276:
2264:
2260:
2258:
2257:
2252:
2232:
2209:
2179:
2170:
2162:
2143:
2141:
2140:
2135:
2096:
2071:
2059:
2057:
2049:
2048:
2039:
2027:
2021:In the limit as
2018:
2016:
2015:
2010:
1985:
1977:
1927:
1919:
1888:
1879:
1871:
1859:
1857:
1849:
1848:
1839:
1827:
1815:
1805:
1803:
1802:
1797:
1792:
1791:
1779:
1778:
1766:
1765:
1753:
1752:
1731:
1730:
1709:
1708:
1674:
1668:
1662:
1658:
1656:
1655:
1650:
1648:
1647:
1629:
1626:
1619:
1618:
1600:
1599:
1596:
1587:
1586:
1571:
1570:
1567:
1549:
1548:
1545:
1523:
1520:
1513:
1512:
1494:
1493:
1490:
1481:
1480:
1465:
1464:
1461:
1443:
1442:
1439:
1401:
1399:
1398:
1393:
1388:
1373:
1372:
1357:
1350:
1344:
1326:
1324:
1323:
1318:
1310:
1308:
1307:
1298:
1287:
1279:
1277:
1276:
1275:
1262:
1258:
1257:
1247:
1238:
1231:
1213:
1205:
1190:
1179:
1177:
1176:
1171:
1122:
1114:
1110:
1108:
1107:
1102:
1052:
1050:
1049:
1048:
1035:
1022:
1021:
1011:
1009:
1007:
999:
998:
989:
977:
963:
887:
880:
873:
514:Superdeterminism
167:Braâket notation
118:
116:
115:
110:
102:
97:
96:
88:
76:
71:
69:
58:
30:
21:
5722:
5721:
5717:
5716:
5715:
5713:
5712:
5711:
5677:
5676:
5671:Delta potential
5661:
5644:
5631:
5628:
5613:10.1119/1.16866
5590:
5589:
5585:
5568:
5564:
5552:D.R. Herschbach
5550:
5546:
5499:
5498:
5494:
5485:
5483:
5475:
5474:
5470:
5466:
5433:Ring wave guide
5390:
5374:
5347:
5342:
5341:
5325:of the Lambert
5297:
5292:
5291:
5238:
5204:
5199:
5198:
5165:
5164:
5150:
5134:
5100:
5092:
5085:
5081:
5065:
5060:
5059:
5052:
5031:
5020:
5019:
4957:
4946:
4939:
4935:
4925:
4906:
4884:
4883:
4879:
4878:
4815:
4810:
4809:
4786:
4785:
4756:
4723:
4722:
4708:
4692:
4683:
4682:
4666:
4661:
4645:
4639:
4638:
4600:
4596:
4585:
4555:
4551:
4540:
4517:
4516:
4481:
4480:
4449:
4448:
4433:
4402:
4401:
4360:
4356:
4325:
4321:
4317:
4313:
4287:
4286:
4214:
4210:
4187:
4186:
4174:
4164:
4154:
4153:
4143:
4136:
4022:
4021:
4002:
3989:
3948:
3939:
3926:
3917:
3903:
3891:
3866:
3856:
3850:
3837:
3827:
3821:
3807:
3795:
3766:
3747:
3746:
3739:
3708:
3699:
3683:
3674:
3660:
3648:
3623:
3613:
3607:
3594:
3584:
3578:
3564:
3552:
3535:
3516:
3515:
3467:
3451:
3442:
3434:
3422:
3387:
3378:
3363:
3349:
3337:
3324:
3323:
3319:
3315:
3311:
3304:
3298:
3294:
3288:
3281:
3275:
3271:
3265:
3261:
3254:
3253:or transmitted
3250:
3243:
3239:
3232:
3222:
3215:
3187:
3183:
3172:
3168:
3161:
3160:
3159:is in units of
3153:
3138:
3103:
3099:
3088:
3084:
3066:
3055:
3045:
3044:
3028:
3027:
3016:
2999:
2998:
2983:
2975:
2961:
2960:
2937:
2924:
2917:
2911:
2910:
2891:
2890:
2874:
2855:
2845:
2823:
2820:
2819:
2800:
2784:
2774:
2752:
2745:
2724:
2723:
2719:
2712:
2706:
2680:
2678:
2670:
2663:
2648:
2641:
2616:
2615:
2599:
2586:
2570:
2563:
2552:
2545:
2535:
2522:
2509:
2496:
2490:
2489:
2478:
2468:
2455:
2442:
2429:
2422:
2416:
2415:
2383:
2370:
2348:
2335:
2322:
2309:
2287:
2277:
2267:
2266:
2262:
2225:
2202:
2172:
2146:
2145:
2050:
2040:
2030:
2029:
2022:
1881:
1850:
1840:
1830:
1829:
1817:
1810:
1783:
1770:
1757:
1744:
1722:
1700:
1680:
1679:
1670:
1664:
1660:
1643:
1642:
1623:
1601:
1591:
1572:
1562:
1540:
1537:
1536:
1517:
1495:
1485:
1466:
1456:
1434:
1427:
1406:
1405:
1360:
1359:
1355:
1346:
1340:
1299:
1288:
1267:
1263:
1249:
1248:
1241:
1240:
1233:
1226:
1220:
1211:
1203:
1198:It is called a
1181:
1132:
1131:
1120:
1117:Planck constant
1115:is the reduced
1112:
1040:
1036:
1013:
1012:
1000:
990:
980:
979:
968:
954:
937:
903:delta potential
891:
862:
861:
860:
625:
617:
616:
562:
561:Advanced topics
554:
553:
552:
504:Hidden-variable
494:de BroglieâBohm
473:
471:Interpretations
463:
462:
461:
431:
423:
422:
421:
379:
371:
370:
369:
336:
292:CHSH inequality
281:
273:
272:
271:
200:Complementarity
194:
186:
185:
184:
152:
123:
62:
46:
45:
28:
23:
22:
15:
12:
11:
5:
5720:
5718:
5710:
5709:
5704:
5699:
5694:
5692:Quantum models
5689:
5679:
5678:
5675:
5674:
5660:
5659:External links
5657:
5656:
5655:
5648:
5642:
5627:
5626:
5583:
5579:J. Chem. Phys.
5562:
5544:
5492:
5467:
5465:
5462:
5461:
5460:
5455:
5450:
5445:
5440:
5435:
5430:
5421:
5416:
5411:
5406:
5401:
5396:
5389:
5386:
5362:
5359:
5354:
5350:
5329:function (see
5323:generalization
5304:
5300:
5267:
5264:
5260:
5256:
5251:
5248:
5245:
5241:
5237:
5234:
5231:
5228:
5225:
5222:
5219:
5216:
5211:
5207:
5183:
5178:
5122:
5118:
5112:
5107:
5103:
5099:
5095:
5091:
5088:
5084:
5080:
5077:
5072:
5068:
5038:
5034:
5030:
5027:
5007:
5002:
4998:
4994:
4989:
4984:
4978:
4975:
4972:
4969:
4964:
4960:
4956:
4953:
4949:
4945:
4942:
4938:
4932:
4928:
4924:
4921:
4918:
4913:
4909:
4905:
4902:
4899:
4896:
4891:
4887:
4882:
4875:
4872:
4867:
4864:
4861:
4858:
4855:
4852:
4849:
4844:
4841:
4836:
4833:
4830:
4827:
4822:
4818:
4793:
4773:
4768:
4763:
4759:
4753:
4750:
4747:
4738:
4735:
4732:
4727:
4721:
4718:
4715:
4712:
4709:
4705:
4702:
4699:
4695:
4691:
4688:
4685:
4684:
4679:
4676:
4673:
4669:
4665:
4662:
4660:
4657:
4654:
4651:
4650:
4648:
4626:
4620:
4614:
4611:
4606:
4603:
4599:
4595:
4592:
4588:
4584:
4581:
4575:
4569:
4566:
4561:
4558:
4554:
4550:
4547:
4543:
4539:
4536:
4533:
4530:
4527:
4524:
4500:
4497:
4494:
4491:
4488:
4468:
4465:
4462:
4459:
4456:
4421:
4418:
4415:
4412:
4409:
4389:
4385:
4380:
4374:
4371:
4366:
4363:
4359:
4355:
4352:
4349:
4345:
4339:
4336:
4331:
4328:
4324:
4320:
4316:
4312:
4309:
4306:
4303:
4300:
4297:
4294:
4274:
4271:
4268:
4265:
4262:
4259:
4256:
4253:
4250:
4247:
4244:
4241:
4238:
4235:
4232:
4229:
4221:
4217:
4213:
4208:
4205:
4202:
4199:
4194:
4190:
4180:
4177:
4171:
4167:
4161:
4135:
4132:
4083:
4080:
4077:
4074:
4071:
4068:
4065:
4062:
4059:
4056:
4053:
4050:
4047:
4044:
4041:
4038:
4035:
4032:
4029:
4000:effective mass
3988:
3985:
3973:
3960:
3955:
3951:
3947:
3933:
3929:
3925:
3914:
3911:
3899:
3888:
3873:
3869:
3863:
3859:
3844:
3840:
3834:
3830:
3818:
3815:
3803:
3792:
3789:
3786:
3783:
3780:
3775:
3770:
3765:
3761:
3757:
3754:
3715:
3711:
3707:
3695:
3690:
3686:
3682:
3671:
3668:
3656:
3645:
3630:
3626:
3620:
3616:
3601:
3597:
3591:
3587:
3575:
3572:
3560:
3549:
3544:
3539:
3534:
3530:
3526:
3523:
3497:
3489:
3486:
3478:
3475:
3463:
3458:
3454:
3450:
3430:
3419:
3416:
3412:
3399:
3394:
3390:
3386:
3374:
3371:
3360:
3357:
3345:
3334:
3331:
3302:
3292:
3287:(reflection),
3279:
3269:
3259:
3248:
3237:
3194:
3190:
3186:
3179:
3175:
3171:
3144:Transmission (
3137:
3130:
3118:
3110:
3106:
3102:
3095:
3091:
3087:
3081:
3078:
3072:
3069:
3062:
3058:
3052:
3048:
3041:
3038:
3035:
3002:
2997:
2990:
2986:
2981:
2978:
2972:
2969:
2966:
2963:
2962:
2959:
2954:
2949:
2940:
2936:
2927:
2923:
2922:
2920:
2894:
2889:
2886:
2883:
2878: if
2875:
2873:
2868:
2865:
2862:
2858:
2848:
2844:
2841:
2838:
2835:
2826:
2822:
2821:
2818:
2815:
2812:
2809:
2804: if
2801:
2799:
2794:
2791:
2787:
2777:
2773:
2770:
2767:
2764:
2755:
2751:
2750:
2748:
2743:
2740:
2737:
2734:
2731:
2717:
2710:
2640:
2633:
2619:
2614:
2611:
2606:
2602:
2598:
2593:
2589:
2585:
2577:
2573:
2569:
2566:
2561:
2558:
2555:
2549:
2546:
2542:
2538:
2534:
2529:
2525:
2521:
2516:
2512:
2508:
2503:
2499:
2495:
2492:
2491:
2488:
2485:
2482:
2479:
2475:
2471:
2467:
2462:
2458:
2454:
2449:
2445:
2441:
2436:
2432:
2428:
2427:
2425:
2401:
2398:
2395:
2390:
2386:
2382:
2377:
2373:
2369:
2366:
2363:
2360:
2355:
2351:
2347:
2342:
2338:
2334:
2329:
2325:
2321:
2316:
2312:
2308:
2305:
2302:
2299:
2293:
2290:
2284:
2280:
2274:
2250:
2247:
2244:
2241:
2238:
2235:
2231:
2228:
2224:
2221:
2218:
2215:
2212:
2208:
2205:
2201:
2198:
2195:
2192:
2188:
2185:
2182:
2178:
2175:
2169:
2166:
2161:
2158:
2154:
2133:
2130:
2127:
2124:
2121:
2118:
2115:
2112:
2109:
2106:
2103:
2099:
2095:
2091:
2087:
2084:
2081:
2078:
2074:
2070:
2066:
2062:
2056:
2053:
2047:
2043:
2037:
2008:
2005:
2002:
1998:
1995:
1992:
1989:
1984:
1981:
1976:
1973:
1969:
1965:
1962:
1959:
1956:
1952:
1949:
1946:
1943:
1940:
1937:
1934:
1931:
1926:
1923:
1918:
1915:
1911:
1907:
1904:
1901:
1897:
1894:
1891:
1887:
1884:
1878:
1875:
1870:
1867:
1863:
1856:
1853:
1847:
1843:
1837:
1795:
1790:
1786:
1782:
1777:
1773:
1769:
1764:
1760:
1756:
1751:
1747:
1743:
1740:
1737:
1734:
1729:
1725:
1721:
1718:
1715:
1712:
1707:
1703:
1699:
1696:
1693:
1690:
1687:
1646:
1641:
1638:
1635:
1632:
1627: if
1624:
1622:
1617:
1614:
1611:
1608:
1604:
1594:
1590:
1585:
1582:
1579:
1575:
1565:
1561:
1558:
1555:
1552:
1543:
1539:
1538:
1535:
1532:
1529:
1526:
1521: if
1518:
1516:
1511:
1508:
1505:
1502:
1498:
1488:
1484:
1479:
1476:
1473:
1469:
1459:
1455:
1452:
1449:
1446:
1437:
1433:
1432:
1430:
1425:
1422:
1419:
1416:
1413:
1391:
1386:
1382:
1379:
1376:
1370:
1367:
1316:
1313:
1306:
1302:
1297:
1294:
1291:
1285:
1282:
1274:
1270:
1266:
1261:
1256:
1252:
1219:
1216:
1169:
1166:
1163:
1160:
1157:
1154:
1151:
1148:
1145:
1142:
1139:
1100:
1097:
1094:
1091:
1088:
1085:
1082:
1079:
1076:
1073:
1070:
1067:
1064:
1061:
1058:
1055:
1047:
1043:
1039:
1034:
1031:
1028:
1025:
1020:
1016:
1006:
1003:
997:
993:
987:
936:
933:
907:potential well
893:
892:
890:
889:
882:
875:
867:
864:
863:
859:
858:
853:
848:
843:
838:
833:
828:
823:
818:
813:
808:
803:
798:
793:
788:
783:
778:
773:
768:
763:
758:
753:
748:
743:
738:
733:
728:
723:
718:
713:
708:
703:
698:
693:
688:
683:
678:
673:
668:
663:
658:
653:
648:
643:
638:
633:
627:
626:
623:
622:
619:
618:
615:
614:
609:
604:
599:
597:Density matrix
594:
589:
584:
579:
574:
569:
563:
560:
559:
556:
555:
551:
550:
545:
540:
535:
530:
525:
520:
519:
518:
517:
516:
501:
496:
491:
486:
481:
475:
474:
469:
468:
465:
464:
460:
459:
454:
449:
444:
439:
433:
432:
429:
428:
425:
424:
420:
419:
414:
409:
404:
399:
394:
388:
387:
386:
380:
377:
376:
373:
372:
368:
367:
362:
357:
351:
350:
349:
348:
347:
345:Delayed-choice
340:Quantum eraser
335:
334:
329:
324:
319:
314:
309:
304:
299:
294:
289:
283:
282:
279:
278:
275:
274:
270:
269:
268:
267:
257:
252:
247:
242:
237:
232:
230:Quantum number
227:
222:
217:
212:
207:
202:
196:
195:
192:
191:
188:
187:
183:
182:
177:
171:
170:
169:
164:
159:
153:
150:
149:
146:
145:
144:
143:
138:
133:
125:
124:
119:
108:
105:
101:
94:
91:
85:
82:
79:
75:
68:
65:
61:
56:
53:
42:
41:
35:
34:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
5719:
5708:
5705:
5703:
5700:
5698:
5695:
5693:
5690:
5688:
5685:
5684:
5682:
5672:
5667:
5663:
5662:
5658:
5653:
5649:
5645:
5639:
5635:
5630:
5629:
5622:
5618:
5614:
5610:
5606:
5602:
5598:
5594:
5587:
5584:
5580:
5576:
5572:
5566:
5563:
5560:
5557:
5553:
5548:
5545:
5541:
5537:
5533:
5529:
5525:
5521:
5516:
5511:
5507:
5503:
5496:
5493:
5482:
5478:
5472:
5469:
5463:
5459:
5456:
5454:
5451:
5449:
5446:
5444:
5441:
5439:
5436:
5434:
5431:
5429:
5425:
5424:Hydrogen atom
5422:
5420:
5417:
5415:
5412:
5410:
5407:
5405:
5402:
5400:
5397:
5395:
5394:Free particle
5392:
5391:
5387:
5385:
5383:
5377:
5360:
5357:
5352:
5348:
5339:
5334:
5332:
5328:
5324:
5320:
5302:
5298:
5289:
5287:
5281:
5265:
5262:
5258:
5249:
5246:
5243:
5239:
5235:
5232:
5229:
5223:
5220:
5217:
5214:
5209:
5205:
5181:
5176:
5162:
5157:
5153:
5148:
5147:
5141:
5137:
5120:
5116:
5110:
5105:
5101:
5097:
5093:
5089:
5086:
5082:
5078:
5075:
5070:
5066:
5055:
5036:
5032:
5028:
5025:
5005:
5000:
4996:
4992:
4987:
4982:
4976:
4970:
4962:
4958:
4954:
4951:
4947:
4943:
4940:
4936:
4930:
4926:
4922:
4919:
4916:
4911:
4903:
4900:
4897:
4889:
4885:
4880:
4873:
4870:
4865:
4859:
4856:
4853:
4847:
4842:
4839:
4834:
4828:
4820:
4816:
4807:
4791:
4771:
4766:
4761:
4757:
4751:
4748:
4745:
4736:
4733:
4730:
4725:
4719:
4716:
4713:
4710:
4703:
4700:
4697:
4693:
4689:
4686:
4677:
4674:
4671:
4667:
4663:
4658:
4655:
4652:
4646:
4624:
4618:
4612:
4609:
4604:
4601:
4597:
4593:
4590:
4586:
4582:
4579:
4573:
4567:
4564:
4559:
4556:
4552:
4548:
4545:
4541:
4537:
4534:
4528:
4522:
4514:
4498:
4495:
4492:
4489:
4486:
4466:
4463:
4460:
4457:
4446:
4440:
4436:
4416:
4413:
4410:
4407:
4387:
4383:
4378:
4372:
4369:
4364:
4361:
4357:
4353:
4350:
4347:
4343:
4337:
4334:
4329:
4326:
4322:
4318:
4314:
4310:
4307:
4304:
4298:
4292:
4272:
4266:
4260:
4257:
4254:
4248:
4242:
4236:
4230:
4227:
4219:
4215:
4211:
4203:
4197:
4192:
4188:
4178:
4175:
4169:
4159:
4146:
4140:
4133:
4131:
4129:
4125:
4121:
4117:
4113:
4112:Hydrogen atom
4108:
4106:
4102:
4097:
4078:
4075:
4072:
4066:
4060:
4054:
4051:
4045:
4042:
4039:
4036:
4033:
4017:
4015:
4011:
4006:
4001:
3997:
3992:
3986:
3984:
3971:
3958:
3953:
3945:
3931:
3927:
3923:
3912:
3909:
3897:
3886:
3871:
3867:
3861:
3842:
3838:
3832:
3828:
3816:
3813:
3801:
3790:
3787:
3784:
3781:
3778:
3773:
3763:
3755:
3752:
3743:
3713:
3709:
3705:
3693:
3688:
3680:
3669:
3666:
3654:
3643:
3628:
3624:
3618:
3614:
3599:
3595:
3589:
3573:
3570:
3558:
3547:
3542:
3532:
3524:
3521:
3513:
3508:
3495:
3487:
3484:
3476:
3473:
3461:
3456:
3448:
3428:
3417:
3414:
3410:
3397:
3392:
3384:
3372:
3369:
3358:
3355:
3343:
3332:
3329:
3308:
3301:
3291:
3285:
3278:
3268:
3258:
3247:
3236:
3229:
3225:
3218:
3192:
3184:
3177:
3173:
3169:
3156:
3151:
3147:
3142:
3135:
3131:
3129:
3116:
3108:
3100:
3093:
3089:
3085:
3079:
3076:
3070:
3067:
3060:
3056:
3050:
3039:
3036:
3033:
3024:
3022:
2995:
2988:
2979:
2976:
2970:
2967:
2964:
2957:
2952:
2947:
2938:
2934:
2925:
2918:
2907:
2887:
2884:
2881:
2871:
2866:
2863:
2860:
2856:
2846:
2842:
2836:
2824:
2816:
2813:
2810:
2807:
2797:
2792:
2789:
2785:
2775:
2771:
2765:
2753:
2746:
2741:
2735:
2729:
2716:
2709:
2704:
2699:
2695:
2688:
2684:
2677:
2673:
2666:
2661:
2651:
2645:
2638:
2635:Bound state (
2634:
2632:
2612:
2604:
2600:
2596:
2591:
2587:
2575:
2567:
2564:
2559:
2556:
2553:
2547:
2540:
2536:
2532:
2527:
2523:
2519:
2514:
2510:
2506:
2501:
2497:
2493:
2486:
2483:
2480:
2473:
2469:
2465:
2460:
2456:
2452:
2447:
2443:
2439:
2434:
2430:
2423:
2412:
2399:
2396:
2388:
2384:
2380:
2375:
2371:
2364:
2361:
2353:
2349:
2345:
2340:
2336:
2332:
2327:
2323:
2319:
2314:
2310:
2306:
2300:
2297:
2291:
2288:
2282:
2272:
2248:
2239:
2236:
2229:
2226:
2222:
2216:
2213:
2206:
2203:
2196:
2193:
2190:
2183:
2176:
2173:
2167:
2164:
2159:
2156:
2152:
2131:
2125:
2119:
2116:
2113:
2104:
2097:
2093:
2089:
2085:
2079:
2072:
2068:
2064:
2054:
2051:
2045:
2035:
2025:
2019:
2006:
2003:
2000:
1993:
1987:
1982:
1979:
1974:
1971:
1967:
1963:
1960:
1957:
1954:
1947:
1941:
1935:
1929:
1924:
1921:
1916:
1913:
1909:
1905:
1902:
1899:
1892:
1885:
1882:
1876:
1873:
1868:
1865:
1861:
1854:
1851:
1845:
1835:
1825:
1821:
1813:
1806:
1793:
1788:
1784:
1780:
1775:
1771:
1767:
1762:
1758:
1754:
1749:
1745:
1741:
1735:
1727:
1723:
1719:
1713:
1705:
1701:
1697:
1691:
1685:
1676:
1673:
1667:
1639:
1636:
1633:
1630:
1620:
1615:
1612:
1609:
1606:
1602:
1592:
1588:
1583:
1580:
1577:
1573:
1563:
1559:
1553:
1541:
1533:
1530:
1527:
1524:
1514:
1509:
1506:
1503:
1500:
1496:
1486:
1482:
1477:
1474:
1471:
1467:
1457:
1453:
1447:
1435:
1428:
1423:
1417:
1411:
1402:
1389:
1380:
1377:
1374:
1368:
1365:
1354:
1349:
1343:
1338:
1334:
1330:
1314:
1311:
1304:
1295:
1292:
1289:
1283:
1280:
1272:
1268:
1264:
1259:
1254:
1250:
1236:
1229:
1223:
1217:
1215:
1209:
1201:
1196:
1194:
1188:
1184:
1167:
1161:
1155:
1152:
1149:
1143:
1137:
1128:
1126:
1118:
1098:
1092:
1086:
1083:
1080:
1074:
1068:
1062:
1056:
1053:
1045:
1041:
1037:
1029:
1023:
1018:
1014:
1004:
1001:
995:
985:
975:
971:
967:
961:
957:
953:
952:wave function
949:
941:
934:
932:
929:
927:
923:
922:limiting case
918:
916:
912:
908:
904:
900:
888:
883:
881:
876:
874:
869:
868:
866:
865:
857:
854:
852:
849:
847:
844:
842:
839:
837:
834:
832:
829:
827:
824:
822:
819:
817:
814:
812:
809:
807:
804:
802:
799:
797:
794:
792:
789:
787:
784:
782:
779:
777:
774:
772:
769:
767:
764:
762:
759:
757:
754:
752:
749:
747:
744:
742:
739:
737:
734:
732:
729:
727:
724:
722:
719:
717:
714:
712:
709:
707:
704:
702:
699:
697:
694:
692:
689:
687:
684:
682:
679:
677:
674:
672:
669:
667:
664:
662:
659:
657:
654:
652:
649:
647:
644:
642:
639:
637:
634:
632:
629:
628:
621:
620:
613:
610:
608:
605:
603:
600:
598:
595:
593:
590:
588:
587:Quantum chaos
585:
583:
580:
578:
575:
573:
570:
568:
565:
564:
558:
557:
549:
546:
544:
543:Transactional
541:
539:
536:
534:
533:Quantum logic
531:
529:
526:
524:
521:
515:
512:
511:
510:
507:
506:
505:
502:
500:
497:
495:
492:
490:
487:
485:
482:
480:
477:
476:
472:
467:
466:
458:
455:
453:
450:
448:
445:
443:
440:
438:
435:
434:
427:
426:
418:
415:
413:
410:
408:
405:
403:
400:
398:
395:
393:
390:
389:
385:
382:
381:
375:
374:
366:
363:
361:
358:
356:
353:
352:
346:
343:
342:
341:
338:
337:
333:
330:
328:
325:
323:
320:
318:
315:
313:
310:
308:
305:
303:
300:
298:
295:
293:
290:
288:
285:
284:
277:
276:
266:
263:
262:
261:
260:Wave function
258:
256:
253:
251:
248:
246:
243:
241:
240:Superposition
238:
236:
233:
231:
228:
226:
223:
221:
218:
216:
213:
211:
208:
206:
203:
201:
198:
197:
190:
189:
181:
178:
176:
173:
172:
168:
165:
163:
160:
158:
155:
154:
148:
147:
142:
139:
137:
134:
132:
129:
128:
127:
126:
122:
89:
83:
66:
63:
59:
51:
44:
43:
40:
36:
32:
31:
19:
5651:
5633:
5596:
5592:
5586:
5565:
5555:
5547:
5508:(11): 1â49,
5505:
5501:
5495:
5484:. Retrieved
5480:
5471:
5375:
5337:
5335:
5326:
5322:
5318:
5285:
5279:
5160:
5155:
5151:
5144:
5139:
5135:
5053:
4805:
4445:atomic units
4438:
4434:
4151:
4144:
4123:
4115:
4109:
4100:
4098:
4018:
4007:
3993:
3990:
3744:
3509:
3306:
3299:
3289:
3283:
3276:
3266:
3256:
3245:
3234:
3230:
3223:
3216:
3213:
3154:
3149:
3145:
3133:
3132:Scattering (
3025:
2908:
2714:
2707:
2702:
2697:
2693:
2686:
2682:
2675:
2671:
2664:
2657:
2649:
2636:
2413:
2023:
2020:
1823:
1819:
1811:
1807:
1677:
1671:
1665:
1403:
1351:, where the
1347:
1341:
1234:
1227:
1224:
1221:
1207:
1199:
1197:
1186:
1182:
1129:
973:
969:
959:
955:
945:
930:
919:
902:
896:
442:KleinâGordon
378:Formulations
215:Energy level
210:Entanglement
193:Fundamentals
180:Interference
131:Introduction
5571:A. Dalgarno
4742:where
4124:double-well
2660:bound state
1353:wave number
831:von Neumann
816:Schrödinger
592:EPR paradox
523:Many-worlds
457:Schrödinger
412:Schrödinger
407:Phase-space
397:Interaction
302:Double-slit
280:Experiments
255:Uncertainty
225:Nonlocality
220:Measurement
205:Decoherence
175:Hamiltonian
5681:Categories
5486:2021-03-29
5464:References
3996:conducting
2144:because
1327:this is a
826:Sommerfeld
741:Heisenberg
736:Gutzwiller
676:de Broglie
624:Scientists
538:Relational
489:Copenhagen
392:Heisenberg
250:Tunnelling
151:Background
5621:0002-9505
5515:1302.0864
5353:−
5244:−
5230:±
5210:±
5106:±
5098:−
5090:±
5071:±
5037:±
4971:λ
4963:±
4952:−
4944:−
4923:λ
4917:−
4904:λ
4866:±
4854:λ
4829:λ
4821:±
4808:equation
4752:−
4717:−
4714:λ
4698:−
4690:λ
4672:−
4656:−
4605:−
4591:−
4546:−
4523:ψ
4493:λ
4455:ℏ
4420:∞
4365:−
4354:δ
4351:λ
4319:δ
4308:−
4261:ψ
4243:ψ
4198:ψ
4166:ℏ
4160:−
4067:ϕ
4055:ψ
4028:Ψ
3950:ℏ
3928:λ
3858:ℏ
3839:λ
3785:−
3710:λ
3685:ℏ
3625:λ
3586:ℏ
3485:−
3477:λ
3453:ℏ
3389:ℏ
3373:λ
3359:−
3189:ℏ
3174:λ
3105:ℏ
3090:λ
3080:−
3057:κ
3047:ℏ
3040:−
2985:ℏ
2980:λ
2971:−
2965:κ
2953:κ
2885:≥
2864:κ
2861:−
2825:ψ
2811:≤
2790:κ
2754:ψ
2730:ψ
2572:ℏ
2560:λ
2533:−
2494:−
2466:−
2453:−
2365:λ
2346:−
2307:−
2279:ℏ
2273:−
2240:ε
2237:−
2227:ψ
2223:−
2217:ε
2204:ψ
2174:ψ
2168:ε
2160:ε
2157:−
2153:∫
2120:ψ
2117:λ
2090:ψ
2086:−
2065:ψ
2042:ℏ
2036:−
1988:ψ
1983:ε
1975:ε
1972:−
1968:∫
1942:ψ
1925:ε
1917:ε
1914:−
1910:∫
1883:ψ
1877:ε
1869:ε
1866:−
1862:∫
1842:ℏ
1836:−
1724:ψ
1702:ψ
1686:ψ
1607:−
1542:ψ
1501:−
1436:ψ
1412:ψ
1385:ℏ
1312:ψ
1301:ℏ
1284:−
1260:ψ
1156:δ
1153:λ
1087:ψ
1069:ψ
1024:ψ
992:ℏ
986:−
966:potential
856:Zeilinger
701:Ehrenfest
430:Equations
107:⟩
104:Ψ
93:^
81:⟩
78:Ψ
55:ℏ
5540:56188533
5388:See also
5288:function
5284:Lambert
5161:ungerade
4447:and set
3512:symmetry
2230:′
2207:′
2177:″
2098:′
2073:′
1886:″
1222:Source:
950:for the
781:Millikan
706:Einstein
691:Davisson
646:Blackett
631:Aharonov
499:Ensemble
479:Bayesian
384:Overview
265:Collapse
245:Symmetry
136:Glossary
5601:Bibcode
5520:Bibcode
5319:unequal
4479:. Here
3136:> 0)
2679:√
2639:< 0)
1191:is the
1123:is the
924:of the
821:Simmons
811:Rydberg
776:Moseley
756:Kramers
746:Hilbert
731:Glauber
726:Feynman
711:Everett
681:Compton
452:Rydberg
141:History
5640:
5619:
5538:
5278:where
5146:gerade
4784:Thus,
4513:ansatz
4400:where
3226:> 0
3219:< 0
3157:> 0
2667:< 0
1818:[â
1237:> 0
1230:< 0
1180:where
1125:energy
1119:, and
1111:where
851:Zeeman
846:Wigner
796:Planck
766:Landau
751:Jordan
402:Matrix
332:Popper
5536:S2CID
5510:arXiv
4103:(see
1826:]
1331:with
905:is a
806:Raman
791:Pauli
786:Onnes
721:Fermi
696:Debye
686:Dirac
651:Bloch
641:Bethe
509:Local
447:Pauli
437:Dirac
235:State
5638:ISBN
5617:ISSN
5506:2012
4496:<
4490:<
4417:<
4411:<
3314:and
1634:>
1528:<
1345:and
1232:and
1195:.
913:- a
901:the
841:Wien
836:Weyl
801:Rabi
771:Laue
761:Lamb
716:Fock
671:Bose
666:Born
661:Bohr
656:Bohm
636:Bell
5609:doi
5528:doi
5426:or
5378:= 0
5333:).
5154:= â
5056:= 1
4437:= ±
4147:= 2
4107:).
4096:.
3295:= 0
3272:= 1
3221:or
2720:= 0
2652:= 0
2026:â 0
1822:, +
1814:= 0
1663:),
1339:of
1210:if
1202:if
978:is
897:In
5683::
5615:.
5607:.
5597:60
5595:.
5577:,
5534:,
5526:,
5518:,
5504:,
5479:.
5338:qR
5138:=
4441:/2
3305:=
3282:=
3023:.
2888:0.
2713:=
2698:iÎș
2696:=
2674:=
2669:,
2400:0.
1828::
5646:.
5623:.
5611::
5603::
5530::
5522::
5512::
5489:.
5376:E
5361:0
5358:=
5349:d
5327:W
5303:+
5299:d
5286:W
5280:W
5266:,
5263:R
5259:/
5255:)
5250:R
5247:q
5240:e
5236:R
5233:q
5227:(
5224:W
5221:+
5218:q
5215:=
5206:d
5182:+
5177:2
5173:H
5156:B
5152:A
5140:B
5136:A
5121:.
5117:]
5111:R
5102:d
5094:e
5087:1
5083:[
5079:q
5076:=
5067:d
5054:λ
5033:d
5029:=
5026:d
5006:,
5001:2
4997:/
4993:1
4988:}
4983:]
4977:R
4974:)
4968:(
4959:d
4955:2
4948:e
4941:1
4937:[
4931:2
4927:q
4920:4
4912:2
4908:)
4901:+
4898:1
4895:(
4890:2
4886:q
4881:{
4874:2
4871:1
4863:)
4860:1
4857:+
4851:(
4848:q
4843:2
4840:1
4835:=
4832:)
4826:(
4817:d
4792:d
4772:.
4767:2
4762:2
4758:d
4749:=
4746:E
4737:,
4734:0
4731:=
4726:|
4720:d
4711:q
4704:R
4701:d
4694:e
4687:q
4678:R
4675:d
4668:e
4664:q
4659:d
4653:q
4647:|
4625:.
4619:|
4613:2
4610:R
4602:x
4598:|
4594:d
4587:e
4583:B
4580:+
4574:|
4568:2
4565:R
4560:+
4557:x
4553:|
4549:d
4542:e
4538:A
4535:=
4532:)
4529:x
4526:(
4499:1
4487:0
4467:1
4464:=
4461:m
4458:=
4439:R
4435:x
4414:R
4408:0
4388:,
4384:]
4379:)
4373:2
4370:R
4362:x
4358:(
4348:+
4344:)
4338:2
4335:R
4330:+
4327:x
4323:(
4315:[
4311:q
4305:=
4302:)
4299:x
4296:(
4293:V
4273:,
4270:)
4267:x
4264:(
4258:E
4255:=
4252:)
4249:x
4246:(
4240:)
4237:x
4234:(
4231:V
4228:+
4220:2
4216:x
4212:d
4207:)
4204:x
4201:(
4193:2
4189:d
4179:m
4176:2
4170:2
4145:R
4101:D
4082:)
4079:z
4076:,
4073:y
4070:(
4064:)
4061:x
4058:(
4052:=
4049:)
4046:z
4043:,
4040:y
4037:,
4034:x
4031:(
4003:m
3972:.
3959:E
3954:2
3946:2
3932:2
3924:m
3913:+
3910:1
3898:1
3887:=
3872:2
3868:k
3862:4
3843:2
3833:2
3829:m
3817:+
3814:1
3802:1
3791:=
3788:R
3782:1
3779:=
3774:2
3769:|
3764:t
3760:|
3756:=
3753:T
3740:λ
3714:2
3706:m
3694:E
3689:2
3681:2
3670:+
3667:1
3655:1
3644:=
3629:2
3619:2
3615:m
3600:2
3596:k
3590:4
3574:+
3571:1
3559:1
3548:=
3543:2
3538:|
3533:r
3529:|
3525:=
3522:R
3496:.
3488:1
3474:m
3462:k
3457:2
3449:i
3429:1
3418:=
3415:r
3411:,
3398:k
3393:2
3385:i
3370:m
3356:1
3344:1
3333:=
3330:t
3320:t
3316:t
3312:r
3307:t
3303:r
3300:B
3293:l
3290:B
3284:r
3280:l
3277:A
3270:r
3267:A
3262:)
3260:r
3257:B
3255:(
3251:)
3249:l
3246:A
3244:(
3240:)
3238:r
3235:A
3233:(
3224:x
3217:x
3193:2
3185:2
3178:2
3170:m
3155:E
3150:R
3146:T
3134:E
3117:.
3109:2
3101:2
3094:2
3086:m
3077:=
3071:m
3068:2
3061:2
3051:2
3037:=
3034:E
3017:λ
2996:,
2989:2
2977:m
2968:=
2958:,
2948:=
2943:r
2939:B
2935:=
2930:l
2926:A
2919:{
2882:x
2872:,
2867:x
2857:e
2851:r
2847:B
2843:=
2840:)
2837:x
2834:(
2829:R
2817:,
2814:0
2808:x
2798:,
2793:x
2786:e
2780:l
2776:A
2772:=
2769:)
2766:x
2763:(
2758:L
2747:{
2742:=
2739:)
2736:x
2733:(
2718:l
2715:B
2711:r
2708:A
2703:x
2694:ħ
2692:/
2689:|
2687:E
2685:|
2683:m
2681:2
2676:i
2672:k
2665:E
2654:.
2650:x
2637:E
2613:.
2610:)
2605:l
2601:A
2597:+
2592:r
2588:A
2584:(
2576:2
2568:k
2565:i
2557:m
2554:2
2548:=
2541:l
2537:B
2528:r
2524:B
2520:+
2515:l
2511:A
2507:+
2502:r
2498:A
2487:,
2484:0
2481:=
2474:l
2470:B
2461:r
2457:B
2448:l
2444:A
2440:+
2435:r
2431:A
2424:{
2397:=
2394:)
2389:l
2385:A
2381:+
2376:r
2372:A
2368:(
2362:+
2359:)
2354:l
2350:B
2341:r
2337:B
2333:+
2328:l
2324:A
2320:+
2315:r
2311:A
2304:(
2301:k
2298:i
2292:m
2289:2
2283:2
2263:Ï
2249:.
2246:]
2243:)
2234:(
2220:)
2214:+
2211:(
2200:[
2197:=
2194:x
2191:d
2187:)
2184:x
2181:(
2165:+
2132:,
2129:)
2126:0
2123:(
2114:+
2111:]
2108:)
2105:0
2102:(
2094:L
2083:)
2080:0
2077:(
2069:R
2061:[
2055:m
2052:2
2046:2
2024:Δ
2007:.
2004:x
2001:d
1997:)
1994:x
1991:(
1980:+
1964:E
1961:=
1958:x
1955:d
1951:)
1948:x
1945:(
1939:)
1936:x
1933:(
1930:V
1922:+
1906:+
1903:x
1900:d
1896:)
1893:x
1890:(
1874:+
1855:m
1852:2
1846:2
1824:Δ
1820:Δ
1812:x
1794:,
1789:l
1785:B
1781:+
1776:r
1772:B
1768:=
1763:l
1759:A
1755:+
1750:r
1746:A
1742:=
1739:)
1736:0
1733:(
1728:R
1720:=
1717:)
1714:0
1711:(
1706:L
1698:=
1695:)
1692:0
1689:(
1672:e
1666:e
1661:k
1640:,
1637:0
1631:x
1621:,
1616:x
1613:k
1610:i
1603:e
1597:l
1593:B
1589:+
1584:x
1581:k
1578:i
1574:e
1568:r
1564:B
1560:=
1557:)
1554:x
1551:(
1546:R
1534:,
1531:0
1525:x
1515:,
1510:x
1507:k
1504:i
1497:e
1491:l
1487:A
1483:+
1478:x
1475:k
1472:i
1468:e
1462:r
1458:A
1454:=
1451:)
1448:x
1445:(
1440:L
1429:{
1424:=
1421:)
1418:x
1415:(
1390:.
1381:E
1378:m
1375:2
1369:=
1366:k
1356:k
1348:e
1342:e
1315:;
1305:2
1296:E
1293:m
1290:2
1281:=
1273:2
1269:x
1265:d
1255:2
1251:d
1235:x
1228:x
1212:λ
1204:λ
1189:)
1187:x
1185:(
1183:ÎŽ
1168:,
1165:)
1162:x
1159:(
1150:=
1147:)
1144:x
1141:(
1138:V
1121:E
1113:ħ
1099:,
1096:)
1093:x
1090:(
1084:E
1081:=
1078:)
1075:x
1072:(
1066:)
1063:x
1060:(
1057:V
1054:+
1046:2
1042:x
1038:d
1033:)
1030:x
1027:(
1019:2
1015:d
1005:m
1002:2
996:2
976:)
974:x
972:(
970:V
962:)
960:x
958:(
956:Ï
886:e
879:t
872:v
100:|
90:H
84:=
74:|
67:t
64:d
60:d
52:i
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.