2240:
3782:
1913:
1434:
132:
5653:
1622:
1078:
2235:{\displaystyle {\begin{aligned}\psi _{L}(0)&=\psi _{C}(0)\\\left.{\frac {d\psi _{L}}{dx}}\right|_{x=0}&=\left.{\frac {d\psi _{C}}{dx}}\right|_{x=0}\\\psi _{C}(a)&=\psi _{R}(a)\\\left.{\frac {d\psi _{C}}{dx}}\right|_{x=a}&=\left.{\frac {d\psi _{R}}{dx}}\right|_{x=a}.\end{aligned}}}
31:
5932:(STM) relies on this tunneling effect. In that case, the barrier is due to the gap between the tip of the STM and the underlying object. Since the tunnel current depends exponentially on the barrier width, this device is extremely sensitive to height variations on the examined sample.
5390:
826:
5925:. 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 barrier potential as above. Electrons may then tunnel from one material to the other giving rise to a current.
5935:
The above model is one-dimensional, while space is three-dimensional. 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; they are
5858:
2788:
3615:
1443:
3400:
4330:
4012:
2576:
1429:{\displaystyle {\begin{cases}\psi _{L}(x)=A_{r}e^{ik_{0}x}+A_{l}e^{-ik_{0}x}&x<0\\\psi _{C}(x)=B_{r}e^{ik_{1}x}+B_{l}e^{-ik_{1}x}&0<x<a\\\psi _{R}(x)=C_{r}e^{ik_{0}x}+C_{l}e^{-ik_{0}x}&x>a\end{cases}}}
706:
5102:
2580:
974:
3404:
5207:
5897:
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. One such example are interfaces between two
5648:{\displaystyle \lim _{x\to 1}f(x)=\lim _{x\to 1}{\frac {\sinh(v_{0}{\sqrt {1-x}})}{(1-x)}}=\lim _{x\to 1}{\frac {{\frac {d}{dx}}\sinh(v_{0}{\sqrt {1-x}})}{{\frac {d}{dx}}{\sqrt {1-x}}}}=v_{0}\cosh(0)=v_{0}}
106:
a particle actually behaving as a matter wave has a non-zero probability of penetrating the barrier and continuing its travel as a wave on the other side. In classical wave-physics, this effect is known as
697:
5344:
379:
4870:
2797:
At this point, it is instructive to compare the situation to the classical case. In both cases, the particle behaves as a free particle outside of the barrier region. A classical particle with energy
2413:
4100:. The transmission is exponentially suppressed with the barrier width, which can be understood from the functional form of the wave function: Outside of the barrier it oscillates with wave vector
5711:
4408:
4092:
3205:
1918:
6012:
6065:
1033:
2310:
3831:
4985:
4188:
3873:
6021:, which can be regarded as a special case of the finite potential barrier. All results from this article immediately apply to the delta potential barrier by taking the limits
6101:
4503:
4697:
4659:
2417:
4738:
3683:
3153:
1780:
5262:
4620:
4445:
3868:
2879:
1617:{\displaystyle {\begin{cases}k_{0}={\sqrt {2mE/\hbar ^{2}}}&x<0\quad {\text{or}}\quad x>a\\k_{1}={\sqrt {2m(E-V_{0})/\hbar ^{2}}}&0<x<a.\end{cases}}}
1074:
594:
561:
408:
193:
226:
162:. The amplitudes and direction of left and right moving waves are indicated. In red, those waves used for the derivation of the reflection and transmission amplitude.
4788:
4164:
3755:
3719:
3113:
3080:
3047:
3014:
1813:
4535:
3648:
432:
5887:
5131:
4943:
4126:
2977:
2946:
2915:
2842:
1848:
1719:
160:
102:
5684:
5379:
5016:
4587:
4561:
1904:
1878:
1747:
1652:
880:
854:
5923:
5704:
5229:
4916:
4896:
3194:
3173:
2815:
1692:
1672:
900:
614:
480:
456:
1035:). In any of these parts, the potential is constant, meaning that the particle is quasi-free, and the solution of the Schrödinger equation can be written as a
5902:
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
5023:
5138:
4168:
If the barrier is much wider than this decay length, the left and right part are virtually independent and tunneling as a consequence is suppressed.
5269:
231:
4793:
3623:
of the model, the amplitudes for incidence from the right are the same as those from the left. Note that these expressions hold for any energy
2314:
4740:
and 1.8 in the above figure). Note that the probabilities and amplitudes as written are for any energy (above/below) the barrier height.
6187:
905:
57:(also called "quantum tunneling") and wave-mechanical reflection. The problem consists of solving the one-dimensional time-independent
821:{\displaystyle V(x)={\begin{cases}0&{\text{if }}x<0\\V_{0}&{\text{if }}0<x<a\\0&{\text{if }}a<x\end{cases}}}
619:
6236:
6202:
2249:
5853:{\displaystyle \lim _{x\to 1}T(x)=\lim _{x\to 1}{\frac {1}{1+{\frac {f(x)^{2}}{4x}}}}={\frac {1}{1+{\frac {v_{0}^{2}}{4}}}}}
2783:{\displaystyle ik_{1}\left(B_{r}e^{iak_{1}}-B_{l}e^{-iak_{1}}\right)=ik_{0}\left(C_{r}e^{iak_{0}}-C_{l}e^{-iak_{0}}\right).}
5940:. The Schrödinger equation may then be reduced to the case considered here by an ansatz for the wave function of the type:
5903:
4950:
3610:{\displaystyle r={\frac {(k_{0}^{2}-k_{1}^{2})\sin(ak_{1})}{2ik_{0}k_{1}\cos(ak_{1})+(k_{0}^{2}+k_{1}^{2})\sin(ak_{1})}}.}
1721:
becomes imaginary and the wave function is exponentially decaying within the barrier. Nevertheless, we keep the notation
4336:
4020:
6151:
6175:
5943:
17:
4450:
1694:
denotes the direction of the velocity vector. Note that, if the energy of the particle is below the barrier height,
2888:
To study the quantum case, consider the following situation: a particle incident on the barrier from the left side
6231:
2983:
To find the amplitudes for reflection and transmission for incidence from the left, we put in the above equations
411:
485:
6024:
982:
5929:
3395:{\displaystyle t={\frac {4k_{0}k_{1}e^{-ia(k_{0}-k_{1})}}{(k_{0}+k_{1})^{2}-e^{2iak_{1}}(k_{0}-k_{1})^{2}}}}
6018:
3788:
2244:
Inserting the wave functions, the boundary conditions give the following restrictions on the coefficients
6226:
6112:
4874:
This expression can be obtained by calculating the transmission coefficient from the constants stated
6070:
5382:
1452:
1087:
730:
112:
108:
4664:
4625:
6221:
5937:
5899:
4702:
3655:
3118:
1752:
700:
5234:
4592:
4417:
3840:
2851:
1046:
566:
384:
165:
120:
58:
6194:
116:
202:
6122:
5889:
in the evaluated value for the limit, the above expression for T is successfully reproduced.
4760:
4134:
3727:
3691:
3085:
3052:
3019:
2986:
1785:
1036:
6186:
4510:
3627:
417:
5865:
5109:
4921:
4325:{\displaystyle T=|t|^{2}={\frac {1}{1+{\frac {V_{0}^{2}\sin ^{2}(k_{1}a)}{4E(E-V_{0})}}}},}
4104:
4007:{\displaystyle T=|t|^{2}={\frac {1}{1+{\frac {V_{0}^{2}\sinh ^{2}(k_{1}a)}{4E(V_{0}-E)}}}}}
2955:
2924:
2893:
2820:
1821:
1697:
138:
80:
5660:
5355:
4992:
8:
5349:
4566:
4540:
1907:
1883:
1857:
1724:
1629:
859:
833:
5908:
5689:
5214:
4901:
4881:
3179:
3158:
2800:
1851:
1677:
1657:
885:
599:
465:
441:
6198:
6171:
6147:
4097:
2571:{\displaystyle B_{r}e^{iak_{1}}+B_{l}e^{-iak_{1}}=C_{r}e^{iak_{0}}+C_{l}e^{-iak_{0}}}
54:
38:
6193:. transl. from the French by Susan Reid Hemley. Wiley-Interscience: Wiley. pp.
111:. The likelihood that the particle will pass through the barrier is given by the
435:
1749:
even though the waves are not propagating anymore in this case. Here we assumed
6117:
3833:= 1, 3, and 7. Dashed: classical result. Solid line: quantum mechanical result.
4447:, the particle may be reflected from the barrier with a non-zero probability
6215:
1040:
66:
4507:
The transmission and reflection probabilities are in fact oscillating with
6185:
Cohen-Tannoudji, Claude; Diu, Bernard; Laloë, Franck; et al. (1996).
902:
position without changing the results. The first term in the
Hamiltonian,
3837:
The surprising result is that for energies less than the barrier height,
1437:
53:
is a standard one-dimensional problem that demonstrates the phenomena of
4414:
Equally surprising is that for energies larger than the barrier height,
4537:. The classical result of perfect transmission without any reflection (
73:
4130:
whereas within the barrier it is exponentially damped over a distance
5097:{\displaystyle f(x)={\frac {\sinh(v_{0}{\sqrt {1-x}})}{\sqrt {1-x}}}}
62:
3620:
30:
969:{\textstyle -{\frac {\hbar ^{2}}{2m}}{\frac {d^{2}}{dx^{2}}}\psi }
5202:{\displaystyle v_{0}={\sqrt {\frac {2mV_{0}a^{2}}{\hbar ^{2}}}}}
3785:
Transmission probability through a finite potential barrier for
199:
The time-independent Schrödinger equation for the wave function
4096:
This effect, which differs from the classical case, is called
115:, whereas the likelihood that it is reflected is given by the
4016:
for the particle to be transmitted through the barrier, with
692:{\displaystyle \Theta (x)=0,\;x<0;\;\Theta (x)=1,\;x>0}
131:
3781:
2180:
2126:
2026:
1972:
1610:
1422:
814:
459:
5339:{\displaystyle T(x)={\frac {1}{1+{\frac {f(x)^{2}}{4x}}}}}
374:{\displaystyle {\hat {H}}\psi (x)=\left\psi (x)=E\psi (x)}
4865:{\displaystyle T={\frac {1}{1+ma^{2}V_{0}/2\hbar ^{2}}}.}
5211:
These definitions can be inserted in the expression for
3759:
so there is a singularity in both of these expressions.
6184:
3762:
2408:{\displaystyle ik_{0}(A_{r}-A_{l})=ik_{1}(B_{r}-B_{l})}
65:
energy barrier. It is usually assumed, as here, that a
4339:
4023:
3791:
908:
6073:
6027:
5946:
5911:
5868:
5714:
5692:
5663:
5393:
5358:
5272:
5237:
5217:
5141:
5112:
5026:
4995:
4953:
4924:
4904:
4884:
4796:
4763:
4705:
4667:
4628:
4595:
4589:) is reproduced not only in the limit of high energy
4569:
4543:
4513:
4453:
4420:
4191:
4137:
4107:
3876:
3843:
3730:
3694:
3658:
3630:
3407:
3208:
3182:
3161:
3121:
3088:
3055:
3022:
2989:
2958:
2927:
2896:
2854:
2823:
2803:
2583:
2420:
2317:
2252:
1916:
1886:
1860:
1824:
1788:
1755:
1727:
1700:
1680:
1660:
1632:
1446:
1081:
1049:
985:
888:
862:
836:
709:
622:
602:
569:
488:
468:
444:
420:
387:
234:
205:
168:
141:
83:
6165:
4622:but also when the energy and barrier width satisfy
4403:{\textstyle k_{1}={\sqrt {2m(E-V_{0})/\hbar ^{2}}}}
4087:{\textstyle k_{1}={\sqrt {2m(V_{0}-E)/\hbar ^{2}}}}
3115:(transmission). We then eliminate the coefficients
135:Scattering at a finite potential barrier of height
6095:
6059:
6006:
5917:
5881:
5852:
5698:
5678:
5647:
5373:
5338:
5256:
5223:
5201:
5125:
5096:
5010:
4979:
4937:
4910:
4890:
4864:
4782:
4732:
4691:
4653:
4614:
4581:
4555:
4529:
4497:
4439:
4402:
4324:
4158:
4120:
4086:
4006:
3862:
3825:
3749:
3713:
3677:
3642:
3609:
3394:
3188:
3167:
3147:
3107:
3074:
3041:
3008:
2971:
2940:
2909:
2873:
2836:
2809:
2782:
2570:
2407:
2304:
2234:
1906:. The wave function and its derivative have to be
1898:
1872:
1842:
1807:
1774:
1741:
1713:
1686:
1666:
1646:
1616:
1428:
1068:
1027:
968:
894:
874:
848:
820:
691:
608:
588:
555:
474:
450:
426:
402:
373:
220:
187:
154:
96:
4878:as for the other cases or by taking the limit of
6213:
5747:
5716:
5503:
5426:
5395:
2848:pass the barrier, and a classical particle with
27:Area, where a potential exhibits a local maximum
6007:{\displaystyle \Psi (x,y,z)=\psi (x)\phi (y,z)}
6141:
2792:
979:The barrier divides the space in three parts (
76:would be reflected if its energy is less than
72:Although classically a particle behaving as a
6017:For another, related model of a barrier, see
123:allows these coefficients to be calculated.
6168:Introduction to Quantum Mechanics (2nd ed.)
5892:
3082:(no incoming particle from the right), and
6146:(1st ed.). University Science Books.
6047:
4989:is defined, which is used in the function
679:
657:
644:
61:for a particle encountering a rectangular
6144:Physical Chemistry - A molecular Approach
6060:{\displaystyle V_{0}\to \infty ,\;a\to 0}
1028:{\displaystyle x<0,0<x<a,x>a}
5862:By plugging in the above expression for
3780:
130:
29:
2305:{\displaystyle A_{r}+A_{l}=B_{r}+B_{l}}
69:impinges on the barrier from the left.
14:
6214:
3826:{\textstyle {\sqrt {2mV_{0}}}a/\hbar }
563:is the barrier potential with height
4980:{\displaystyle x={\frac {E}{V_{0}}}}
3763:Analysis of the obtained expressions
1039:of left and right moving waves (see
882:. The barrier can be shifted to any
24:
6041:
5947:
830:The barrier is positioned between
658:
623:
532:
517:
25:
6248:
3820:
5231:which was obtained for the case
4757:The transmission probability at
3870:there is a non-zero probability
3155:from the equation and solve for
6142:McQuarrie DA, Simon JD (1997).
6096:{\displaystyle V_{0}a=\lambda }
2817:larger than the barrier height
1513:
1507:
482:the energy of the particle and
6135:
6051:
6038:
6001:
5989:
5983:
5977:
5968:
5950:
5787:
5780:
5754:
5740:
5734:
5723:
5706:approaches 1 can be obtained:
5673:
5667:
5629:
5623:
5568:
5542:
5510:
5493:
5481:
5476:
5450:
5433:
5419:
5413:
5402:
5368:
5362:
5313:
5306:
5282:
5276:
5077:
5051:
5036:
5030:
5005:
4999:
4945:. For this purpose the ratio
4498:{\displaystyle R=|r|^{2}=1-T.}
4470:
4461:
4380:
4361:
4310:
4291:
4280:
4264:
4208:
4199:
4064:
4045:
3995:
3976:
3965:
3949:
3893:
3884:
3598:
3582:
3573:
3537:
3531:
3515:
3478:
3462:
3453:
3417:
3380:
3353:
3315:
3288:
3281:
3255:
2881:incident on the barrier would
2402:
2376:
2357:
2331:
2117:
2111:
2091:
2085:
1963:
1957:
1937:
1931:
1567:
1548:
1440:are related to the energy via
1330:
1324:
1215:
1209:
1106:
1100:
719:
713:
667:
661:
632:
626:
550:
547:
535:
526:
520:
514:
498:
492:
394:
368:
362:
350:
344:
333:
327:
256:
250:
241:
215:
209:
126:
13:
1:
6237:Quantum mechanical potentials
6128:
5930:scanning tunneling microscope
6166:Griffiths, David J. (2004).
6019:Delta potential barrier (QM)
4692:{\displaystyle n=1,2,\dots }
4654:{\displaystyle k_{1}a=n\pi }
4171:
3767:
7:
6106:
4875:
4733:{\displaystyle E/V_{0}=1.2}
3678:{\displaystyle E\neq V_{0}}
3148:{\displaystyle B_{l},B_{r}}
2793:Transmission and reflection
1775:{\displaystyle E\neq V_{0}}
121:Schrödinger's wave-equation
10:
6253:
6113:Morse/Long-range potential
5348:Now, when calculating the
5257:{\displaystyle E<V_{0}}
4615:{\displaystyle E\gg V_{0}}
4440:{\displaystyle E>V_{0}}
3863:{\displaystyle E<V_{0}}
2874:{\displaystyle E<V_{0}}
1850:have to be found from the
1069:{\displaystyle E>V_{0}}
589:{\displaystyle V_{0}>0}
556:{\displaystyle V(x)=V_{0}}
403:{\displaystyle {\hat {H}}}
188:{\displaystyle E>V_{0}}
5381:as x approaches 1 (using
4743:
55:wave-mechanical tunneling
5893:Remarks and applications
1854:of the wave function at
221:{\displaystyle \psi (x)}
113:transmission coefficient
109:evanescent wave coupling
5133:is defined as follows:
4783:{\displaystyle E=V_{0}}
4159:{\displaystyle 1/k_{1}}
3750:{\displaystyle k_{1}=0}
3714:{\displaystyle E=V_{0}}
3108:{\displaystyle C_{r}=t}
3075:{\displaystyle C_{l}=0}
3042:{\displaystyle A_{l}=r}
3009:{\displaystyle A_{r}=1}
1808:{\displaystyle E=V_{0}}
976:is the kinetic energy.
701:Heaviside step function
6097:
6061:
6008:
5919:
5883:
5854:
5700:
5680:
5649:
5375:
5340:
5258:
5225:
5203:
5127:
5098:
5012:
4981:
4939:
4912:
4892:
4866:
4784:
4734:
4693:
4655:
4616:
4583:
4557:
4531:
4530:{\displaystyle k_{1}a}
4499:
4441:
4404:
4326:
4160:
4122:
4088:
4008:
3864:
3834:
3827:
3751:
3715:
3679:
3644:
3643:{\displaystyle E>0}
3611:
3396:
3190:
3169:
3149:
3109:
3076:
3043:
3010:
2973:
2942:
2911:
2875:
2838:
2811:
2784:
2572:
2409:
2306:
2236:
1900:
1874:
1844:
1809:
1776:
1743:
1715:
1688:
1668:
1648:
1618:
1430:
1070:
1029:
970:
896:
876:
850:
822:
693:
610:
590:
557:
476:
452:
428:
427:{\displaystyle \hbar }
404:
375:
222:
196:
195:for this illustration.
189:
156:
117:reflection coefficient
98:
34:
6123:Finite potential well
6098:
6062:
6009:
5920:
5884:
5882:{\displaystyle v_{0}}
5855:
5701:
5681:
5650:
5376:
5341:
5259:
5226:
5204:
5128:
5126:{\displaystyle v_{0}}
5106:In the last equation
5099:
5013:
4982:
4940:
4938:{\displaystyle V_{0}}
4913:
4893:
4867:
4785:
4735:
4694:
4656:
4617:
4584:
4558:
4532:
4500:
4442:
4405:
4327:
4161:
4123:
4121:{\displaystyle k_{0}}
4089:
4009:
3865:
3828:
3784:
3752:
3716:
3680:
3645:
3612:
3397:
3191:
3170:
3150:
3110:
3077:
3044:
3016:(incoming particle),
3011:
2974:
2972:{\displaystyle C_{r}}
2943:
2941:{\displaystyle A_{l}}
2912:
2910:{\displaystyle A_{r}}
2876:
2839:
2837:{\displaystyle V_{0}}
2812:
2785:
2573:
2410:
2307:
2237:
1901:
1875:
1845:
1843:{\displaystyle A,B,C}
1810:
1777:
1744:
1716:
1714:{\displaystyle k_{1}}
1689:
1669:
1649:
1619:
1431:
1071:
1030:
971:
897:
877:
851:
823:
694:
611:
591:
558:
477:
453:
429:
405:
376:
223:
190:
157:
155:{\displaystyle V_{0}}
134:
99:
97:{\displaystyle V_{0}}
33:
6232:Schrödinger equation
6071:
6025:
5944:
5909:
5866:
5712:
5690:
5679:{\displaystyle T(x)}
5661:
5391:
5374:{\displaystyle f(x)}
5356:
5270:
5235:
5215:
5139:
5110:
5024:
5011:{\displaystyle f(x)}
4993:
4951:
4922:
4902:
4882:
4794:
4761:
4703:
4665:
4626:
4593:
4567:
4541:
4511:
4451:
4418:
4337:
4189:
4135:
4105:
4021:
3874:
3841:
3789:
3728:
3692:
3656:
3628:
3405:
3206:
3180:
3159:
3119:
3086:
3053:
3020:
2987:
2956:
2925:
2919:It may be reflected
2894:
2852:
2821:
2801:
2581:
2418:
2315:
2250:
1914:
1884:
1858:
1822:
1786:
1753:
1725:
1698:
1678:
1658:
1654:on the coefficients
1630:
1444:
1079:
1047:
983:
906:
886:
860:
834:
707:
620:
600:
567:
486:
466:
442:
418:
385:
232:
203:
166:
139:
81:
59:Schrödinger equation
5928:The operation of a
5841:
4582:{\displaystyle R=0}
4556:{\displaystyle T=1}
4250:
3935:
3572:
3554:
3452:
3434:
1899:{\displaystyle x=a}
1873:{\displaystyle x=0}
1852:boundary conditions
1742:{\displaystyle r/l}
1647:{\displaystyle r/l}
875:{\displaystyle x=a}
849:{\displaystyle x=0}
6093:
6057:
6004:
5915:
5879:
5850:
5827:
5761:
5730:
5696:
5676:
5657:also the limit of
5645:
5517:
5440:
5409:
5371:
5336:
5254:
5221:
5199:
5123:
5094:
5008:
4977:
4935:
4908:
4888:
4862:
4780:
4730:
4689:
4651:
4612:
4579:
4553:
4527:
4495:
4437:
4400:
4322:
4236:
4156:
4118:
4084:
4004:
3921:
3860:
3835:
3823:
3747:
3711:
3675:
3640:
3619:Due to the mirror
3607:
3558:
3540:
3438:
3420:
3392:
3186:
3165:
3145:
3105:
3072:
3039:
3006:
2969:
2938:
2907:
2871:
2834:
2807:
2780:
2568:
2405:
2302:
2232:
2230:
1896:
1870:
1840:
1815:is treated below.
1805:
1772:
1739:
1711:
1684:
1664:
1644:
1614:
1609:
1426:
1421:
1066:
1025:
966:
892:
872:
846:
818:
813:
689:
606:
586:
553:
472:
448:
424:
400:
371:
218:
197:
185:
152:
94:
35:
6227:Scattering theory
6204:978-0-471-56952-7
6189:Quantum mechanics
6170:. Prentice Hall.
5918:{\displaystyle m}
5848:
5845:
5808:
5805:
5746:
5715:
5699:{\displaystyle x}
5602:
5599:
5586:
5566:
5534:
5502:
5497:
5474:
5425:
5394:
5334:
5331:
5224:{\displaystyle T}
5197:
5196:
5092:
5091:
5075:
4975:
4911:{\displaystyle E}
4891:{\displaystyle T}
4857:
4398:
4317:
4314:
4098:quantum tunneling
4082:
4002:
3999:
3810:
3602:
3390:
3189:{\displaystyle t}
3168:{\displaystyle r}
2810:{\displaystyle E}
2207:
2153:
2053:
1999:
1818:The coefficients
1687:{\displaystyle B}
1667:{\displaystyle A}
1585:
1511:
1494:
961:
932:
895:{\displaystyle x}
800:
771:
741:
609:{\displaystyle a}
475:{\displaystyle E}
451:{\displaystyle m}
434:is the (reduced)
397:
319:
290:
244:
51:potential barrier
39:quantum mechanics
18:Potential barrier
16:(Redirected from
6244:
6208:
6192:
6181:
6158:
6157:
6139:
6102:
6100:
6099:
6094:
6083:
6082:
6066:
6064:
6063:
6058:
6037:
6036:
6013:
6011:
6010:
6005:
5924:
5922:
5921:
5916:
5888:
5886:
5885:
5880:
5878:
5877:
5859:
5857:
5856:
5851:
5849:
5847:
5846:
5840:
5835:
5826:
5814:
5809:
5807:
5806:
5804:
5796:
5795:
5794:
5775:
5763:
5760:
5729:
5705:
5703:
5702:
5697:
5685:
5683:
5682:
5677:
5654:
5652:
5651:
5646:
5644:
5643:
5616:
5615:
5603:
5601:
5600:
5589:
5587:
5585:
5574:
5571:
5567:
5556:
5554:
5553:
5535:
5533:
5522:
5519:
5516:
5498:
5496:
5479:
5475:
5464:
5462:
5461:
5442:
5439:
5408:
5383:L'HĂ´pital's rule
5380:
5378:
5377:
5372:
5345:
5343:
5342:
5337:
5335:
5333:
5332:
5330:
5322:
5321:
5320:
5301:
5289:
5263:
5261:
5260:
5255:
5253:
5252:
5230:
5228:
5227:
5222:
5208:
5206:
5205:
5200:
5198:
5195:
5194:
5185:
5184:
5183:
5174:
5173:
5157:
5156:
5151:
5150:
5132:
5130:
5129:
5124:
5122:
5121:
5103:
5101:
5100:
5095:
5093:
5081:
5080:
5076:
5065:
5063:
5062:
5043:
5017:
5015:
5014:
5009:
4986:
4984:
4983:
4978:
4976:
4974:
4973:
4961:
4944:
4942:
4941:
4936:
4934:
4933:
4917:
4915:
4914:
4909:
4897:
4895:
4894:
4889:
4871:
4869:
4868:
4863:
4858:
4856:
4855:
4854:
4842:
4837:
4836:
4827:
4826:
4804:
4789:
4787:
4786:
4781:
4779:
4778:
4739:
4737:
4736:
4731:
4723:
4722:
4713:
4699:(see peaks near
4698:
4696:
4695:
4690:
4660:
4658:
4657:
4652:
4638:
4637:
4621:
4619:
4618:
4613:
4611:
4610:
4588:
4586:
4585:
4580:
4562:
4560:
4559:
4554:
4536:
4534:
4533:
4528:
4523:
4522:
4504:
4502:
4501:
4496:
4479:
4478:
4473:
4464:
4446:
4444:
4443:
4438:
4436:
4435:
4411:
4409:
4407:
4406:
4401:
4399:
4397:
4396:
4387:
4379:
4378:
4354:
4349:
4348:
4331:
4329:
4328:
4323:
4318:
4316:
4315:
4313:
4309:
4308:
4283:
4276:
4275:
4260:
4259:
4249:
4244:
4234:
4222:
4217:
4216:
4211:
4202:
4167:
4165:
4163:
4162:
4157:
4155:
4154:
4145:
4129:
4127:
4125:
4124:
4119:
4117:
4116:
4095:
4093:
4091:
4090:
4085:
4083:
4081:
4080:
4071:
4057:
4056:
4038:
4033:
4032:
4013:
4011:
4010:
4005:
4003:
4001:
4000:
3998:
3988:
3987:
3968:
3961:
3960:
3945:
3944:
3934:
3929:
3919:
3907:
3902:
3901:
3896:
3887:
3869:
3867:
3866:
3861:
3859:
3858:
3832:
3830:
3829:
3824:
3819:
3811:
3809:
3808:
3793:
3758:
3756:
3754:
3753:
3748:
3740:
3739:
3722:
3720:
3718:
3717:
3712:
3710:
3709:
3686:
3684:
3682:
3681:
3676:
3674:
3673:
3651:
3649:
3647:
3646:
3641:
3616:
3614:
3613:
3608:
3603:
3601:
3597:
3596:
3571:
3566:
3553:
3548:
3530:
3529:
3508:
3507:
3498:
3497:
3481:
3477:
3476:
3451:
3446:
3433:
3428:
3415:
3401:
3399:
3398:
3393:
3391:
3389:
3388:
3387:
3378:
3377:
3365:
3364:
3352:
3351:
3350:
3349:
3323:
3322:
3313:
3312:
3300:
3299:
3286:
3285:
3284:
3280:
3279:
3267:
3266:
3240:
3239:
3230:
3229:
3216:
3197:
3195:
3193:
3192:
3187:
3174:
3172:
3171:
3166:
3154:
3152:
3151:
3146:
3144:
3143:
3131:
3130:
3114:
3112:
3111:
3106:
3098:
3097:
3081:
3079:
3078:
3073:
3065:
3064:
3048:
3046:
3045:
3040:
3032:
3031:
3015:
3013:
3012:
3007:
2999:
2998:
2980:
2978:
2976:
2975:
2970:
2968:
2967:
2949:
2947:
2945:
2944:
2939:
2937:
2936:
2918:
2916:
2914:
2913:
2908:
2906:
2905:
2880:
2878:
2877:
2872:
2870:
2869:
2843:
2841:
2840:
2835:
2833:
2832:
2816:
2814:
2813:
2808:
2789:
2787:
2786:
2781:
2776:
2772:
2771:
2770:
2769:
2768:
2745:
2744:
2732:
2731:
2730:
2729:
2709:
2708:
2694:
2693:
2678:
2674:
2673:
2672:
2671:
2670:
2647:
2646:
2634:
2633:
2632:
2631:
2611:
2610:
2596:
2595:
2577:
2575:
2574:
2569:
2567:
2566:
2565:
2564:
2541:
2540:
2528:
2527:
2526:
2525:
2505:
2504:
2492:
2491:
2490:
2489:
2466:
2465:
2453:
2452:
2451:
2450:
2430:
2429:
2414:
2412:
2411:
2406:
2401:
2400:
2388:
2387:
2375:
2374:
2356:
2355:
2343:
2342:
2330:
2329:
2311:
2309:
2308:
2303:
2301:
2300:
2288:
2287:
2275:
2274:
2262:
2261:
2241:
2239:
2238:
2233:
2231:
2224:
2223:
2212:
2208:
2206:
2198:
2197:
2196:
2183:
2170:
2169:
2158:
2154:
2152:
2144:
2143:
2142:
2129:
2110:
2109:
2084:
2083:
2070:
2069:
2058:
2054:
2052:
2044:
2043:
2042:
2029:
2016:
2015:
2004:
2000:
1998:
1990:
1989:
1988:
1975:
1956:
1955:
1930:
1929:
1905:
1903:
1902:
1897:
1879:
1877:
1876:
1871:
1849:
1847:
1846:
1841:
1814:
1812:
1811:
1806:
1804:
1803:
1781:
1779:
1778:
1773:
1771:
1770:
1748:
1746:
1745:
1740:
1735:
1720:
1718:
1717:
1712:
1710:
1709:
1693:
1691:
1690:
1685:
1673:
1671:
1670:
1665:
1653:
1651:
1650:
1645:
1640:
1623:
1621:
1620:
1615:
1613:
1612:
1586:
1584:
1583:
1574:
1566:
1565:
1541:
1536:
1535:
1512:
1509:
1495:
1493:
1492:
1483:
1469:
1464:
1463:
1435:
1433:
1432:
1427:
1425:
1424:
1407:
1406:
1402:
1401:
1381:
1380:
1368:
1367:
1363:
1362:
1345:
1344:
1323:
1322:
1292:
1291:
1287:
1286:
1266:
1265:
1253:
1252:
1248:
1247:
1230:
1229:
1208:
1207:
1183:
1182:
1178:
1177:
1157:
1156:
1144:
1143:
1139:
1138:
1121:
1120:
1099:
1098:
1075:
1073:
1072:
1067:
1065:
1064:
1034:
1032:
1031:
1026:
975:
973:
972:
967:
962:
960:
959:
958:
945:
944:
935:
933:
931:
923:
922:
913:
901:
899:
898:
893:
881:
879:
878:
873:
855:
853:
852:
847:
827:
825:
824:
819:
817:
816:
801:
798:
772:
769:
765:
764:
742:
739:
698:
696:
695:
690:
615:
613:
612:
607:
595:
593:
592:
587:
579:
578:
562:
560:
559:
554:
513:
512:
481:
479:
478:
473:
457:
455:
454:
449:
433:
431:
430:
425:
409:
407:
406:
401:
399:
398:
390:
380:
378:
377:
372:
340:
336:
320:
318:
317:
316:
303:
302:
293:
291:
289:
281:
280:
271:
246:
245:
237:
227:
225:
224:
219:
194:
192:
191:
186:
184:
183:
161:
159:
158:
153:
151:
150:
105:
103:
101:
100:
95:
93:
92:
21:
6252:
6251:
6247:
6246:
6245:
6243:
6242:
6241:
6212:
6211:
6205:
6178:
6162:
6161:
6154:
6140:
6136:
6131:
6109:
6078:
6074:
6072:
6069:
6068:
6032:
6028:
6026:
6023:
6022:
5945:
5942:
5941:
5910:
5907:
5906:
5895:
5873:
5869:
5867:
5864:
5863:
5836:
5831:
5825:
5818:
5813:
5797:
5790:
5786:
5776:
5774:
5767:
5762:
5750:
5719:
5713:
5710:
5709:
5691:
5688:
5687:
5662:
5659:
5658:
5639:
5635:
5611:
5607:
5588:
5578:
5573:
5572:
5555:
5549:
5545:
5526:
5521:
5520:
5518:
5506:
5480:
5463:
5457:
5453:
5443:
5441:
5429:
5398:
5392:
5389:
5388:
5357:
5354:
5353:
5323:
5316:
5312:
5302:
5300:
5293:
5288:
5271:
5268:
5267:
5248:
5244:
5236:
5233:
5232:
5216:
5213:
5212:
5190:
5186:
5179:
5175:
5169:
5165:
5158:
5155:
5146:
5142:
5140:
5137:
5136:
5117:
5113:
5111:
5108:
5107:
5064:
5058:
5054:
5044:
5042:
5025:
5022:
5021:
4994:
4991:
4990:
4969:
4965:
4960:
4952:
4949:
4948:
4929:
4925:
4923:
4920:
4919:
4903:
4900:
4899:
4883:
4880:
4879:
4850:
4846:
4838:
4832:
4828:
4822:
4818:
4808:
4803:
4795:
4792:
4791:
4774:
4770:
4762:
4759:
4758:
4755:
4753:
4718:
4714:
4709:
4704:
4701:
4700:
4666:
4663:
4662:
4633:
4629:
4627:
4624:
4623:
4606:
4602:
4594:
4591:
4590:
4568:
4565:
4564:
4542:
4539:
4538:
4518:
4514:
4512:
4509:
4508:
4474:
4469:
4468:
4460:
4452:
4449:
4448:
4431:
4427:
4419:
4416:
4415:
4392:
4388:
4383:
4374:
4370:
4353:
4344:
4340:
4338:
4335:
4334:
4333:
4304:
4300:
4284:
4271:
4267:
4255:
4251:
4245:
4240:
4235:
4233:
4226:
4221:
4212:
4207:
4206:
4198:
4190:
4187:
4186:
4183:
4181:
4150:
4146:
4141:
4136:
4133:
4132:
4131:
4112:
4108:
4106:
4103:
4102:
4101:
4076:
4072:
4067:
4052:
4048:
4037:
4028:
4024:
4022:
4019:
4018:
4017:
3983:
3979:
3969:
3956:
3952:
3940:
3936:
3930:
3925:
3920:
3918:
3911:
3906:
3897:
3892:
3891:
3883:
3875:
3872:
3871:
3854:
3850:
3842:
3839:
3838:
3815:
3804:
3800:
3792:
3790:
3787:
3786:
3779:
3777:
3765:
3735:
3731:
3729:
3726:
3725:
3724:
3705:
3701:
3693:
3690:
3689:
3688:
3669:
3665:
3657:
3654:
3653:
3652:
3629:
3626:
3625:
3624:
3592:
3588:
3567:
3562:
3549:
3544:
3525:
3521:
3503:
3499:
3493:
3489:
3482:
3472:
3468:
3447:
3442:
3429:
3424:
3416:
3414:
3406:
3403:
3402:
3383:
3379:
3373:
3369:
3360:
3356:
3345:
3341:
3331:
3327:
3318:
3314:
3308:
3304:
3295:
3291:
3287:
3275:
3271:
3262:
3258:
3245:
3241:
3235:
3231:
3225:
3221:
3217:
3215:
3207:
3204:
3203:
3200:The result is:
3181:
3178:
3177:
3176:
3160:
3157:
3156:
3139:
3135:
3126:
3122:
3120:
3117:
3116:
3093:
3089:
3087:
3084:
3083:
3060:
3056:
3054:
3051:
3050:
3027:
3023:
3021:
3018:
3017:
2994:
2990:
2988:
2985:
2984:
2963:
2959:
2957:
2954:
2953:
2951:
2950:or transmitted
2932:
2928:
2926:
2923:
2922:
2920:
2901:
2897:
2895:
2892:
2891:
2889:
2885:get reflected.
2865:
2861:
2853:
2850:
2849:
2828:
2824:
2822:
2819:
2818:
2802:
2799:
2798:
2795:
2764:
2760:
2750:
2746:
2740:
2736:
2725:
2721:
2714:
2710:
2704:
2700:
2699:
2695:
2689:
2685:
2666:
2662:
2652:
2648:
2642:
2638:
2627:
2623:
2616:
2612:
2606:
2602:
2601:
2597:
2591:
2587:
2582:
2579:
2578:
2560:
2556:
2546:
2542:
2536:
2532:
2521:
2517:
2510:
2506:
2500:
2496:
2485:
2481:
2471:
2467:
2461:
2457:
2446:
2442:
2435:
2431:
2425:
2421:
2419:
2416:
2415:
2396:
2392:
2383:
2379:
2370:
2366:
2351:
2347:
2338:
2334:
2325:
2321:
2316:
2313:
2312:
2296:
2292:
2283:
2279:
2270:
2266:
2257:
2253:
2251:
2248:
2247:
2229:
2228:
2213:
2199:
2192:
2188:
2184:
2182:
2179:
2178:
2171:
2159:
2145:
2138:
2134:
2130:
2128:
2125:
2124:
2121:
2120:
2105:
2101:
2094:
2079:
2075:
2072:
2071:
2059:
2045:
2038:
2034:
2030:
2028:
2025:
2024:
2017:
2005:
1991:
1984:
1980:
1976:
1974:
1971:
1970:
1967:
1966:
1951:
1947:
1940:
1925:
1921:
1917:
1915:
1912:
1911:
1910:everywhere, so
1885:
1882:
1881:
1859:
1856:
1855:
1823:
1820:
1819:
1799:
1795:
1787:
1784:
1783:
1766:
1762:
1754:
1751:
1750:
1731:
1726:
1723:
1722:
1705:
1701:
1699:
1696:
1695:
1679:
1676:
1675:
1659:
1656:
1655:
1636:
1631:
1628:
1627:
1608:
1607:
1587:
1579:
1575:
1570:
1561:
1557:
1540:
1531:
1527:
1524:
1523:
1508:
1496:
1488:
1484:
1479:
1468:
1459:
1455:
1448:
1447:
1445:
1442:
1441:
1420:
1419:
1408:
1397:
1393:
1386:
1382:
1376:
1372:
1358:
1354:
1350:
1346:
1340:
1336:
1318:
1314:
1311:
1310:
1293:
1282:
1278:
1271:
1267:
1261:
1257:
1243:
1239:
1235:
1231:
1225:
1221:
1203:
1199:
1196:
1195:
1184:
1173:
1169:
1162:
1158:
1152:
1148:
1134:
1130:
1126:
1122:
1116:
1112:
1094:
1090:
1083:
1082:
1080:
1077:
1076:
1060:
1056:
1048:
1045:
1044:
984:
981:
980:
954:
950:
946:
940:
936:
934:
924:
918:
914:
912:
907:
904:
903:
887:
884:
883:
861:
858:
857:
835:
832:
831:
812:
811:
797:
795:
789:
788:
768:
766:
760:
756:
753:
752:
738:
736:
726:
725:
708:
705:
704:
621:
618:
617:
601:
598:
597:
574:
570:
568:
565:
564:
508:
504:
487:
484:
483:
467:
464:
463:
443:
440:
439:
436:Planck constant
419:
416:
415:
389:
388:
386:
383:
382:
312:
308:
304:
298:
294:
292:
282:
276:
272:
270:
266:
262:
236:
235:
233:
230:
229:
204:
201:
200:
179:
175:
167:
164:
163:
146:
142:
140:
137:
136:
129:
88:
84:
82:
79:
78:
77:
45:(or, at times,
28:
23:
22:
15:
12:
11:
5:
6250:
6240:
6239:
6234:
6229:
6224:
6222:Quantum models
6210:
6209:
6203:
6182:
6176:
6160:
6159:
6153:978-0935702996
6152:
6133:
6132:
6130:
6127:
6126:
6125:
6120:
6118:Step potential
6115:
6108:
6105:
6092:
6089:
6086:
6081:
6077:
6067:while keeping
6056:
6053:
6050:
6046:
6043:
6040:
6035:
6031:
6003:
6000:
5997:
5994:
5991:
5988:
5985:
5982:
5979:
5976:
5973:
5970:
5967:
5964:
5961:
5958:
5955:
5952:
5949:
5914:
5904:effective mass
5894:
5891:
5876:
5872:
5844:
5839:
5834:
5830:
5824:
5821:
5817:
5812:
5803:
5800:
5793:
5789:
5785:
5782:
5779:
5773:
5770:
5766:
5759:
5756:
5753:
5749:
5745:
5742:
5739:
5736:
5733:
5728:
5725:
5722:
5718:
5695:
5675:
5672:
5669:
5666:
5642:
5638:
5634:
5631:
5628:
5625:
5622:
5619:
5614:
5610:
5606:
5598:
5595:
5592:
5584:
5581:
5577:
5570:
5565:
5562:
5559:
5552:
5548:
5544:
5541:
5538:
5532:
5529:
5525:
5515:
5512:
5509:
5505:
5501:
5495:
5492:
5489:
5486:
5483:
5478:
5473:
5470:
5467:
5460:
5456:
5452:
5449:
5446:
5438:
5435:
5432:
5428:
5424:
5421:
5418:
5415:
5412:
5407:
5404:
5401:
5397:
5370:
5367:
5364:
5361:
5329:
5326:
5319:
5315:
5311:
5308:
5305:
5299:
5296:
5292:
5287:
5284:
5281:
5278:
5275:
5251:
5247:
5243:
5240:
5220:
5193:
5189:
5182:
5178:
5172:
5168:
5164:
5161:
5154:
5149:
5145:
5120:
5116:
5090:
5087:
5084:
5079:
5074:
5071:
5068:
5061:
5057:
5053:
5050:
5047:
5041:
5038:
5035:
5032:
5029:
5007:
5004:
5001:
4998:
4972:
4968:
4964:
4959:
4956:
4932:
4928:
4907:
4887:
4861:
4853:
4849:
4845:
4841:
4835:
4831:
4825:
4821:
4817:
4814:
4811:
4807:
4802:
4799:
4777:
4773:
4769:
4766:
4754:
4751:
4742:
4729:
4726:
4721:
4717:
4712:
4708:
4688:
4685:
4682:
4679:
4676:
4673:
4670:
4650:
4647:
4644:
4641:
4636:
4632:
4609:
4605:
4601:
4598:
4578:
4575:
4572:
4552:
4549:
4546:
4526:
4521:
4517:
4494:
4491:
4488:
4485:
4482:
4477:
4472:
4467:
4463:
4459:
4456:
4434:
4430:
4426:
4423:
4395:
4391:
4386:
4382:
4377:
4373:
4369:
4366:
4363:
4360:
4357:
4352:
4347:
4343:
4321:
4312:
4307:
4303:
4299:
4296:
4293:
4290:
4287:
4282:
4279:
4274:
4270:
4266:
4263:
4258:
4254:
4248:
4243:
4239:
4232:
4229:
4225:
4220:
4215:
4210:
4205:
4201:
4197:
4194:
4182:
4179:
4170:
4153:
4149:
4144:
4140:
4115:
4111:
4079:
4075:
4070:
4066:
4063:
4060:
4055:
4051:
4047:
4044:
4041:
4036:
4031:
4027:
3997:
3994:
3991:
3986:
3982:
3978:
3975:
3972:
3967:
3964:
3959:
3955:
3951:
3948:
3943:
3939:
3933:
3928:
3924:
3917:
3914:
3910:
3905:
3900:
3895:
3890:
3886:
3882:
3879:
3857:
3853:
3849:
3846:
3822:
3818:
3814:
3807:
3803:
3799:
3796:
3778:
3775:
3766:
3764:
3761:
3746:
3743:
3738:
3734:
3708:
3704:
3700:
3697:
3672:
3668:
3664:
3661:
3639:
3636:
3633:
3606:
3600:
3595:
3591:
3587:
3584:
3581:
3578:
3575:
3570:
3565:
3561:
3557:
3552:
3547:
3543:
3539:
3536:
3533:
3528:
3524:
3520:
3517:
3514:
3511:
3506:
3502:
3496:
3492:
3488:
3485:
3480:
3475:
3471:
3467:
3464:
3461:
3458:
3455:
3450:
3445:
3441:
3437:
3432:
3427:
3423:
3419:
3413:
3410:
3386:
3382:
3376:
3372:
3368:
3363:
3359:
3355:
3348:
3344:
3340:
3337:
3334:
3330:
3326:
3321:
3317:
3311:
3307:
3303:
3298:
3294:
3290:
3283:
3278:
3274:
3270:
3265:
3261:
3257:
3254:
3251:
3248:
3244:
3238:
3234:
3228:
3224:
3220:
3214:
3211:
3185:
3164:
3142:
3138:
3134:
3129:
3125:
3104:
3101:
3096:
3092:
3071:
3068:
3063:
3059:
3049:(reflection),
3038:
3035:
3030:
3026:
3005:
3002:
2997:
2993:
2966:
2962:
2935:
2931:
2904:
2900:
2868:
2864:
2860:
2857:
2831:
2827:
2806:
2794:
2791:
2779:
2775:
2767:
2763:
2759:
2756:
2753:
2749:
2743:
2739:
2735:
2728:
2724:
2720:
2717:
2713:
2707:
2703:
2698:
2692:
2688:
2684:
2681:
2677:
2669:
2665:
2661:
2658:
2655:
2651:
2645:
2641:
2637:
2630:
2626:
2622:
2619:
2615:
2609:
2605:
2600:
2594:
2590:
2586:
2563:
2559:
2555:
2552:
2549:
2545:
2539:
2535:
2531:
2524:
2520:
2516:
2513:
2509:
2503:
2499:
2495:
2488:
2484:
2480:
2477:
2474:
2470:
2464:
2460:
2456:
2449:
2445:
2441:
2438:
2434:
2428:
2424:
2404:
2399:
2395:
2391:
2386:
2382:
2378:
2373:
2369:
2365:
2362:
2359:
2354:
2350:
2346:
2341:
2337:
2333:
2328:
2324:
2320:
2299:
2295:
2291:
2286:
2282:
2278:
2273:
2269:
2265:
2260:
2256:
2227:
2222:
2219:
2216:
2211:
2205:
2202:
2195:
2191:
2187:
2181:
2177:
2174:
2172:
2168:
2165:
2162:
2157:
2151:
2148:
2141:
2137:
2133:
2127:
2123:
2122:
2119:
2116:
2113:
2108:
2104:
2100:
2097:
2095:
2093:
2090:
2087:
2082:
2078:
2074:
2073:
2068:
2065:
2062:
2057:
2051:
2048:
2041:
2037:
2033:
2027:
2023:
2020:
2018:
2014:
2011:
2008:
2003:
1997:
1994:
1987:
1983:
1979:
1973:
1969:
1968:
1965:
1962:
1959:
1954:
1950:
1946:
1943:
1941:
1939:
1936:
1933:
1928:
1924:
1920:
1919:
1895:
1892:
1889:
1869:
1866:
1863:
1839:
1836:
1833:
1830:
1827:
1802:
1798:
1794:
1791:
1769:
1765:
1761:
1758:
1738:
1734:
1730:
1708:
1704:
1683:
1663:
1643:
1639:
1635:
1611:
1606:
1603:
1600:
1597:
1594:
1591:
1588:
1582:
1578:
1573:
1569:
1564:
1560:
1556:
1553:
1550:
1547:
1544:
1539:
1534:
1530:
1526:
1525:
1522:
1519:
1516:
1506:
1503:
1500:
1497:
1491:
1487:
1482:
1478:
1475:
1472:
1467:
1462:
1458:
1454:
1453:
1451:
1423:
1418:
1415:
1412:
1409:
1405:
1400:
1396:
1392:
1389:
1385:
1379:
1375:
1371:
1366:
1361:
1357:
1353:
1349:
1343:
1339:
1335:
1332:
1329:
1326:
1321:
1317:
1313:
1312:
1309:
1306:
1303:
1300:
1297:
1294:
1290:
1285:
1281:
1277:
1274:
1270:
1264:
1260:
1256:
1251:
1246:
1242:
1238:
1234:
1228:
1224:
1220:
1217:
1214:
1211:
1206:
1202:
1198:
1197:
1194:
1191:
1188:
1185:
1181:
1176:
1172:
1168:
1165:
1161:
1155:
1151:
1147:
1142:
1137:
1133:
1129:
1125:
1119:
1115:
1111:
1108:
1105:
1102:
1097:
1093:
1089:
1088:
1086:
1063:
1059:
1055:
1052:
1024:
1021:
1018:
1015:
1012:
1009:
1006:
1003:
1000:
997:
994:
991:
988:
965:
957:
953:
949:
943:
939:
930:
927:
921:
917:
911:
891:
871:
868:
865:
845:
842:
839:
815:
810:
807:
804:
796:
794:
791:
790:
787:
784:
781:
778:
775:
767:
763:
759:
755:
754:
751:
748:
745:
737:
735:
732:
731:
729:
724:
721:
718:
715:
712:
688:
685:
682:
678:
675:
672:
669:
666:
663:
660:
656:
653:
650:
647:
643:
640:
637:
634:
631:
628:
625:
605:
585:
582:
577:
573:
552:
549:
546:
543:
540:
537:
534:
531:
528:
525:
522:
519:
516:
511:
507:
503:
500:
497:
494:
491:
471:
447:
423:
396:
393:
370:
367:
364:
361:
358:
355:
352:
349:
346:
343:
339:
335:
332:
329:
326:
323:
315:
311:
307:
301:
297:
288:
285:
279:
275:
269:
265:
261:
258:
255:
252:
249:
243:
240:
217:
214:
211:
208:
182:
178:
174:
171:
149:
145:
128:
125:
91:
87:
26:
9:
6:
4:
3:
2:
6249:
6238:
6235:
6233:
6230:
6228:
6225:
6223:
6220:
6219:
6217:
6206:
6200:
6196:
6191:
6190:
6183:
6179:
6177:0-13-111892-7
6173:
6169:
6164:
6163:
6155:
6149:
6145:
6138:
6134:
6124:
6121:
6119:
6116:
6114:
6111:
6110:
6104:
6090:
6087:
6084:
6079:
6075:
6054:
6048:
6044:
6033:
6029:
6020:
6015:
5998:
5995:
5992:
5986:
5980:
5974:
5971:
5965:
5962:
5959:
5956:
5953:
5939:
5933:
5931:
5926:
5912:
5905:
5901:
5890:
5874:
5870:
5860:
5842:
5837:
5832:
5828:
5822:
5819:
5815:
5810:
5801:
5798:
5791:
5783:
5777:
5771:
5768:
5764:
5757:
5751:
5743:
5737:
5731:
5726:
5720:
5707:
5693:
5670:
5664:
5655:
5640:
5636:
5632:
5626:
5620:
5617:
5612:
5608:
5604:
5596:
5593:
5590:
5582:
5579:
5575:
5563:
5560:
5557:
5550:
5546:
5539:
5536:
5530:
5527:
5523:
5513:
5507:
5499:
5490:
5487:
5484:
5471:
5468:
5465:
5458:
5454:
5447:
5444:
5436:
5430:
5422:
5416:
5410:
5405:
5399:
5386:
5384:
5365:
5359:
5351:
5346:
5327:
5324:
5317:
5309:
5303:
5297:
5294:
5290:
5285:
5279:
5273:
5265:
5249:
5245:
5241:
5238:
5218:
5209:
5191:
5187:
5180:
5176:
5170:
5166:
5162:
5159:
5152:
5147:
5143:
5134:
5118:
5114:
5104:
5088:
5085:
5082:
5072:
5069:
5066:
5059:
5055:
5048:
5045:
5039:
5033:
5027:
5019:
5002:
4996:
4987:
4970:
4966:
4962:
4957:
4954:
4946:
4930:
4926:
4905:
4885:
4877:
4872:
4859:
4851:
4847:
4843:
4839:
4833:
4829:
4823:
4819:
4815:
4812:
4809:
4805:
4800:
4797:
4775:
4771:
4767:
4764:
4750:
4746:
4741:
4727:
4724:
4719:
4715:
4710:
4706:
4686:
4683:
4680:
4677:
4674:
4671:
4668:
4648:
4645:
4642:
4639:
4634:
4630:
4607:
4603:
4599:
4596:
4576:
4573:
4570:
4550:
4547:
4544:
4524:
4519:
4515:
4505:
4492:
4489:
4486:
4483:
4480:
4475:
4465:
4457:
4454:
4432:
4428:
4424:
4421:
4412:
4393:
4389:
4384:
4375:
4371:
4367:
4364:
4358:
4355:
4350:
4345:
4341:
4319:
4305:
4301:
4297:
4294:
4288:
4285:
4277:
4272:
4268:
4261:
4256:
4252:
4246:
4241:
4237:
4230:
4227:
4223:
4218:
4213:
4203:
4195:
4192:
4185:In this case
4178:
4174:
4169:
4151:
4147:
4142:
4138:
4113:
4109:
4099:
4077:
4073:
4068:
4061:
4058:
4053:
4049:
4042:
4039:
4034:
4029:
4025:
4014:
3992:
3989:
3984:
3980:
3973:
3970:
3962:
3957:
3953:
3946:
3941:
3937:
3931:
3926:
3922:
3915:
3912:
3908:
3903:
3898:
3888:
3880:
3877:
3855:
3851:
3847:
3844:
3816:
3812:
3805:
3801:
3797:
3794:
3783:
3774:
3770:
3760:
3744:
3741:
3736:
3732:
3706:
3702:
3698:
3695:
3670:
3666:
3662:
3659:
3637:
3634:
3631:
3622:
3617:
3604:
3593:
3589:
3585:
3579:
3576:
3568:
3563:
3559:
3555:
3550:
3545:
3541:
3534:
3526:
3522:
3518:
3512:
3509:
3504:
3500:
3494:
3490:
3486:
3483:
3473:
3469:
3465:
3459:
3456:
3448:
3443:
3439:
3435:
3430:
3425:
3421:
3411:
3408:
3384:
3374:
3370:
3366:
3361:
3357:
3346:
3342:
3338:
3335:
3332:
3328:
3324:
3319:
3309:
3305:
3301:
3296:
3292:
3276:
3272:
3268:
3263:
3259:
3252:
3249:
3246:
3242:
3236:
3232:
3226:
3222:
3218:
3212:
3209:
3201:
3198:
3183:
3162:
3140:
3136:
3132:
3127:
3123:
3102:
3099:
3094:
3090:
3069:
3066:
3061:
3057:
3036:
3033:
3028:
3024:
3003:
3000:
2995:
2991:
2981:
2964:
2960:
2933:
2929:
2902:
2898:
2886:
2884:
2866:
2862:
2858:
2855:
2847:
2829:
2825:
2804:
2790:
2777:
2773:
2765:
2761:
2757:
2754:
2751:
2747:
2741:
2737:
2733:
2726:
2722:
2718:
2715:
2711:
2705:
2701:
2696:
2690:
2686:
2682:
2679:
2675:
2667:
2663:
2659:
2656:
2653:
2649:
2643:
2639:
2635:
2628:
2624:
2620:
2617:
2613:
2607:
2603:
2598:
2592:
2588:
2584:
2561:
2557:
2553:
2550:
2547:
2543:
2537:
2533:
2529:
2522:
2518:
2514:
2511:
2507:
2501:
2497:
2493:
2486:
2482:
2478:
2475:
2472:
2468:
2462:
2458:
2454:
2447:
2443:
2439:
2436:
2432:
2426:
2422:
2397:
2393:
2389:
2384:
2380:
2371:
2367:
2363:
2360:
2352:
2348:
2344:
2339:
2335:
2326:
2322:
2318:
2297:
2293:
2289:
2284:
2280:
2276:
2271:
2267:
2263:
2258:
2254:
2245:
2242:
2225:
2220:
2217:
2214:
2209:
2203:
2200:
2193:
2189:
2185:
2175:
2173:
2166:
2163:
2160:
2155:
2149:
2146:
2139:
2135:
2131:
2114:
2106:
2102:
2098:
2096:
2088:
2080:
2076:
2066:
2063:
2060:
2055:
2049:
2046:
2039:
2035:
2031:
2021:
2019:
2012:
2009:
2006:
2001:
1995:
1992:
1985:
1981:
1977:
1960:
1952:
1948:
1944:
1942:
1934:
1926:
1922:
1909:
1893:
1890:
1887:
1867:
1864:
1861:
1853:
1837:
1834:
1831:
1828:
1825:
1816:
1800:
1796:
1792:
1789:
1767:
1763:
1759:
1756:
1736:
1732:
1728:
1706:
1702:
1681:
1661:
1641:
1637:
1633:
1624:
1604:
1601:
1598:
1595:
1592:
1589:
1580:
1576:
1571:
1562:
1558:
1554:
1551:
1545:
1542:
1537:
1532:
1528:
1520:
1517:
1514:
1504:
1501:
1498:
1489:
1485:
1480:
1476:
1473:
1470:
1465:
1460:
1456:
1449:
1439:
1416:
1413:
1410:
1403:
1398:
1394:
1390:
1387:
1383:
1377:
1373:
1369:
1364:
1359:
1355:
1351:
1347:
1341:
1337:
1333:
1327:
1319:
1315:
1307:
1304:
1301:
1298:
1295:
1288:
1283:
1279:
1275:
1272:
1268:
1262:
1258:
1254:
1249:
1244:
1240:
1236:
1232:
1226:
1222:
1218:
1212:
1204:
1200:
1192:
1189:
1186:
1179:
1174:
1170:
1166:
1163:
1159:
1153:
1149:
1145:
1140:
1135:
1131:
1127:
1123:
1117:
1113:
1109:
1103:
1095:
1091:
1084:
1061:
1057:
1053:
1050:
1042:
1041:free particle
1038:
1037:superposition
1022:
1019:
1016:
1013:
1010:
1007:
1004:
1001:
998:
995:
992:
989:
986:
977:
963:
955:
951:
947:
941:
937:
928:
925:
919:
915:
909:
889:
869:
866:
863:
843:
840:
837:
828:
808:
805:
802:
792:
785:
782:
779:
776:
773:
761:
757:
749:
746:
743:
733:
727:
722:
716:
710:
702:
686:
683:
680:
676:
673:
670:
664:
654:
651:
648:
645:
641:
638:
635:
629:
603:
583:
580:
575:
571:
544:
541:
538:
529:
523:
509:
505:
501:
495:
489:
469:
461:
445:
437:
421:
413:
391:
365:
359:
356:
353:
347:
341:
337:
330:
324:
321:
313:
309:
305:
299:
295:
286:
283:
277:
273:
267:
263:
259:
253:
247:
238:
212:
206:
180:
176:
172:
169:
147:
143:
133:
124:
122:
118:
114:
110:
89:
85:
75:
70:
68:
67:free particle
64:
60:
56:
52:
48:
44:
40:
32:
19:
6188:
6167:
6143:
6137:
6016:
5934:
5927:
5896:
5861:
5708:
5656:
5387:
5347:
5266:
5210:
5135:
5105:
5020:
4988:
4947:
4873:
4756:
4748:
4744:
4506:
4413:
4184:
4176:
4172:
4015:
3836:
3772:
3768:
3618:
3202:
3199:
2982:
2887:
2882:
2845:
2796:
2246:
2243:
1817:
1625:
1438:wave numbers
978:
829:
198:
71:
50:
46:
42:
36:
4918:approaches
1782:. The case
412:Hamiltonian
127:Calculation
43:rectangular
6216:Categories
6129:References
6103:constant.
5900:conducting
1908:continuous
1626:The index
1436:where the
596:and width
74:point mass
6091:λ
6052:→
6042:∞
6039:→
5987:ϕ
5975:ψ
5948:Ψ
5938:separable
5755:→
5724:→
5621:
5594:−
5561:−
5540:
5511:→
5488:−
5469:−
5448:
5434:→
5403:→
5188:ℏ
5086:−
5070:−
5049:
4848:ℏ
4687:…
4649:π
4600:≫
4487:−
4390:ℏ
4368:−
4298:−
4262:
4074:ℏ
4059:−
3990:−
3947:
3821:ℏ
3663:≠
3580:
3513:
3460:
3436:−
3367:−
3325:−
3269:−
3247:−
2752:−
2734:−
2654:−
2636:−
2548:−
2473:−
2390:−
2345:−
2190:ψ
2136:ψ
2103:ψ
2077:ψ
2036:ψ
1982:ψ
1949:ψ
1923:ψ
1760:≠
1577:ℏ
1555:−
1486:ℏ
1388:−
1316:ψ
1273:−
1201:ψ
1164:−
1092:ψ
964:ψ
916:ℏ
910:−
659:Θ
624:Θ
542:−
533:Θ
530:−
518:Θ
422:ℏ
395:^
360:ψ
342:ψ
274:ℏ
268:−
248:ψ
242:^
207:ψ
63:potential
6107:See also
4661:, where
3621:symmetry
799:if
770:if
740:if
703:, i.e.,
699:is the
458:is the
410:is the
41:, the
6201:
6197:–233.
6174:
6150:
4332:where
2883:always
2846:always
2844:would
1043:). If
381:where
228:reads
47:square
5350:limit
4876:above
4790:is
4175:>
3771:<
3723:then
6199:ISBN
6172:ISBN
6148:ISBN
5618:cosh
5537:sinh
5445:sinh
5385:),
5242:<
5046:sinh
4425:>
3938:sinh
3848:<
3635:>
3175:and
2859:<
1880:and
1674:and
1599:<
1593:<
1518:>
1502:<
1414:>
1305:<
1299:<
1190:<
1054:>
1020:>
1008:<
1002:<
990:<
856:and
806:<
783:<
777:<
747:<
684:>
649:<
581:>
460:mass
173:>
6195:231
5748:lim
5717:lim
5686:as
5504:lim
5427:lim
5396:lim
5352:of
4898:as
4728:1.2
4253:sin
3687:If
3577:sin
3510:cos
3457:sin
119:.
37:In
6218::
6014:.
5264:.
5018::
4747:=
4563:,
2979:).
2917:).
1510:or
616:.
462:,
438:,
414:,
49:)
6207:.
6180:.
6156:.
6088:=
6085:a
6080:0
6076:V
6055:0
6049:a
6045:,
6034:0
6030:V
6002:)
5999:z
5996:,
5993:y
5990:(
5984:)
5981:x
5978:(
5972:=
5969:)
5966:z
5963:,
5960:y
5957:,
5954:x
5951:(
5913:m
5875:0
5871:v
5843:4
5838:2
5833:0
5829:v
5823:+
5820:1
5816:1
5811:=
5802:x
5799:4
5792:2
5788:)
5784:x
5781:(
5778:f
5772:+
5769:1
5765:1
5758:1
5752:x
5744:=
5741:)
5738:x
5735:(
5732:T
5727:1
5721:x
5694:x
5674:)
5671:x
5668:(
5665:T
5641:0
5637:v
5633:=
5630:)
5627:0
5624:(
5613:0
5609:v
5605:=
5597:x
5591:1
5583:x
5580:d
5576:d
5569:)
5564:x
5558:1
5551:0
5547:v
5543:(
5531:x
5528:d
5524:d
5514:1
5508:x
5500:=
5494:)
5491:x
5485:1
5482:(
5477:)
5472:x
5466:1
5459:0
5455:v
5451:(
5437:1
5431:x
5423:=
5420:)
5417:x
5414:(
5411:f
5406:1
5400:x
5369:)
5366:x
5363:(
5360:f
5328:x
5325:4
5318:2
5314:)
5310:x
5307:(
5304:f
5298:+
5295:1
5291:1
5286:=
5283:)
5280:x
5277:(
5274:T
5250:0
5246:V
5239:E
5219:T
5192:2
5181:2
5177:a
5171:0
5167:V
5163:m
5160:2
5153:=
5148:0
5144:v
5119:0
5115:v
5089:x
5083:1
5078:)
5073:x
5067:1
5060:0
5056:v
5052:(
5040:=
5037:)
5034:x
5031:(
5028:f
5006:)
5003:x
5000:(
4997:f
4971:0
4967:V
4963:E
4958:=
4955:x
4931:0
4927:V
4906:E
4886:T
4860:.
4852:2
4844:2
4840:/
4834:0
4830:V
4824:2
4820:a
4816:m
4813:+
4810:1
4806:1
4801:=
4798:T
4776:0
4772:V
4768:=
4765:E
4752:0
4749:V
4745:E
4725:=
4720:0
4716:V
4711:/
4707:E
4684:,
4681:2
4678:,
4675:1
4672:=
4669:n
4646:n
4643:=
4640:a
4635:1
4631:k
4608:0
4604:V
4597:E
4577:0
4574:=
4571:R
4551:1
4548:=
4545:T
4525:a
4520:1
4516:k
4493:.
4490:T
4484:1
4481:=
4476:2
4471:|
4466:r
4462:|
4458:=
4455:R
4433:0
4429:V
4422:E
4410:.
4394:2
4385:/
4381:)
4376:0
4372:V
4365:E
4362:(
4359:m
4356:2
4351:=
4346:1
4342:k
4320:,
4311:)
4306:0
4302:V
4295:E
4292:(
4289:E
4286:4
4281:)
4278:a
4273:1
4269:k
4265:(
4257:2
4247:2
4242:0
4238:V
4231:+
4228:1
4224:1
4219:=
4214:2
4209:|
4204:t
4200:|
4196:=
4193:T
4180:0
4177:V
4173:E
4166:.
4152:1
4148:k
4143:/
4139:1
4128:,
4114:0
4110:k
4094:.
4078:2
4069:/
4065:)
4062:E
4054:0
4050:V
4046:(
4043:m
4040:2
4035:=
4030:1
4026:k
3996:)
3993:E
3985:0
3981:V
3977:(
3974:E
3971:4
3966:)
3963:a
3958:1
3954:k
3950:(
3942:2
3932:2
3927:0
3923:V
3916:+
3913:1
3909:1
3904:=
3899:2
3894:|
3889:t
3885:|
3881:=
3878:T
3856:0
3852:V
3845:E
3817:/
3813:a
3806:0
3802:V
3798:m
3795:2
3776:0
3773:V
3769:E
3757:,
3745:0
3742:=
3737:1
3733:k
3721:,
3707:0
3703:V
3699:=
3696:E
3685:.
3671:0
3667:V
3660:E
3650:,
3638:0
3632:E
3605:.
3599:)
3594:1
3590:k
3586:a
3583:(
3574:)
3569:2
3564:1
3560:k
3556:+
3551:2
3546:0
3542:k
3538:(
3535:+
3532:)
3527:1
3523:k
3519:a
3516:(
3505:1
3501:k
3495:0
3491:k
3487:i
3484:2
3479:)
3474:1
3470:k
3466:a
3463:(
3454:)
3449:2
3444:1
3440:k
3431:2
3426:0
3422:k
3418:(
3412:=
3409:r
3385:2
3381:)
3375:1
3371:k
3362:0
3358:k
3354:(
3347:1
3343:k
3339:a
3336:i
3333:2
3329:e
3320:2
3316:)
3310:1
3306:k
3302:+
3297:0
3293:k
3289:(
3282:)
3277:1
3273:k
3264:0
3260:k
3256:(
3253:a
3250:i
3243:e
3237:1
3233:k
3227:0
3223:k
3219:4
3213:=
3210:t
3196:.
3184:t
3163:r
3141:r
3137:B
3133:,
3128:l
3124:B
3103:t
3100:=
3095:r
3091:C
3070:0
3067:=
3062:l
3058:C
3037:r
3034:=
3029:l
3025:A
3004:1
3001:=
2996:r
2992:A
2965:r
2961:C
2952:(
2948:)
2934:l
2930:A
2921:(
2903:r
2899:A
2890:(
2867:0
2863:V
2856:E
2830:0
2826:V
2805:E
2778:.
2774:)
2766:0
2762:k
2758:a
2755:i
2748:e
2742:l
2738:C
2727:0
2723:k
2719:a
2716:i
2712:e
2706:r
2702:C
2697:(
2691:0
2687:k
2683:i
2680:=
2676:)
2668:1
2664:k
2660:a
2657:i
2650:e
2644:l
2640:B
2629:1
2625:k
2621:a
2618:i
2614:e
2608:r
2604:B
2599:(
2593:1
2589:k
2585:i
2562:0
2558:k
2554:a
2551:i
2544:e
2538:l
2534:C
2530:+
2523:0
2519:k
2515:a
2512:i
2508:e
2502:r
2498:C
2494:=
2487:1
2483:k
2479:a
2476:i
2469:e
2463:l
2459:B
2455:+
2448:1
2444:k
2440:a
2437:i
2433:e
2427:r
2423:B
2403:)
2398:l
2394:B
2385:r
2381:B
2377:(
2372:1
2368:k
2364:i
2361:=
2358:)
2353:l
2349:A
2340:r
2336:A
2332:(
2327:0
2323:k
2319:i
2298:l
2294:B
2290:+
2285:r
2281:B
2277:=
2272:l
2268:A
2264:+
2259:r
2255:A
2226:.
2221:a
2218:=
2215:x
2210:|
2204:x
2201:d
2194:R
2186:d
2176:=
2167:a
2164:=
2161:x
2156:|
2150:x
2147:d
2140:C
2132:d
2118:)
2115:a
2112:(
2107:R
2099:=
2092:)
2089:a
2086:(
2081:C
2067:0
2064:=
2061:x
2056:|
2050:x
2047:d
2040:C
2032:d
2022:=
2013:0
2010:=
2007:x
2002:|
1996:x
1993:d
1986:L
1978:d
1964:)
1961:0
1958:(
1953:C
1945:=
1938:)
1935:0
1932:(
1927:L
1894:a
1891:=
1888:x
1868:0
1865:=
1862:x
1838:C
1835:,
1832:B
1829:,
1826:A
1801:0
1797:V
1793:=
1790:E
1768:0
1764:V
1757:E
1737:l
1733:/
1729:r
1707:1
1703:k
1682:B
1662:A
1642:l
1638:/
1634:r
1605:.
1602:a
1596:x
1590:0
1581:2
1572:/
1568:)
1563:0
1559:V
1552:E
1549:(
1546:m
1543:2
1538:=
1533:1
1529:k
1521:a
1515:x
1505:0
1499:x
1490:2
1481:/
1477:E
1474:m
1471:2
1466:=
1461:0
1457:k
1450:{
1417:a
1411:x
1404:x
1399:0
1395:k
1391:i
1384:e
1378:l
1374:C
1370:+
1365:x
1360:0
1356:k
1352:i
1348:e
1342:r
1338:C
1334:=
1331:)
1328:x
1325:(
1320:R
1308:a
1302:x
1296:0
1289:x
1284:1
1280:k
1276:i
1269:e
1263:l
1259:B
1255:+
1250:x
1245:1
1241:k
1237:i
1233:e
1227:r
1223:B
1219:=
1216:)
1213:x
1210:(
1205:C
1193:0
1187:x
1180:x
1175:0
1171:k
1167:i
1160:e
1154:l
1150:A
1146:+
1141:x
1136:0
1132:k
1128:i
1124:e
1118:r
1114:A
1110:=
1107:)
1104:x
1101:(
1096:L
1085:{
1062:0
1058:V
1051:E
1023:a
1017:x
1014:,
1011:a
1005:x
999:0
996:,
993:0
987:x
956:2
952:x
948:d
942:2
938:d
929:m
926:2
920:2
890:x
870:a
867:=
864:x
844:0
841:=
838:x
809:x
803:a
793:0
786:a
780:x
774:0
762:0
758:V
750:0
744:x
734:0
728:{
723:=
720:)
717:x
714:(
711:V
687:0
681:x
677:,
674:1
671:=
668:)
665:x
662:(
655:;
652:0
646:x
642:,
639:0
636:=
633:)
630:x
627:(
604:a
584:0
576:0
572:V
551:]
548:)
545:a
539:x
536:(
527:)
524:x
521:(
515:[
510:0
506:V
502:=
499:)
496:x
493:(
490:V
470:E
446:m
392:H
369:)
366:x
363:(
357:E
354:=
351:)
348:x
345:(
338:]
334:)
331:x
328:(
325:V
322:+
314:2
310:x
306:d
300:2
296:d
287:m
284:2
278:2
264:[
260:=
257:)
254:x
251:(
239:H
216:)
213:x
210:(
181:0
177:V
170:E
148:0
144:V
104:,
90:0
86:V
20:)
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