2483:
1748:
2277:
1461:
995:
2031:
3788:
is an example of a
Hermitian operator whose eigenfunctions form an orthonormal basis. When the Hamiltonian does not depend explicitly on time, general solutions of the Schrödinger equation are linear combinations of the stationary states multiplied by the oscillatory
2821:
659:
Eigenfunctions can be expressed as column vectors and linear operators can be expressed as matrices, although they may have infinite dimensions. As a result, many of the concepts related to eigenvectors of matrices carry over to the study of eigenfunctions.
2997:
2632:
516:
is a parameter that depends on the boundary conditions. Note that in this case the eigenfunction is itself a function of its associated eigenvalue λ, which can take any real or complex value. In particular, note that for λ = 0 the eigenfunction
2468:
As a consequence, in many important cases, the eigenfunctions of the
Hermitian operator form an orthonormal basis. In these cases, an arbitrary function can be expressed as a linear combination of the eigenfunctions of the Hermitian operator.
4032:
3917:
3393:
1993:
1275:
820:
2486:
The shape of a standing wave in a string fixed at its boundaries is an example of an eigenfunction of a differential operator. The admissible eigenvalues are governed by the length of the string and determine the frequency of
1441:
3263:
3477:
1743:{\displaystyle {\begin{aligned}\sum _{j=1}^{n}c_{j}\int _{\Omega }\ u_{i}^{*}(t)u_{j}(t)dt&=\sum _{j=1}^{n}b_{j}\int _{\Omega }\ u_{i}^{*}(t)Du_{j}(t)dt,\\c_{i}&=\sum _{j=1}^{n}b_{j}A_{ij}.\end{aligned}}}
757:
3661:
3570:
2686:
2036:
3126:
1466:
1100:
601:
401:
2449:. Depending on whether the spectrum is discrete or continuous, the eigenfunctions can be normalized by setting the inner product of the eigenfunctions equal to either a Kronecker delta or a
2887:
2536:
2272:{\displaystyle {\begin{aligned}\langle u_{i},Du_{j}\rangle &=\langle Du_{i},u_{j}\rangle ,\\\int _{\Omega }dt\ u_{i}^{*}(t)Du_{j}(t)&=\int _{\Omega }dt\ u_{j}(t)^{*}.\end{aligned}}}
2396:
20:
2867:
3922:
3753:
3314:
507:
1862:
1144:
446:
229:
649:
336:
115:
142:
3803:
1302:
3170:
4036:
The success of the Schrödinger equation in explaining the spectral characteristics of hydrogen is considered one of the greatest triumphs of 20th century physics.
287:
278:
Each value of λ corresponds to one or more eigenfunctions. If multiple linearly independent eigenfunctions have the same eigenvalue, the eigenvalue is said to be
253:) may also be subject to boundary conditions. Because of the boundary conditions, the possible values of λ are generally limited, for example to a discrete set
75:
3401:
674:
4101:
3593:
3518:
990:{\displaystyle \langle u_{i},u_{j}\rangle =\int _{\Omega }\ u_{i}^{*}(t)u_{j}(t)dt=\delta _{ij}={\begin{cases}1&i=j\\0&i\neq j\end{cases}},}
3079:
1134:. In some special cases, such as the coefficients of the Fourier series of a sinusoidal function, this column vector has finite dimension.
1020:
341:
3692:
454:
145:
543:
2816:{\displaystyle {\frac {d^{2}}{dx^{2}}}X=-{\frac {\omega ^{2}}{c^{2}}}X,\qquad {\frac {d^{2}}{dt^{2}}}T=-\omega ^{2}T.}
298:
A widely used class of linear operators acting on infinite dimensional spaces are differential operators on the space
282:
and the maximum number of linearly independent eigenfunctions associated with the same eigenvalue is the eigenvalue's
2349:
199:
3479:
can be solved by separation of variables if the
Hamiltonian does not depend explicitly on time. In that case, the
606:
88:
4388:
4369:
4347:
4331:
4319:
3396:
2828:
279:
272:
2531:
24:
651:
where λ = 2 is the only eigenvalue of the differential equation that also satisfies the boundary condition.
4445:
4091:
152:
82:
4096:
2992:{\displaystyle X(x)=\sin \left({\frac {\omega x}{c}}+\varphi \right),\qquad T(t)=\sin(\omega t+\psi ),}
2627:{\displaystyle {\frac {\partial ^{2}h}{\partial t^{2}}}=c^{2}{\frac {\partial ^{2}h}{\partial x^{2}}},}
2457:
3781:. They represent allowable energy states of the system and may be constrained by boundary conditions.
4111:
942:
408:
175:
acts upon it, is simply scaled by some scalar value called an eigenvalue. In the special case where
4027:{\displaystyle \Psi (\mathbf {r} ,t)=\int dE\,c_{E}\varphi _{E}(\mathbf {r} )e^{{-iEt}/{\hbar }}.}
3912:{\textstyle \Psi (\mathbf {r} ,t)=\sum _{k}c_{k}\varphi _{k}(\mathbf {r} )e^{{-iE_{k}t}/{\hbar }}}
3388:{\displaystyle i\hbar {\frac {\partial }{\partial t}}\Psi (\mathbf {r} ,t)=H\Psi (\mathbf {r} ,t)}
2646:
309:
4342:. Translated, edited, and with additions by D. ter Haar (2nd ed.). Oxford: Pergamon Press.
3309:
4106:
124:
55:
1988:{\displaystyle A_{ij}=\langle u_{i},Du_{j}\rangle =\int _{\Omega }dt\ u_{i}^{*}(t)Du_{j}(t).}
1801:
1270:{\displaystyle A_{ij}=\langle u_{i},Du_{j}\rangle =\int _{\Omega }\ u_{i}^{*}(t)Du_{j}(t)dt.}
2450:
2446:
449:
118:
2436:, orthogonal eigenfunctions can always be chosen that span the eigenspace associated with
8:
3167:
is any integer. Thus, the clamped string supports a family of standing waves of the form
1445:
Taking the inner product of each side of this equation with an arbitrary basis function
3010:
If we impose boundary conditions, for example that the ends of the string are fixed at
60:
2464:
Its eigenfunctions form a basis of the function space on which the operator is defined
4384:
4365:
4343:
4327:
4315:
4045:
3305:
2511:
775:
768:
302:
of infinitely differentiable real or complex functions of a real or complex argument
4407:
3771:
2507:
2003:
267:, … or to a continuous set over some range. The set of all possible eigenvalues of
28:
4430:
4399:
1436:{\displaystyle Df(t)=\sum _{j=1}^{n}c_{j}u_{j}(t)=\sum _{j=1}^{n}b_{j}Du_{j}(t).}
1011:
1007:
3258:{\displaystyle h(x,t)=\sin \left({\frac {n\pi x}{L}}\right)\sin(\omega _{n}t).}
1103:
51:
3679:
Both of these differential equations are eigenvalue equations with eigenvalue
4439:
3480:
2635:
2523:
1809:
664:
2506:
denote the transverse displacement of a stressed elastic chord, such as the
4357:
1760:
written in summation notation and is a matrix equivalent of the operator
1017:
Functions can be written as a linear combination of the basis functions,
36:
3472:{\displaystyle H=-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}+V(\mathbf {r} ,t)}
2642:
is a constant speed that depends on the tension and mass of the string.
1795:
19:
44:
405:
This differential equation can be solved by multiplying both sides by
4287:
1137:
Additionally, define a matrix representation of the linear operator
752:{\displaystyle \langle f,g\rangle =\int _{\Omega }\ f^{*}(t)g(t)dt,}
179:
is defined on a function space, the eigenvectors are referred to as
3774:
of the quantum mechanical system, each with a corresponding energy
3763:) is the time-independent Schrödinger equation. The eigenfunctions
3293:
3281:
654:
167:
defined on some vector space is a nonzero vector in the domain of
3656:{\displaystyle i\hbar {\frac {\partial T(t)}{\partial t}}=ET(t).}
3565:{\displaystyle H\varphi (\mathbf {r} )=E\varphi (\mathbf {r} ),}
2482:
3049:, we constrain the eigenvalues. For these boundary conditions,
275:, which may be discrete, continuous, or a combination of both.
1287:) either as a linear combination of the basis functions or as
4214:
4212:
4210:
4197:
4195:
4193:
4150:
4148:
3683:. As shown in an earlier example, the solution of Equation (
2321:), …. This Hermitian operator has the following properties:
980:
549:
4207:
4190:
4145:
3121:{\displaystyle \sin \left({\frac {\omega L}{c}}\right)=0.}
2825:
Each of these is an eigenvalue equation with eigenvalues
2683:, we can form a pair of ordinary differential equations:
2427:. For degenerate eigenfunctions with the same eigenvalue
1812:
with an orthonormal basis given by the set of functions {
148:
that limit the allowable eigenvalues and eigenfunctions.
4241:
4239:
4180:
4178:
4165:
4163:
4135:
4133:
4131:
509:
is the eigenfunction of the derivative operator, where
4356:
4293:
4059:
that, when input into the system, produces a response
3806:
2831:
546:
411:
312:
144:
The solutions to this equation may also be subject to
81:, is only multiplied by some scaling factor called an
4275:
4263:
3925:
3695:
3596:
3521:
3404:
3317:
3267:
In the example of a string instrument, the frequency
3173:
3082:
2890:
2689:
2539:
2352:
2034:
1865:
1796:
Eigenvalues and eigenfunctions of
Hermitian operators
1464:
1305:
1147:
1023:
823:
677:
609:
457:
344:
202:
127:
91:
63:
4251:
4236:
4224:
4175:
4160:
4128:
2346:
Its eigenfunctions obey an orthogonality condition,
85:. As an equation, this condition can be written as
4102:Spectral theory of ordinary differential equations
4026:
3911:
3747:
3655:
3564:
3471:
3387:
3257:
3120:
2991:
2861:
2815:
2626:
2390:
2271:
1987:
1742:
1435:
1269:
1095:{\displaystyle f(t)=\sum _{j=1}^{n}b_{j}u_{j}(t),}
1094:
989:
751:
643:
596:{\textstyle \left.{\frac {df}{dt}}\right|_{t=0}=2}
595:
501:
440:
395:
330:
223:
136:
109:
69:
27:is, at any point in time, an eigenfunction of the
4360:; Rabenstein, Rudolf; Stenger, Alexander (2001).
1800:Many of the operators encountered in physics are
396:{\displaystyle {\frac {d}{dt}}f(t)=\lambda f(t).}
4437:
655:Link to eigenvalues and eigenvectors of matrices
306:. For example, consider the derivative operator
247:where λ is a scalar. The solutions to Equation (
163:In general, an eigenvector of a linear operator
2884:, the equations are satisfied by the functions
3919:or, for a system with a continuous spectrum,
2391:{\displaystyle \langle f_{i},f_{j}\rangle =0}
1851:may be infinite. In this basis, the operator
1010:and can be thought of as the elements of the
4378:
4218:
4201:
4154:
2379:
2353:
2107:
2078:
2068:
2039:
1911:
1882:
1193:
1164:
850:
824:
817:may be infinite. For the orthonormal basis,
690:
678:
4048:, an eigenfunction of a system is a signal
1995:integrated over some range of interest for
759:integrated over some range of interest for
4309:
2862:{\textstyle -{\frac {\omega ^{2}}{c^{2}}}}
2664:can be written as the product of the form
2645:This problem is amenable to the method of
16:Mathematical function of a linear operator
4397:
4169:
3958:
3748:{\displaystyle T(t)=e^{{-iEt}/{\hbar }}.}
3510:leads to the two differential equations,
1772:) expressed in the orthonormal basis. If
3130:This last boundary condition constrains
2481:
536:) is subject to the boundary conditions
502:{\displaystyle f(t)=f_{0}e^{\lambda t},}
18:
4337:
4281:
4269:
4257:
4245:
4230:
4184:
4139:
3299:
77:in that space that, when acted upon by
4438:
4039:
2634:which is called the (one-dimensional)
2456:For many Hermitian operators, notably
2409:The second condition always holds for
2526:portions of the string, the function
2299:, … and corresponding eigenfunctions
293:
4379:Kusse, Bruce; Westwig, Erik (1998).
4294:Girod, Rabenstein & Stenger 2001
3587:
3512:
2522:. Applying the laws of mechanics to
2477:
193:
1808:acts on a function space that is a
448:and integrating. Its solution, the
13:
4310:Courant, Richard; Hilbert, David.
3926:
3807:
3623:
3606:
3437:
3365:
3339:
3330:
3326:
2876:, respectively. For any values of
2605:
2591:
2558:
2544:
2194:
2122:
1922:
1752:This is the matrix multiplication
1605:
1505:
1204:
861:
774:Suppose the function space has an
701:
14:
4457:
4423:
3600:
3321:
158:
4383:. New York: Wiley Interscience.
4112:Fourier transform eigenfunctions
4082:is a complex scalar eigenvalue.
3983:
3933:
3864:
3814:
3770:of the Hamiltonian operator are
3552:
3532:
3456:
3372:
3346:
2514:, as a function of the position
2281:Consider the Hermitian operator
4312:Methods of Mathematical Physics
2946:
2758:
2472:
778:given by the set of functions {
667:in the function space on which
441:{\textstyle {\frac {dt}{f(t)}}}
4302:
3987:
3979:
3943:
3929:
3868:
3860:
3824:
3810:
3705:
3699:
3647:
3641:
3618:
3612:
3556:
3548:
3536:
3528:
3466:
3452:
3382:
3368:
3356:
3342:
3249:
3233:
3189:
3177:
3007:are arbitrary real constants.
2983:
2968:
2956:
2950:
2900:
2894:
2253:
2249:
2243:
2227:
2224:
2218:
2179:
2173:
2157:
2151:
1979:
1973:
1957:
1951:
1804:. Suppose the linear operator
1656:
1650:
1634:
1628:
1553:
1547:
1534:
1528:
1427:
1421:
1371:
1365:
1318:
1312:
1255:
1249:
1233:
1227:
1086:
1080:
1033:
1027:
909:
903:
890:
884:
737:
731:
725:
719:
619:
613:
467:
461:
432:
426:
387:
381:
369:
363:
151:An eigenfunction is a type of
1:
2532:partial differential equation
2518:along the string and of time
1291:acting upon the expansion of
224:{\displaystyle Df=\lambda f,}
191:if it satisfies the equation
4122:
4092:Eigenvalues and eigenvectors
1855:has a matrix representation
644:{\displaystyle f(t)=e^{2t},}
528:Suppose in the example that
331:{\textstyle {\frac {d}{dt}}}
110:{\displaystyle Df=\lambda f}
7:
4398:Wasserman, Eric W. (2016).
4085:
3759:
3685:
3669:
3578:
2445:, for example by using the
2010:is a Hermitian operator if
249:
237:
10:
4462:
2325:Its eigenvalues are real,
1279:We can write the function
4429:More images (non-GPL) at
3784:The Hamiltonian operator
2458:Sturm–Liouville operators
1780:) is an eigenfunction of
1764:acting upon the function
338:with eigenvalue equation
137:{\displaystyle \lambda .}
4219:Kusse & Westwig 1998
4202:Kusse & Westwig 1998
4155:Kusse & Westwig 1998
4117:
3276:is the frequency of the
1784:with eigenvalue λ, then
271:is sometimes called its
4364:(2nd ed.). Wiley.
4338:Davydov, A. S. (1976).
4097:Hilbert–Schmidt theorem
2999:where the phase angles
2647:separation of variables
1123:can be stacked into an
187:is an eigenfunction of
4314:. Vol. 1. Wiley.
4107:Fixed point combinator
4028:
3913:
3749:
3657:
3566:
3473:
3389:
3284:, which is called the
3259:
3122:
3065:, so the phase angles
2993:
2863:
2817:
2628:
2488:
2460:, a third property is
2392:
2273:
1989:
1744:
1709:
1589:
1489:
1437:
1397:
1344:
1271:
1102:for example through a
1096:
1059:
991:
753:
645:
597:
503:
442:
397:
332:
288:geometric multiplicity
225:
183:. That is, a function
138:
111:
71:
32:
25:vibrating drum problem
4029:
3914:
3750:
3689:) is the exponential
3658:
3567:
3474:
3390:
3260:
3123:
2994:
2864:
2818:
2629:
2485:
2393:
2274:
1990:
1745:
1689:
1569:
1469:
1438:
1377:
1324:
1272:
1097:
1039:
992:
754:
646:
598:
504:
443:
398:
333:
226:
139:
112:
72:
23:This solution of the
22:
4381:Mathematical Physics
3923:
3804:
3693:
3594:
3519:
3402:
3397:Hamiltonian operator
3315:
3310:Schrödinger equation
3300:Schrödinger equation
3171:
3080:
2888:
2829:
2687:
2649:. If we assume that
2537:
2451:Dirac delta function
2447:Gram-Schmidt process
2350:
2032:
1863:
1462:
1303:
1145:
1114:). The coefficients
1021:
821:
675:
607:
603:. We then find that
544:
455:
450:exponential function
409:
342:
310:
284:degree of degeneracy
200:
125:
89:
61:
4446:Functional analysis
4362:Signals and systems
4046:signals and systems
4040:Signals and systems
2150:
1950:
1627:
1527:
1226:
1127:by 1 column vector
883:
146:boundary conditions
4024:
3909:
3839:
3745:
3653:
3562:
3469:
3385:
3255:
3118:
2989:
2859:
2813:
2624:
2489:
2388:
2269:
2267:
2136:
2004:Hermitian matrices
1985:
1936:
1740:
1738:
1613:
1513:
1433:
1267:
1212:
1092:
987:
979:
869:
749:
641:
593:
499:
438:
393:
328:
294:Derivative example
221:
134:
107:
67:
33:
4340:Quantum Mechanics
3830:
3772:stationary states
3677:
3676:
3630:
3586:
3585:
3434:
3337:
3306:quantum mechanics
3221:
3106:
2930:
2857:
2786:
2750:
2717:
2619:
2572:
2512:string instrument
2508:vibrating strings
2478:Vibrating strings
2285:with eigenvalues
2207:
2135:
1935:
1612:
1512:
1211:
1104:Fourier expansion
868:
776:orthonormal basis
769:complex conjugate
708:
569:
525:) is a constant.
436:
358:
326:
245:
244:
70:{\displaystyle f}
4453:
4418:
4416:
4414:
4408:Wolfram Research
4394:
4375:
4353:
4325:
4297:
4291:
4285:
4279:
4273:
4267:
4261:
4255:
4249:
4243:
4234:
4228:
4222:
4216:
4205:
4199:
4188:
4182:
4173:
4167:
4158:
4152:
4143:
4137:
4081:
4077:
4058:
4044:In the study of
4033:
4031:
4030:
4025:
4020:
4019:
4018:
4013:
4008:
3986:
3978:
3977:
3968:
3967:
3936:
3918:
3916:
3915:
3910:
3908:
3907:
3906:
3901:
3896:
3892:
3891:
3867:
3859:
3858:
3849:
3848:
3838:
3817:
3799:
3787:
3780:
3769:
3754:
3752:
3751:
3746:
3741:
3740:
3739:
3734:
3729:
3682:
3671:
3662:
3660:
3659:
3654:
3631:
3629:
3621:
3604:
3588:
3580:
3571:
3569:
3568:
3563:
3555:
3535:
3513:
3509:
3478:
3476:
3475:
3470:
3459:
3445:
3444:
3435:
3433:
3425:
3424:
3415:
3394:
3392:
3391:
3386:
3375:
3349:
3338:
3336:
3325:
3291:
3279:
3275:
3264:
3262:
3261:
3256:
3245:
3244:
3226:
3222:
3217:
3206:
3166:
3162:
3161:
3159:
3158:
3153:
3150:
3134:to take a value
3133:
3127:
3125:
3124:
3119:
3111:
3107:
3102:
3094:
3075:
3064:
3056:
3048:
3041:
3026:
3016:
3006:
3002:
2998:
2996:
2995:
2990:
2942:
2938:
2931:
2926:
2918:
2883:
2879:
2875:
2868:
2866:
2865:
2860:
2858:
2856:
2855:
2846:
2845:
2836:
2822:
2820:
2819:
2814:
2806:
2805:
2787:
2785:
2784:
2783:
2770:
2769:
2760:
2751:
2749:
2748:
2739:
2738:
2729:
2718:
2716:
2715:
2714:
2701:
2700:
2691:
2682:
2663:
2641:
2633:
2631:
2630:
2625:
2620:
2618:
2617:
2616:
2603:
2599:
2598:
2588:
2586:
2585:
2573:
2571:
2570:
2569:
2556:
2552:
2551:
2541:
2529:
2521:
2517:
2505:
2453:, respectively.
2397:
2395:
2394:
2389:
2378:
2377:
2365:
2364:
2278:
2276:
2275:
2270:
2268:
2261:
2260:
2242:
2241:
2217:
2216:
2205:
2198:
2197:
2172:
2171:
2149:
2144:
2133:
2126:
2125:
2106:
2105:
2093:
2092:
2067:
2066:
2051:
2050:
2002:By analogy with
1994:
1992:
1991:
1986:
1972:
1971:
1949:
1944:
1933:
1926:
1925:
1910:
1909:
1894:
1893:
1878:
1877:
1749:
1747:
1746:
1741:
1739:
1732:
1731:
1719:
1718:
1708:
1703:
1681:
1680:
1649:
1648:
1626:
1621:
1610:
1609:
1608:
1599:
1598:
1588:
1583:
1546:
1545:
1526:
1521:
1510:
1509:
1508:
1499:
1498:
1488:
1483:
1442:
1440:
1439:
1434:
1420:
1419:
1407:
1406:
1396:
1391:
1364:
1363:
1354:
1353:
1343:
1338:
1276:
1274:
1273:
1268:
1248:
1247:
1225:
1220:
1209:
1208:
1207:
1192:
1191:
1176:
1175:
1160:
1159:
1133:
1101:
1099:
1098:
1093:
1079:
1078:
1069:
1068:
1058:
1053:
996:
994:
993:
988:
983:
982:
933:
932:
902:
901:
882:
877:
866:
865:
864:
849:
848:
836:
835:
758:
756:
755:
750:
718:
717:
706:
705:
704:
650:
648:
647:
642:
637:
636:
602:
600:
599:
594:
586:
585:
574:
570:
568:
560:
552:
508:
506:
505:
500:
495:
494:
482:
481:
447:
445:
444:
439:
437:
435:
421:
413:
402:
400:
399:
394:
359:
357:
346:
337:
335:
334:
329:
327:
325:
314:
239:
230:
228:
227:
222:
194:
143:
141:
140:
135:
116:
114:
113:
108:
76:
74:
73:
68:
54:is any non-zero
50:defined on some
29:Laplace operator
4461:
4460:
4456:
4455:
4454:
4452:
4451:
4450:
4436:
4435:
4426:
4421:
4412:
4410:
4400:"Eigenfunction"
4391:
4372:
4350:
4322:
4305:
4300:
4292:
4288:
4280:
4276:
4268:
4264:
4256:
4252:
4244:
4237:
4229:
4225:
4217:
4208:
4200:
4191:
4183:
4176:
4168:
4161:
4153:
4146:
4138:
4129:
4125:
4120:
4088:
4079:
4060:
4049:
4042:
4014:
4009:
3995:
3994:
3990:
3982:
3973:
3969:
3963:
3959:
3932:
3924:
3921:
3920:
3902:
3897:
3887:
3883:
3876:
3875:
3871:
3863:
3854:
3850:
3844:
3840:
3834:
3813:
3805:
3802:
3801:
3790:
3785:
3779:
3775:
3768:
3764:
3735:
3730:
3716:
3715:
3711:
3694:
3691:
3690:
3680:
3622:
3605:
3603:
3595:
3592:
3591:
3551:
3531:
3520:
3517:
3516:
3483:
3455:
3440:
3436:
3426:
3420:
3416:
3414:
3403:
3400:
3399:
3371:
3345:
3329:
3324:
3316:
3313:
3312:
3302:
3285:
3277:
3273:
3268:
3240:
3236:
3207:
3205:
3201:
3172:
3169:
3168:
3164:
3154:
3151:
3146:
3145:
3143:
3140:
3135:
3131:
3095:
3093:
3089:
3081:
3078:
3077:
3066:
3058:
3050:
3043:
3028:
3018:
3011:
3004:
3000:
2919:
2917:
2916:
2912:
2889:
2886:
2885:
2881:
2877:
2870:
2851:
2847:
2841:
2837:
2835:
2830:
2827:
2826:
2801:
2797:
2779:
2775:
2771:
2765:
2761:
2759:
2744:
2740:
2734:
2730:
2728:
2710:
2706:
2702:
2696:
2692:
2690:
2688:
2685:
2684:
2665:
2650:
2639:
2612:
2608:
2604:
2594:
2590:
2589:
2587:
2581:
2577:
2565:
2561:
2557:
2547:
2543:
2542:
2540:
2538:
2535:
2534:
2527:
2519:
2515:
2492:
2480:
2475:
2444:
2435:
2426:
2417:
2373:
2369:
2360:
2356:
2351:
2348:
2347:
2342:
2333:
2316:
2305:
2298:
2291:
2266:
2265:
2256:
2252:
2237:
2233:
2212:
2208:
2193:
2189:
2182:
2167:
2163:
2145:
2140:
2121:
2117:
2114:
2113:
2101:
2097:
2088:
2084:
2071:
2062:
2058:
2046:
2042:
2035:
2033:
2030:
2029:
2027:
2018:
1967:
1963:
1945:
1940:
1921:
1917:
1905:
1901:
1889:
1885:
1870:
1866:
1864:
1861:
1860:
1842:
1829:
1818:
1798:
1737:
1736:
1724:
1720:
1714:
1710:
1704:
1693:
1682:
1676:
1672:
1669:
1668:
1644:
1640:
1622:
1617:
1604:
1600:
1594:
1590:
1584:
1573:
1562:
1541:
1537:
1522:
1517:
1504:
1500:
1494:
1490:
1484:
1473:
1465:
1463:
1460:
1459:
1453:
1415:
1411:
1402:
1398:
1392:
1381:
1359:
1355:
1349:
1345:
1339:
1328:
1304:
1301:
1300:
1243:
1239:
1221:
1216:
1203:
1199:
1187:
1183:
1171:
1167:
1152:
1148:
1146:
1143:
1142:
1128:
1122:
1074:
1070:
1064:
1060:
1054:
1043:
1022:
1019:
1018:
1012:identity matrix
1008:Kronecker delta
1005:
978:
977:
966:
960:
959:
948:
938:
937:
925:
921:
897:
893:
878:
873:
860:
856:
844:
840:
831:
827:
822:
819:
818:
808:
795:
784:
713:
709:
700:
696:
676:
673:
672:
657:
629:
625:
608:
605:
604:
575:
561:
553:
551:
548:
547:
545:
542:
541:
515:
487:
483:
477:
473:
456:
453:
452:
422:
414:
412:
410:
407:
406:
350:
345:
343:
340:
339:
318:
313:
311:
308:
307:
296:
266:
259:
201:
198:
197:
161:
126:
123:
122:
90:
87:
86:
62:
59:
58:
45:linear operator
17:
12:
11:
5:
4459:
4449:
4448:
4434:
4433:
4425:
4424:External links
4422:
4420:
4419:
4395:
4389:
4376:
4370:
4354:
4348:
4335:
4320:
4306:
4304:
4301:
4299:
4298:
4286:
4274:
4262:
4250:
4235:
4223:
4221:, p. 436.
4206:
4204:, p. 437.
4189:
4174:
4170:Wasserman 2016
4159:
4157:, p. 435.
4144:
4126:
4124:
4121:
4119:
4116:
4115:
4114:
4109:
4104:
4099:
4094:
4087:
4084:
4041:
4038:
4023:
4017:
4012:
4007:
4004:
4001:
3998:
3993:
3989:
3985:
3981:
3976:
3972:
3966:
3962:
3957:
3954:
3951:
3948:
3945:
3942:
3939:
3935:
3931:
3928:
3905:
3900:
3895:
3890:
3886:
3882:
3879:
3874:
3870:
3866:
3862:
3857:
3853:
3847:
3843:
3837:
3833:
3829:
3826:
3823:
3820:
3816:
3812:
3809:
3777:
3766:
3744:
3738:
3733:
3728:
3725:
3722:
3719:
3714:
3710:
3707:
3704:
3701:
3698:
3675:
3674:
3665:
3663:
3652:
3649:
3646:
3643:
3640:
3637:
3634:
3628:
3625:
3620:
3617:
3614:
3611:
3608:
3602:
3599:
3584:
3583:
3574:
3572:
3561:
3558:
3554:
3550:
3547:
3544:
3541:
3538:
3534:
3530:
3527:
3524:
3468:
3465:
3462:
3458:
3454:
3451:
3448:
3443:
3439:
3432:
3429:
3423:
3419:
3413:
3410:
3407:
3384:
3381:
3378:
3374:
3370:
3367:
3364:
3361:
3358:
3355:
3352:
3348:
3344:
3341:
3335:
3332:
3328:
3323:
3320:
3301:
3298:
3271:
3254:
3251:
3248:
3243:
3239:
3235:
3232:
3229:
3225:
3220:
3216:
3213:
3210:
3204:
3200:
3197:
3194:
3191:
3188:
3185:
3182:
3179:
3176:
3138:
3117:
3114:
3110:
3105:
3101:
3098:
3092:
3088:
3085:
2988:
2985:
2982:
2979:
2976:
2973:
2970:
2967:
2964:
2961:
2958:
2955:
2952:
2949:
2945:
2941:
2937:
2934:
2929:
2925:
2922:
2915:
2911:
2908:
2905:
2902:
2899:
2896:
2893:
2854:
2850:
2844:
2840:
2834:
2812:
2809:
2804:
2800:
2796:
2793:
2790:
2782:
2778:
2774:
2768:
2764:
2757:
2754:
2747:
2743:
2737:
2733:
2727:
2724:
2721:
2713:
2709:
2705:
2699:
2695:
2623:
2615:
2611:
2607:
2602:
2597:
2593:
2584:
2580:
2576:
2568:
2564:
2560:
2555:
2550:
2546:
2530:satisfies the
2479:
2476:
2474:
2471:
2466:
2465:
2440:
2431:
2422:
2413:
2407:
2406:
2387:
2384:
2381:
2376:
2372:
2368:
2363:
2359:
2355:
2344:
2338:
2329:
2314:
2303:
2296:
2289:
2264:
2259:
2255:
2251:
2248:
2245:
2240:
2236:
2232:
2229:
2226:
2223:
2220:
2215:
2211:
2204:
2201:
2196:
2192:
2188:
2185:
2183:
2181:
2178:
2175:
2170:
2166:
2162:
2159:
2156:
2153:
2148:
2143:
2139:
2132:
2129:
2124:
2120:
2116:
2115:
2112:
2109:
2104:
2100:
2096:
2091:
2087:
2083:
2080:
2077:
2074:
2072:
2070:
2065:
2061:
2057:
2054:
2049:
2045:
2041:
2038:
2037:
2023:
2014:
1984:
1981:
1978:
1975:
1970:
1966:
1962:
1959:
1956:
1953:
1948:
1943:
1939:
1932:
1929:
1924:
1920:
1916:
1913:
1908:
1904:
1900:
1897:
1892:
1888:
1884:
1881:
1876:
1873:
1869:
1859:with elements
1838:
1827:
1816:
1797:
1794:
1735:
1730:
1727:
1723:
1717:
1713:
1707:
1702:
1699:
1696:
1692:
1688:
1685:
1683:
1679:
1675:
1671:
1670:
1667:
1664:
1661:
1658:
1655:
1652:
1647:
1643:
1639:
1636:
1633:
1630:
1625:
1620:
1616:
1607:
1603:
1597:
1593:
1587:
1582:
1579:
1576:
1572:
1568:
1565:
1563:
1561:
1558:
1555:
1552:
1549:
1544:
1540:
1536:
1533:
1530:
1525:
1520:
1516:
1507:
1503:
1497:
1493:
1487:
1482:
1479:
1476:
1472:
1468:
1467:
1449:
1432:
1429:
1426:
1423:
1418:
1414:
1410:
1405:
1401:
1395:
1390:
1387:
1384:
1380:
1376:
1373:
1370:
1367:
1362:
1358:
1352:
1348:
1342:
1337:
1334:
1331:
1327:
1323:
1320:
1317:
1314:
1311:
1308:
1266:
1263:
1260:
1257:
1254:
1251:
1246:
1242:
1238:
1235:
1232:
1229:
1224:
1219:
1215:
1206:
1202:
1198:
1195:
1190:
1186:
1182:
1179:
1174:
1170:
1166:
1163:
1158:
1155:
1151:
1141:with elements
1118:
1091:
1088:
1085:
1082:
1077:
1073:
1067:
1063:
1057:
1052:
1049:
1046:
1042:
1038:
1035:
1032:
1029:
1026:
1001:
986:
981:
976:
973:
970:
967:
965:
962:
961:
958:
955:
952:
949:
947:
944:
943:
941:
936:
931:
928:
924:
920:
917:
914:
911:
908:
905:
900:
896:
892:
889:
886:
881:
876:
872:
863:
859:
855:
852:
847:
843:
839:
834:
830:
826:
804:
793:
782:
763:called Ω. The
748:
745:
742:
739:
736:
733:
730:
727:
724:
721:
716:
712:
703:
699:
695:
692:
689:
686:
683:
680:
671:is defined as
656:
653:
640:
635:
632:
628:
624:
621:
618:
615:
612:
592:
589:
584:
581:
578:
573:
567:
564:
559:
556:
550:
513:
498:
493:
490:
486:
480:
476:
472:
469:
466:
463:
460:
434:
431:
428:
425:
420:
417:
392:
389:
386:
383:
380:
377:
374:
371:
368:
365:
362:
356:
353:
349:
324:
321:
317:
295:
292:
264:
257:
243:
242:
233:
231:
220:
217:
214:
211:
208:
205:
181:eigenfunctions
160:
159:Eigenfunctions
157:
133:
130:
106:
103:
100:
97:
94:
66:
52:function space
15:
9:
6:
4:
3:
2:
4458:
4447:
4444:
4443:
4441:
4432:
4431:Atom in a Box
4428:
4427:
4409:
4405:
4401:
4396:
4392:
4386:
4382:
4377:
4373:
4367:
4363:
4359:
4355:
4351:
4345:
4341:
4336:
4333:
4329:
4323:
4317:
4313:
4308:
4307:
4296:, p. 49.
4295:
4290:
4284:, p. 52.
4283:
4278:
4272:, p. 51.
4271:
4266:
4260:, p. 25.
4259:
4254:
4248:, p. 29.
4247:
4242:
4240:
4233:, p. 24.
4232:
4227:
4220:
4215:
4213:
4211:
4203:
4198:
4196:
4194:
4187:, p. 21.
4186:
4181:
4179:
4171:
4166:
4164:
4156:
4151:
4149:
4142:, p. 20.
4141:
4136:
4134:
4132:
4127:
4113:
4110:
4108:
4105:
4103:
4100:
4098:
4095:
4093:
4090:
4089:
4083:
4075:
4071:
4067:
4063:
4056:
4052:
4047:
4037:
4034:
4021:
4015:
4010:
4005:
4002:
3999:
3996:
3991:
3974:
3970:
3964:
3960:
3955:
3952:
3949:
3946:
3940:
3937:
3903:
3898:
3893:
3888:
3884:
3880:
3877:
3872:
3855:
3851:
3845:
3841:
3835:
3831:
3827:
3821:
3818:
3797:
3793:
3782:
3773:
3762:
3761:
3755:
3742:
3736:
3731:
3726:
3723:
3720:
3717:
3712:
3708:
3702:
3696:
3688:
3687:
3673:
3666:
3664:
3650:
3644:
3638:
3635:
3632:
3626:
3615:
3609:
3597:
3590:
3589:
3582:
3575:
3573:
3559:
3545:
3542:
3539:
3525:
3522:
3515:
3514:
3511:
3507:
3503:
3499:
3495:
3491:
3487:
3482:
3481:wave function
3463:
3460:
3449:
3446:
3441:
3430:
3427:
3421:
3417:
3411:
3408:
3405:
3398:
3379:
3376:
3362:
3359:
3353:
3350:
3333:
3318:
3311:
3307:
3297:
3295:
3289:
3283:
3274:
3265:
3252:
3246:
3241:
3237:
3230:
3227:
3223:
3218:
3214:
3211:
3208:
3202:
3198:
3195:
3192:
3186:
3183:
3180:
3174:
3157:
3149:
3141:
3128:
3115:
3112:
3108:
3103:
3099:
3096:
3090:
3086:
3083:
3073:
3069:
3062:
3054:
3046:
3039:
3035:
3031:
3025:
3021:
3014:
3008:
2986:
2980:
2977:
2974:
2971:
2965:
2962:
2959:
2953:
2947:
2943:
2939:
2935:
2932:
2927:
2923:
2920:
2913:
2909:
2906:
2903:
2897:
2891:
2874:
2852:
2848:
2842:
2838:
2832:
2823:
2810:
2807:
2802:
2798:
2794:
2791:
2788:
2780:
2776:
2772:
2766:
2762:
2755:
2752:
2745:
2741:
2735:
2731:
2725:
2722:
2719:
2711:
2707:
2703:
2697:
2693:
2680:
2676:
2672:
2668:
2661:
2657:
2653:
2648:
2643:
2637:
2636:wave equation
2621:
2613:
2609:
2600:
2595:
2582:
2578:
2574:
2566:
2562:
2553:
2548:
2533:
2525:
2524:infinitesimal
2513:
2509:
2503:
2499:
2495:
2484:
2470:
2463:
2462:
2461:
2459:
2454:
2452:
2448:
2443:
2439:
2434:
2430:
2425:
2421:
2416:
2412:
2405:
2401:
2385:
2382:
2374:
2370:
2366:
2361:
2357:
2345:
2341:
2337:
2332:
2328:
2324:
2323:
2322:
2320:
2313:
2309:
2302:
2295:
2288:
2284:
2279:
2262:
2257:
2246:
2238:
2234:
2230:
2221:
2213:
2209:
2202:
2199:
2190:
2186:
2184:
2176:
2168:
2164:
2160:
2154:
2146:
2141:
2137:
2130:
2127:
2118:
2110:
2102:
2098:
2094:
2089:
2085:
2081:
2075:
2073:
2063:
2059:
2055:
2052:
2047:
2043:
2026:
2022:
2017:
2013:
2009:
2005:
2000:
1998:
1982:
1976:
1968:
1964:
1960:
1954:
1946:
1941:
1937:
1930:
1927:
1918:
1914:
1906:
1902:
1898:
1895:
1890:
1886:
1879:
1874:
1871:
1867:
1858:
1854:
1850:
1846:
1841:
1837:
1833:
1826:
1822:
1815:
1811:
1810:Hilbert space
1807:
1803:
1793:
1791:
1787:
1783:
1779:
1775:
1771:
1767:
1763:
1759:
1755:
1750:
1733:
1728:
1725:
1721:
1715:
1711:
1705:
1700:
1697:
1694:
1690:
1686:
1684:
1677:
1673:
1665:
1662:
1659:
1653:
1645:
1641:
1637:
1631:
1623:
1618:
1614:
1601:
1595:
1591:
1585:
1580:
1577:
1574:
1570:
1566:
1564:
1559:
1556:
1550:
1542:
1538:
1531:
1523:
1518:
1514:
1501:
1495:
1491:
1485:
1480:
1477:
1474:
1470:
1457:
1452:
1448:
1443:
1430:
1424:
1416:
1412:
1408:
1403:
1399:
1393:
1388:
1385:
1382:
1378:
1374:
1368:
1360:
1356:
1350:
1346:
1340:
1335:
1332:
1329:
1325:
1321:
1315:
1309:
1306:
1298:
1294:
1290:
1286:
1282:
1277:
1264:
1261:
1258:
1252:
1244:
1240:
1236:
1230:
1222:
1217:
1213:
1200:
1196:
1188:
1184:
1180:
1177:
1172:
1168:
1161:
1156:
1153:
1149:
1140:
1135:
1131:
1126:
1121:
1117:
1113:
1109:
1105:
1089:
1083:
1075:
1071:
1065:
1061:
1055:
1050:
1047:
1044:
1040:
1036:
1030:
1024:
1015:
1013:
1009:
1004:
1000:
984:
974:
971:
968:
963:
956:
953:
950:
945:
939:
934:
929:
926:
922:
918:
915:
912:
906:
898:
894:
887:
879:
874:
870:
857:
853:
845:
841:
837:
832:
828:
816:
812:
807:
803:
799:
792:
788:
781:
777:
772:
770:
766:
762:
746:
743:
740:
734:
728:
722:
714:
710:
697:
693:
687:
684:
681:
670:
666:
665:inner product
661:
652:
638:
633:
630:
626:
622:
616:
610:
590:
587:
582:
579:
576:
571:
565:
562:
557:
554:
539:
535:
531:
526:
524:
520:
512:
496:
491:
488:
484:
478:
474:
470:
464:
458:
451:
429:
423:
418:
415:
403:
390:
384:
378:
375:
372:
366:
360:
354:
351:
347:
322:
319:
315:
305:
301:
291:
289:
285:
281:
276:
274:
270:
263:
256:
252:
251:
241:
234:
232:
218:
215:
212:
209:
206:
203:
196:
195:
192:
190:
186:
182:
178:
174:
170:
166:
156:
154:
149:
147:
131:
128:
120:
104:
101:
98:
95:
92:
84:
80:
64:
57:
53:
49:
46:
42:
41:eigenfunction
38:
30:
26:
21:
4411:. Retrieved
4403:
4380:
4361:
4358:Girod, Bernd
4339:
4311:
4289:
4282:Davydov 1976
4277:
4270:Davydov 1976
4265:
4258:Davydov 1976
4253:
4246:Davydov 1976
4231:Davydov 1976
4226:
4185:Davydov 1976
4140:Davydov 1976
4073:
4069:
4065:
4061:
4054:
4050:
4043:
4035:
3795:
3791:
3783:
3758:
3756:
3684:
3678:
3667:
3576:
3505:
3501:
3497:
3493:
3489:
3485:
3303:
3287:
3269:
3266:
3155:
3147:
3136:
3129:
3071:
3067:
3060:
3052:
3044:
3037:
3033:
3029:
3023:
3019:
3012:
3009:
2872:
2824:
2678:
2674:
2670:
2666:
2659:
2655:
2651:
2644:
2501:
2497:
2493:
2490:
2487:oscillation.
2473:Applications
2467:
2455:
2441:
2437:
2432:
2428:
2423:
2419:
2414:
2410:
2408:
2403:
2399:
2339:
2335:
2330:
2326:
2318:
2311:
2307:
2300:
2293:
2286:
2282:
2280:
2024:
2020:
2015:
2011:
2007:
2001:
1996:
1856:
1852:
1848:
1844:
1839:
1835:
1831:
1824:
1820:
1813:
1805:
1799:
1789:
1785:
1781:
1777:
1773:
1769:
1765:
1761:
1757:
1753:
1751:
1455:
1450:
1446:
1444:
1296:
1292:
1288:
1284:
1280:
1278:
1138:
1136:
1129:
1124:
1119:
1115:
1111:
1107:
1016:
1002:
998:
814:
810:
805:
801:
797:
790:
786:
779:
773:
767:denotes the
764:
760:
668:
662:
658:
540:(0) = 1 and
537:
533:
529:
527:
522:
518:
510:
404:
303:
299:
297:
283:
277:
268:
261:
254:
248:
246:
235:
188:
184:
180:
176:
172:
168:
164:
162:
150:
78:
47:
40:
34:
4390:047115431-8
4371:047198800-6
4349:008020438-4
4332:047150439-4
4326:(Volume 2:
4321:047150447-5
4303:Works cited
3042:, and that
1999:denoted Ω.
663:Define the
171:that, when
153:eigenvector
121:eigenvalue
37:mathematics
3757:Equation (
1847:)}, where
813:)}, where
280:degenerate
83:eigenvalue
31:on a disk.
4413:April 12,
4404:MathWorld
4123:Citations
4016:ℏ
3997:−
3971:φ
3950:∫
3927:Ψ
3904:ℏ
3878:−
3852:φ
3832:∑
3808:Ψ
3737:ℏ
3718:−
3624:∂
3607:∂
3601:ℏ
3546:φ
3526:φ
3438:∇
3418:ℏ
3412:−
3395:with the
3366:Ψ
3340:Ψ
3331:∂
3327:∂
3322:ℏ
3238:ω
3231:
3212:π
3199:
3097:ω
3087:
3027:, namely
2981:ψ
2972:ω
2966:
2936:φ
2921:ω
2910:
2839:ω
2833:−
2799:ω
2795:−
2732:ω
2726:−
2606:∂
2592:∂
2559:∂
2545:∂
2380:⟩
2354:⟨
2258:∗
2195:Ω
2191:∫
2147:∗
2123:Ω
2119:∫
2108:⟩
2079:⟨
2069:⟩
2040:⟨
1947:∗
1923:Ω
1919:∫
1912:⟩
1883:⟨
1802:Hermitian
1691:∑
1624:∗
1606:Ω
1602:∫
1571:∑
1524:∗
1506:Ω
1502:∫
1471:∑
1379:∑
1326:∑
1223:∗
1205:Ω
1201:∫
1194:⟩
1165:⟨
1041:∑
972:≠
923:δ
880:∗
862:Ω
858:∫
851:⟩
825:⟨
715:∗
702:Ω
698:∫
691:⟩
679:⟨
489:λ
376:λ
213:λ
129:λ
117:for some
102:λ
4440:Category
4086:See also
4078:, where
3294:overtone
3282:harmonic
3163:, where
273:spectrum
56:function
3160:
3144:
3047:(0) = 0
2638:. Here
2028:*, or:
1006:is the
4387:
4368:
4346:
4330:
4318:
3308:, the
3076:, and
3032:(0) =
2206:
2134:
1934:
1834:), …,
1611:
1511:
1210:
997:where
867:
800:), …,
707:
119:scalar
4118:Notes
3063:) = 0
3055:) = 0
3040:) = 0
2510:of a
43:of a
39:, an
4415:2016
4385:ISBN
4366:ISBN
4344:ISBN
4328:ISBN
4316:ISBN
4068:) =
3492:) =
3292:-th
3290:− 1)
3280:-th
3059:sin(
3057:and
3051:sin(
3017:and
3003:and
2880:and
2869:and
2491:Let
3304:In
3228:sin
3196:sin
3148:ncπ
3084:sin
3074:= 0
3015:= 0
2963:sin
2907:sin
2398:if
2310:),
1823:),
1458:),
1299:),
1106:of
789:),
286:or
35:In
4442::
4406:.
4402:.
4238:^
4209:^
4192:^
4177:^
4162:^
4147:^
4130:^
4070:λf
3800:,
3484:Ψ(
3296:.
3142:=
3116:0.
3070:=
3022:=
2658:,
2500:,
2418:≠
2402:≠
2334:=
2292:,
2025:ji
2019:=
2016:ij
2006:,
1792:.
1790:λb
1788:=
1786:Ab
1756:=
1754:Ab
1281:Df
1132:=
1014:.
1003:ij
771:.
290:.
260:,
155:.
4417:.
4393:.
4374:.
4352:.
4334:)
4324:.
4172:.
4080:λ
4076:)
4074:t
4072:(
4066:t
4064:(
4062:y
4057:)
4055:t
4053:(
4051:f
4022:.
4011:/
4006:t
4003:E
4000:i
3992:e
3988:)
3984:r
3980:(
3975:E
3965:E
3961:c
3956:E
3953:d
3947:=
3944:)
3941:t
3938:,
3934:r
3930:(
3899:/
3894:t
3889:k
3885:E
3881:i
3873:e
3869:)
3865:r
3861:(
3856:k
3846:k
3842:c
3836:k
3828:=
3825:)
3822:t
3819:,
3815:r
3811:(
3798:)
3796:t
3794:(
3792:T
3786:H
3778:k
3776:E
3767:k
3765:φ
3760:2
3743:.
3732:/
3727:t
3724:E
3721:i
3713:e
3709:=
3706:)
3703:t
3700:(
3697:T
3686:3
3681:E
3672:)
3670:3
3668:(
3651:.
3648:)
3645:t
3642:(
3639:T
3636:E
3633:=
3627:t
3619:)
3616:t
3613:(
3610:T
3598:i
3581:)
3579:2
3577:(
3560:,
3557:)
3553:r
3549:(
3543:E
3540:=
3537:)
3533:r
3529:(
3523:H
3508:)
3506:t
3504:(
3502:T
3500:)
3498:r
3496:(
3494:φ
3490:t
3488:,
3486:r
3467:)
3464:t
3461:,
3457:r
3453:(
3450:V
3447:+
3442:2
3431:m
3428:2
3422:2
3409:=
3406:H
3383:)
3380:t
3377:,
3373:r
3369:(
3363:H
3360:=
3357:)
3354:t
3351:,
3347:r
3343:(
3334:t
3319:i
3288:n
3286:(
3278:n
3272:n
3270:ω
3253:.
3250:)
3247:t
3242:n
3234:(
3224:)
3219:L
3215:x
3209:n
3203:(
3193:=
3190:)
3187:t
3184:,
3181:x
3178:(
3175:h
3165:n
3156:L
3152:/
3139:n
3137:ω
3132:ω
3113:=
3109:)
3104:c
3100:L
3091:(
3072:ψ
3068:φ
3061:ψ
3053:φ
3045:T
3038:L
3036:(
3034:X
3030:X
3024:L
3020:x
3013:x
3005:ψ
3001:φ
2987:,
2984:)
2978:+
2975:t
2969:(
2960:=
2957:)
2954:t
2951:(
2948:T
2944:,
2940:)
2933:+
2928:c
2924:x
2914:(
2904:=
2901:)
2898:x
2895:(
2892:X
2882:c
2878:ω
2873:ω
2871:−
2853:2
2849:c
2843:2
2811:.
2808:T
2803:2
2792:=
2789:T
2781:2
2777:t
2773:d
2767:2
2763:d
2756:,
2753:X
2746:2
2742:c
2736:2
2723:=
2720:X
2712:2
2708:x
2704:d
2698:2
2694:d
2681:)
2679:t
2677:(
2675:T
2673:)
2671:x
2669:(
2667:X
2662:)
2660:t
2656:x
2654:(
2652:h
2640:c
2622:,
2614:2
2610:x
2601:h
2596:2
2583:2
2579:c
2575:=
2567:2
2563:t
2554:h
2549:2
2528:h
2520:t
2516:x
2504:)
2502:t
2498:x
2496:(
2494:h
2442:i
2438:λ
2433:i
2429:λ
2424:j
2420:λ
2415:i
2411:λ
2404:j
2400:i
2386:0
2383:=
2375:j
2371:f
2367:,
2362:i
2358:f
2343:*
2340:i
2336:λ
2331:i
2327:λ
2319:t
2317:(
2315:2
2312:f
2308:t
2306:(
2304:1
2301:f
2297:2
2294:λ
2290:1
2287:λ
2283:D
2263:.
2254:]
2250:)
2247:t
2244:(
2239:i
2235:u
2231:D
2228:[
2225:)
2222:t
2219:(
2214:j
2210:u
2203:t
2200:d
2187:=
2180:)
2177:t
2174:(
2169:j
2165:u
2161:D
2158:)
2155:t
2152:(
2142:i
2138:u
2131:t
2128:d
2111:,
2103:j
2099:u
2095:,
2090:i
2086:u
2082:D
2076:=
2064:j
2060:u
2056:D
2053:,
2048:i
2044:u
2021:A
2012:A
2008:D
1997:t
1983:.
1980:)
1977:t
1974:(
1969:j
1965:u
1961:D
1958:)
1955:t
1952:(
1942:i
1938:u
1931:t
1928:d
1915:=
1907:j
1903:u
1899:D
1896:,
1891:i
1887:u
1880:=
1875:j
1872:i
1868:A
1857:A
1853:D
1849:n
1845:t
1843:(
1840:n
1836:u
1832:t
1830:(
1828:2
1825:u
1821:t
1819:(
1817:1
1814:u
1806:D
1782:D
1778:t
1776:(
1774:f
1770:t
1768:(
1766:f
1762:D
1758:c
1734:.
1729:j
1726:i
1722:A
1716:j
1712:b
1706:n
1701:1
1698:=
1695:j
1687:=
1678:i
1674:c
1666:,
1663:t
1660:d
1657:)
1654:t
1651:(
1646:j
1642:u
1638:D
1635:)
1632:t
1629:(
1619:i
1615:u
1596:j
1592:b
1586:n
1581:1
1578:=
1575:j
1567:=
1560:t
1557:d
1554:)
1551:t
1548:(
1543:j
1539:u
1535:)
1532:t
1529:(
1519:i
1515:u
1496:j
1492:c
1486:n
1481:1
1478:=
1475:j
1456:t
1454:(
1451:i
1447:u
1431:.
1428:)
1425:t
1422:(
1417:j
1413:u
1409:D
1404:j
1400:b
1394:n
1389:1
1386:=
1383:j
1375:=
1372:)
1369:t
1366:(
1361:j
1357:u
1351:j
1347:c
1341:n
1336:1
1333:=
1330:j
1322:=
1319:)
1316:t
1313:(
1310:f
1307:D
1297:t
1295:(
1293:f
1289:D
1285:t
1283:(
1265:.
1262:t
1259:d
1256:)
1253:t
1250:(
1245:j
1241:u
1237:D
1234:)
1231:t
1228:(
1218:i
1214:u
1197:=
1189:j
1185:u
1181:D
1178:,
1173:i
1169:u
1162:=
1157:j
1154:i
1150:A
1139:D
1130:b
1125:n
1120:j
1116:b
1112:t
1110:(
1108:f
1090:,
1087:)
1084:t
1081:(
1076:j
1072:u
1066:j
1062:b
1056:n
1051:1
1048:=
1045:j
1037:=
1034:)
1031:t
1028:(
1025:f
999:δ
985:,
975:j
969:i
964:0
957:j
954:=
951:i
946:1
940:{
935:=
930:j
927:i
919:=
916:t
913:d
910:)
907:t
904:(
899:j
895:u
891:)
888:t
885:(
875:i
871:u
854:=
846:j
842:u
838:,
833:i
829:u
815:n
811:t
809:(
806:n
802:u
798:t
796:(
794:2
791:u
787:t
785:(
783:1
780:u
765:*
761:t
747:,
744:t
741:d
738:)
735:t
732:(
729:g
726:)
723:t
720:(
711:f
694:=
688:g
685:,
682:f
669:D
639:,
634:t
631:2
627:e
623:=
620:)
617:t
614:(
611:f
591:2
588:=
583:0
580:=
577:t
572:|
566:t
563:d
558:f
555:d
538:f
534:t
532:(
530:f
523:t
521:(
519:f
514:0
511:f
497:,
492:t
485:e
479:0
475:f
471:=
468:)
465:t
462:(
459:f
433:)
430:t
427:(
424:f
419:t
416:d
391:.
388:)
385:t
382:(
379:f
373:=
370:)
367:t
364:(
361:f
355:t
352:d
348:d
323:t
320:d
316:d
304:t
300:C
269:D
265:2
262:λ
258:1
255:λ
250:1
240:)
238:1
236:(
219:,
216:f
210:=
207:f
204:D
189:D
185:f
177:D
173:D
169:D
165:D
132:.
105:f
99:=
96:f
93:D
79:D
65:f
48:D
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