972:
704:
967:{\displaystyle {\begin{aligned}\nabla \psi &=\mathbf {e} _{x}{\frac {\partial \psi }{\partial x}}+\mathbf {e} _{y}{\frac {\partial \psi }{\partial y}}+\mathbf {e} _{z}{\frac {\partial \psi }{\partial z}}\\&={\frac {i}{\hbar }}\left(p_{x}\mathbf {e} _{x}+p_{y}\mathbf {e} _{y}+p_{z}\mathbf {e} _{z}\right)\psi \\&={\frac {i}{\hbar }}\mathbf {p} \psi \end{aligned}}}
3607:
3265:
1835:
2658:
3448:
545:
3112:
1656:
2880:
3437:
620:
of other states, when this momentum operator acts on the entire superimposed wave, it yields the momentum eigenvalues for each plane wave component. These new components then superimpose to form the new state, in general not a multiple of the old wave function.
1975:
2516:
2511:
2374:
2266:
1598:
2068:
2160:
699:
2760:
1233:
418:
237:
2957:
3758:
1326:
3103:
603:
104:
2773:
1410:
3332:
1840:
401:
1446:, there is no way to make the momentum operator Hermitian. This is closely related to the fact that a semi-infinite interval cannot have translational symmetryâmore specifically, it does not have
3016:
1121:
1053:
709:
3602:{\displaystyle {\hat {P}}_{\mu }=\left(-{\frac {1}{c}}{\hat {E}},\mathbf {\hat {p}} \right)=-i\hbar \left({\frac {1}{c}}{\frac {\partial }{\partial t}},\nabla \right)=-i\hbar \partial _{\mu }}
3260:{\displaystyle {\hat {P}}_{\mu }=\left({\frac {1}{c}}{\hat {E}},-\mathbf {\hat {p}} \right)=i\hbar \left({\frac {1}{c}}{\frac {\partial }{\partial t}},\nabla \right)=i\hbar \partial _{\mu }}
2424:
3618:
1830:{\displaystyle \psi (x)=\langle x|\psi \rangle =\int \!\!dp~\langle x|p\rangle \langle p|\psi \rangle =\int \!\!dp~{e^{ixp/\hbar }{\tilde {\psi }}(p) \over {\sqrt {2\pi \hbar }}},}
159:
3306:, since energy and momentum combine into the 4-momentum vector above, momentum and energy operators correspond to space and time derivatives, and they need to be first order
2397:
2270:
1620:
2165:
1478:
175:) since the wave function is also a function of time. The "hat" indicates an operator. The "application" of the operator on a differentiable wave function is as follows:
1980:
2075:
3914:
636:
2689:
1157:
2653:{\displaystyle \int dx|x+\varepsilon \rangle \langle x|\psi \rangle =\int dx|x\rangle \langle x-\varepsilon |\psi \rangle =\int dx|x\rangle \psi (x-\varepsilon )}
178:
2899:
4014:
3047:
552:
53:
1360:
3623:
2403:
1450:
3840:
327:
3693:
1267:
4128:
3980:
3825:
3660:
2962:
1082:
1014:
4248:
540:{\displaystyle {\frac {\partial \psi (x,t)}{\partial x}}={\frac {ip}{\hbar }}e^{{\frac {i}{\hbar }}(px-Et)}={\frac {ip}{\hbar }}\psi .}
1633:
defines limits on how accurately the momentum and position of a single observable system can be known at once. In quantum mechanics,
3679:
284:
At the time quantum mechanics was developed in the 1920s, the momentum operator was found by many theoretical physicists, including
4170:
3633:
4149:
1057:
This momentum operator is in position space because the partial derivatives were taken with respect to the spatial variables.
4217:
4007:
1466:
1837:
so the tilde represents the
Fourier transform, in converting from coordinate space to momentum space. It then holds that
3861:
Bonneau, G., Faraut, J., Valent, G. (2001). "Self-adjoint extensions of operators and the teaching of quantum mechanics".
2875:{\displaystyle T(\varepsilon )=1-\varepsilon {d \over dx}=1-{i \over \hbar }\varepsilon \left(-i\hbar {d \over dx}\right)}
1638:
3432:{\displaystyle \gamma ^{\mu }{\hat {P}}_{\mu }=i\hbar \gamma ^{\mu }\partial _{\mu }={\hat {P}}=i\hbar \partial \!\!\!/}
321:
3628:
3303:
1970:{\displaystyle {\hat {p}}=\int \!\!dp~|p\rangle p\langle p|=-i\hbar \int \!\!dx~|x\rangle {\frac {d}{dx}}\langle x|~,}
1066:
633:
is used instead of one partial derivative. In three dimensions, the plane wave solution to Schrödinger's equation is:
4273:
1436:
4175:
4000:
2381:
4196:
605:
so the momentum of the particle and the value that is measured when a particle is in a plane wave state is the
3847:
for a specific mathematical discussion and proof for the case of a single, uncharged, spin-zero particle. See
4180:
4086:
2671:
1138:
254:
113:
1435:(more technically, in math terminology a "self-adjoint operator") when it acts on physical (in particular,
4044:
133:
3848:
4222:
4212:
3908:
2893:
1630:
1246:
617:
300:. Its existence and form is sometimes taken as one of the foundational postulates of quantum mechanics.
270:
47:
1650:
3837:
3937:
3880:
3782:
3295:
2506:{\displaystyle T(\varepsilon )|\psi \rangle =\int dxT(\varepsilon )|x\rangle \langle x|\psi \rangle }
2393:
1250:
293:
2885:
261:
is a multiplication operator in the position representation. Note that the definition above is the
1603:
4023:
3896:
3870:
3311:
3307:
1432:
274:
262:
249:
expressed in the momentum representation, the action of the operator is simply multiplication by
43:
39:
1339:, a gauge invariant physical quantity, can be expressed in terms of the canonical momentum, the
1418:. For electrically neutral particles, the canonical momentum is equal to the kinetic momentum.
4123:
3976:
3953:
3821:
3800:
3675:
3656:
2667:
1634:
1627:
1623:
1258:
1238:
289:
258:
31:
4227:
4049:
3945:
3888:
3790:
3041:
1447:
1442:(In certain artificial situations, such as the quantum states on the semi-infinite interval
1415:
1348:
1340:
1336:
1329:
616:, the momentum operator is also linear, and because any wave function can be expressed as a
278:
162:
4243:
4144:
4091:
3844:
3026:
1475:
to an arbitrary state in either the position or momentum basis, one can easily show that:
1072:
613:
266:
3941:
3884:
3786:
4154:
4054:
3326:
3318:
3299:
1148:
1076:
123:
4267:
2763:
2679:
2675:
2369:{\displaystyle \langle x|{\hat {p}}|x'\rangle =-i\hbar {\frac {d}{dx}}\delta (x-x'),}
1254:
297:
242:
3900:
2261:{\displaystyle \langle p|{\hat {x}}|p'\rangle =i\hbar {\frac {d}{dp}}\delta (p-p'),}
1593:{\displaystyle \left={\hat {x}}{\hat {p}}-{\hat {p}}{\hat {x}}=i\hbar \mathbb {I} ,}
415:
is the particle energy. The first order partial derivative with respect to space is
2063:{\displaystyle \langle x|{\hat {p}}|\psi \rangle =-i\hbar {\frac {d}{dx}}\psi (x).}
1977:
that is, the momentum acting in coordinate space corresponds to spatial frequency,
17:
2155:{\displaystyle \langle p|{\hat {x}}|\psi \rangle =i\hbar {\frac {d}{dp}}\psi (p),}
3322:
1008:
694:{\displaystyle \psi =e^{{\frac {i}{\hbar }}(\mathbf {p} \cdot \mathbf {r} -Et)}}
2755:{\displaystyle \psi (x-\varepsilon )=\psi (x)-\varepsilon {\frac {d\psi }{dx}}}
2421:
represents the length of the translation. It satisfies the following identity:
1228:{\displaystyle {\hat {p}}={\hat {p}}_{x}=-i\hbar {\partial \over \partial x}.}
4070:
3795:
3770:
3277:
3030:
2072:
An analogous result applies for the position operator in the momentum basis,
1472:
606:
317:
285:
246:
3957:
3804:
629:
The derivation in three dimensions is the same, except the gradient operator
232:{\displaystyle {\hat {p}}\psi =-i\hbar {\frac {\partial \psi }{\partial x}}}
50:. For the case of one particle in one spatial dimension, the definition is:
46:. The momentum operator is, in the position representation, an example of a
2952:{\displaystyle T(\varepsilon )=1-{\frac {i}{\hbar }}\varepsilon {\hat {p}}}
308:
The momentum and energy operators can be constructed in the following way.
3875:
3753:{\textstyle -i\hbar \int dx\left|x\right\rangle \partial _{x}\langle x|.}
2889:
1321:{\textstyle {\hat {p}}\psi =-i\hbar {\frac {\partial \psi }{\partial x}}}
1128:
3294:
preceding the 3-momentum operator. This operator occurs in relativistic
3992:
3949:
3892:
3034:
3820:(2nd Edition), R. Resnick, R. Eisberg, John Wiley & Sons, 1985,
3655:(2nd Edition), R. Resnick, R. Eisberg, John Wiley & Sons, 1985,
269:
and not a measurable physical quantity for charged particles in an
3098:{\displaystyle P_{\mu }=\left({\frac {E}{c}},-\mathbf {p} \right)}
598:{\displaystyle {\hat {p}}=-i\hbar {\frac {\partial }{\partial x}}}
99:{\displaystyle {\hat {p}}=-i\hbar {\frac {\partial }{\partial x}}}
3818:
Quantum
Physics of Atoms, Molecules, Solids, Nuclei and Particles
3653:
Quantum
Physics of Atoms, Molecules, Solids, Nuclei and Particles
2896:, so the relation between translation and momentum operators is:
1405:{\displaystyle \mathbf {\hat {P}} =-i\hbar \nabla -q\mathbf {A} }
1079:, the momentum operator can be written in the position basis as:
1328:
will change its value. Therefore, the canonical momentum is not
1261:
130:
is the spatial coordinate, and a partial derivative (denoted by
3996:
1454:
630:
396:{\displaystyle \psi (x,t)=e^{{\frac {i}{\hbar }}(px-Et)},}
3975:(3rd ed.). Cambridge: Cambridge University Press.
3696:
3619:
Mathematical descriptions of the electromagnetic field
1270:
3690:
In the position coordinate representation, that is,
3451:
3335:
3115:
3050:
2965:
2902:
2776:
2692:
2519:
2427:
2273:
2168:
2078:
1983:
1843:
1659:
1606:
1481:
1363:
1160:
1085:
1017:
707:
639:
555:
421:
330:
181:
136:
56:
3928:
4236:
4205:
4189:
4163:
4137:
4111:
4104:
4079:
4063:
4037:
4030:
3325:of the 4-momentum is given by contracting with the
3011:{\displaystyle {\hat {p}}=-i\hbar {\frac {d}{dx}}.}
1116:{\displaystyle \mathbf {\hat {p}} =-i\hbar \nabla }
1048:{\displaystyle \mathbf {\hat {p}} =-i\hbar \nabla }
3752:
3601:
3431:
3259:
3097:
3010:
2951:
2874:
2754:
2652:
2505:
2368:
2260:
2154:
2062:
1969:
1829:
1614:
1592:
1404:
1320:
1227:
1115:
1047:
966:
693:
597:
539:
395:
231:
153:
98:
3769:Zinn-Justin, Jean; Guida, Riccardo (2008-12-04).
3513:
3423:
3422:
3421:
3177:
3025:Inserting the 3d momentum operator above and the
1914:
1913:
1863:
1862:
1750:
1749:
1699:
1698:
1370:
1092:
1024:
1332:, and hence not a measurable physical quantity.
2398:Stone's theorem on one-parameter unitary groups
4008:
8:
3913:: CS1 maint: multiple names: authors list (
3736:
2629:
2606:
2586:
2583:
2560:
2546:
2543:
2500:
2486:
2483:
2448:
2310:
2274:
2205:
2169:
2110:
2079:
2015:
1984:
1950:
1932:
1887:
1881:
1740:
1726:
1723:
1709:
1689:
1675:
3971:Sakurai, Jun John; Napolitano, Jim (2021).
4108:
4034:
4015:
4001:
3993:
2388:Derivation from infinitesimal translations
3874:
3794:
3742:
3730:
3695:
3593:
3551:
3541:
3508:
3507:
3493:
3492:
3482:
3465:
3454:
3453:
3450:
3424:
3398:
3397:
3388:
3378:
3359:
3348:
3347:
3340:
3334:
3251:
3212:
3202:
3172:
3171:
3154:
3153:
3143:
3129:
3118:
3117:
3114:
3085:
3069:
3055:
3049:
2990:
2967:
2966:
2964:
2938:
2937:
2924:
2901:
2852:
2825:
2801:
2775:
2732:
2691:
2621:
2598:
2575:
2552:
2529:
2518:
2492:
2475:
2440:
2426:
2325:
2297:
2286:
2285:
2280:
2272:
2217:
2192:
2181:
2180:
2175:
2167:
2122:
2102:
2091:
2090:
2085:
2077:
2030:
2007:
1996:
1995:
1990:
1982:
1956:
1935:
1924:
1893:
1873:
1845:
1844:
1842:
1809:
1788:
1787:
1777:
1767:
1760:
1732:
1715:
1681:
1658:
1608:
1607:
1605:
1583:
1582:
1562:
1561:
1550:
1549:
1535:
1534:
1523:
1522:
1503:
1502:
1488:
1487:
1480:
1397:
1365:
1364:
1362:
1298:
1272:
1271:
1269:
1207:
1189:
1178:
1177:
1162:
1161:
1159:
1087:
1086:
1084:
1019:
1018:
1016:
952:
942:
918:
913:
906:
893:
888:
881:
868:
863:
856:
837:
807:
801:
796:
772:
766:
761:
737:
731:
726:
708:
706:
672:
664:
651:
650:
638:
580:
557:
556:
554:
516:
480:
479:
460:
422:
420:
357:
356:
329:
209:
183:
182:
180:
140:
135:
81:
58:
57:
55:
3674:, D. McMahon, Mc Graw Hill (USA), 2006,
3624:Translation operator (quantum mechanics)
1011:for the three spatial dimensions, hence
3703:
3645:
3586:
3533:
3415:
3371:
3244:
3194:
2987:
2929:
2849:
2830:
2322:
2214:
2119:
2027:
1907:
1817:
1782:
1579:
1385:
1295:
1204:
1154:In one spatial dimension, this becomes
1107:
1039:
947:
842:
656:
577:
549:This suggests the operator equivalence
526:
485:
470:
362:
206:
78:
3906:
2162:leading to further useful relations,
316:Starting in one dimension, using the
7:
3849:Lecture notes 4 by Robert Littlejohn
3838:Lecture notes 1 by Robert Littlejohn
154:{\displaystyle \partial /\partial x}
3727:
3590:
3569:
3557:
3553:
3418:
3385:
3248:
3230:
3218:
3214:
1649:The following discussion uses the
1431:The momentum operator is always a
1388:
1309:
1301:
1213:
1209:
1110:
1042:
818:
810:
783:
775:
748:
740:
712:
612:Since the partial derivative is a
586:
582:
448:
425:
407:is interpreted as momentum in the
304:Origin from de Broglie plane waves
220:
212:
145:
137:
87:
83:
25:
3510:
3174:
3086:
1398:
1367:
1089:
1021:
953:
914:
889:
864:
797:
762:
727:
673:
665:
1414:The expression above is called
1237:This is the expression for the
4218:Hanbury Brown and Twiss effect
3743:
3498:
3459:
3403:
3353:
3159:
3123:
2972:
2943:
2912:
2906:
2786:
2780:
2723:
2717:
2708:
2696:
2647:
2635:
2622:
2599:
2576:
2553:
2530:
2493:
2476:
2472:
2466:
2441:
2437:
2431:
2360:
2343:
2298:
2291:
2281:
2252:
2235:
2193:
2186:
2176:
2146:
2140:
2103:
2096:
2086:
2054:
2048:
2008:
2001:
1991:
1957:
1925:
1894:
1874:
1850:
1805:
1799:
1793:
1733:
1716:
1682:
1669:
1663:
1567:
1555:
1540:
1528:
1508:
1493:
1467:Canonical commutation relation
1461:Canonical commutation relation
1277:
1183:
1167:
1071:For a single particle with no
686:
661:
562:
508:
490:
443:
431:
385:
367:
346:
334:
188:
63:
1:
3672:Quantum Mechanics Demystified
27:Operator in quantum mechanics
1615:{\displaystyle \mathbb {I} }
3863:American Journal of Physics
3634:PauliâLubanski pseudovector
3629:Relativistic wave equations
3443:(â + + +)
3304:relativistic wave equations
3039:(+ â â â)
1067:Position and momentum space
1061:Definition (position space)
324:of a single free particle,
4290:
2391:
1464:
1264:group transformation, and
1064:
4249:Creation and annihilation
3796:10.4249/scholarpedia.8287
1241:. For a charged particle
4197:Transition dipole moment
3973:Modern quantum mechanics
3445:, the operator would be
4087:Anti-symmetric operator
4080:Operators for operators
2678:), one may expand in a
1139:reduced Planck constant
609:of the above operator.
255:multiplication operator
245:consisting of momentum
161:) is used instead of a
114:reduced Planck constant
3930:Zeitschrift fĂŒr Physik
3754:
3603:
3433:
3261:
3099:
3012:
2953:
2876:
2756:
2674:in some domain of the
2662:Assuming the function
2654:
2507:
2382:Dirac's delta function
2370:
2262:
2156:
2064:
1971:
1831:
1616:
1594:
1406:
1322:
1229:
1117:
1049:
968:
695:
599:
541:
397:
322:Schrödinger's equation
233:
155:
100:
3851:for the general case.
3755:
3604:
3441:If the signature was
3434:
3262:
3100:
3013:
2954:
2877:
2757:
2655:
2508:
2371:
2263:
2157:
2065:
1972:
1832:
1631:uncertainty principle
1617:
1595:
1465:Further information:
1451:translation operators
1407:
1323:
1253:, the position space
1247:electromagnetic field
1230:
1118:
1050:
969:
696:
600:
542:
398:
271:electromagnetic field
234:
156:
101:
48:differential operator
4024:Operators in physics
3694:
3449:
3333:
3296:quantum field theory
3113:
3048:
2963:
2900:
2892:is the generator of
2884:As it is known from
2774:
2690:
2517:
2425:
2404:translation operator
2271:
2166:
2076:
1981:
1841:
1657:
1604:
1479:
1361:
1268:
1251:gauge transformation
1158:
1083:
1015:
705:
701:and the gradient is
637:
553:
419:
328:
277:is not equal to the
273:. In that case, the
179:
134:
54:
42:associated with the
3942:1925ZPhy...34..858B
3885:2001AmJPh..69..322B
3787:2008SchpJ...3.8287Z
3308:partial derivatives
3107:4-momentum operator
3021:4-momentum operator
2886:classical mechanics
1639:conjugate variables
18:4-momentum operator
4223:Quantum correlator
3950:10.1007/BF01328531
3843:2012-06-17 at the
3771:"Gauge invariance"
3750:
3599:
3429:
3312:Lorentz covariance
3257:
3095:
3008:
2949:
2872:
2752:
2650:
2503:
2366:
2258:
2152:
2060:
1967:
1827:
1653:. One may write
1612:
1590:
1439:) quantum states.
1433:Hermitian operator
1402:
1318:
1239:canonical momentum
1225:
1113:
1045:
964:
962:
691:
595:
537:
393:
275:canonical momentum
263:canonical momentum
229:
151:
96:
4274:Quantum mechanics
4261:
4260:
4257:
4256:
4244:Casimir invariant
4100:
4099:
3982:978-1-108-47322-4
3893:10.1119/1.1328351
3826:978-0-471-87373-0
3661:978-0-471-87373-0
3564:
3549:
3516:
3501:
3490:
3462:
3406:
3356:
3225:
3210:
3180:
3162:
3151:
3126:
3077:
3003:
2975:
2946:
2932:
2865:
2833:
2814:
2750:
2394:Noether's theorem
2338:
2294:
2230:
2189:
2135:
2099:
2043:
2004:
1963:
1948:
1923:
1872:
1853:
1822:
1820:
1796:
1759:
1708:
1645:Fourier transform
1637:and momentum are
1570:
1558:
1543:
1531:
1511:
1496:
1373:
1316:
1280:
1220:
1186:
1170:
1095:
1027:
950:
845:
825:
790:
755:
659:
593:
565:
529:
488:
473:
455:
365:
294:Erwin Schrödinger
290:Arnold Sommerfeld
259:position operator
227:
191:
94:
66:
36:momentum operator
32:quantum mechanics
16:(Redirected from
4281:
4237:Particle physics
4190:Electromagnetism
4164:Angular momentum
4109:
4035:
4017:
4010:
4003:
3994:
3987:
3986:
3968:
3962:
3961:
3925:
3919:
3918:
3912:
3904:
3878:
3876:quant-ph/0103153
3858:
3852:
3834:
3828:
3815:
3809:
3808:
3798:
3766:
3760:
3759:
3757:
3756:
3751:
3746:
3735:
3734:
3725:
3688:
3682:
3669:
3663:
3650:
3608:
3606:
3605:
3600:
3598:
3597:
3576:
3572:
3565:
3563:
3552:
3550:
3542:
3523:
3519:
3518:
3517:
3509:
3503:
3502:
3494:
3491:
3483:
3470:
3469:
3464:
3463:
3455:
3444:
3438:
3436:
3435:
3430:
3428:
3408:
3407:
3399:
3393:
3392:
3383:
3382:
3364:
3363:
3358:
3357:
3349:
3345:
3344:
3293:
3286:
3275:
3266:
3264:
3263:
3258:
3256:
3255:
3237:
3233:
3226:
3224:
3213:
3211:
3203:
3187:
3183:
3182:
3181:
3173:
3164:
3163:
3155:
3152:
3144:
3134:
3133:
3128:
3127:
3119:
3104:
3102:
3101:
3096:
3094:
3090:
3089:
3078:
3070:
3060:
3059:
3042:metric signature
3040:
3017:
3015:
3014:
3009:
3004:
3002:
2991:
2977:
2976:
2968:
2958:
2956:
2955:
2950:
2948:
2947:
2939:
2933:
2925:
2881:
2879:
2878:
2873:
2871:
2867:
2866:
2864:
2853:
2834:
2826:
2815:
2813:
2802:
2769:
2761:
2759:
2758:
2753:
2751:
2749:
2741:
2733:
2685:
2665:
2659:
2657:
2656:
2651:
2625:
2602:
2579:
2556:
2533:
2512:
2510:
2509:
2504:
2496:
2479:
2444:
2420:
2416:
2379:
2375:
2373:
2372:
2367:
2359:
2339:
2337:
2326:
2309:
2301:
2296:
2295:
2287:
2284:
2267:
2265:
2264:
2259:
2251:
2231:
2229:
2218:
2204:
2196:
2191:
2190:
2182:
2179:
2161:
2159:
2158:
2153:
2136:
2134:
2123:
2106:
2101:
2100:
2092:
2089:
2069:
2067:
2066:
2061:
2044:
2042:
2031:
2011:
2006:
2005:
1997:
1994:
1976:
1974:
1973:
1968:
1961:
1960:
1949:
1947:
1936:
1928:
1921:
1897:
1877:
1870:
1855:
1854:
1846:
1836:
1834:
1833:
1828:
1823:
1821:
1810:
1808:
1798:
1797:
1789:
1786:
1785:
1781:
1761:
1757:
1736:
1719:
1706:
1685:
1651:braâket notation
1621:
1619:
1618:
1613:
1611:
1599:
1597:
1596:
1591:
1586:
1572:
1571:
1563:
1560:
1559:
1551:
1545:
1544:
1536:
1533:
1532:
1524:
1518:
1514:
1513:
1512:
1504:
1498:
1497:
1489:
1471:By applying the
1445:
1416:minimal coupling
1411:
1409:
1408:
1403:
1401:
1375:
1374:
1366:
1356:
1349:vector potential
1346:
1341:scalar potential
1337:kinetic momentum
1327:
1325:
1324:
1319:
1317:
1315:
1307:
1299:
1282:
1281:
1273:
1244:
1234:
1232:
1231:
1226:
1221:
1219:
1208:
1194:
1193:
1188:
1187:
1179:
1172:
1171:
1163:
1146:
1136:
1126:
1122:
1120:
1119:
1114:
1097:
1096:
1088:
1054:
1052:
1051:
1046:
1029:
1028:
1020:
1006:
995:
984:
973:
971:
970:
965:
963:
956:
951:
943:
935:
928:
924:
923:
922:
917:
911:
910:
898:
897:
892:
886:
885:
873:
872:
867:
861:
860:
846:
838:
830:
826:
824:
816:
808:
806:
805:
800:
791:
789:
781:
773:
771:
770:
765:
756:
754:
746:
738:
736:
735:
730:
700:
698:
697:
692:
690:
689:
676:
668:
660:
652:
625:Three dimensions
604:
602:
601:
596:
594:
592:
581:
567:
566:
558:
546:
544:
543:
538:
530:
525:
517:
512:
511:
489:
481:
474:
469:
461:
456:
454:
446:
423:
414:
410:
406:
402:
400:
399:
394:
389:
388:
366:
358:
279:kinetic momentum
252:
238:
236:
235:
230:
228:
226:
218:
210:
193:
192:
184:
174:
163:total derivative
160:
158:
157:
152:
144:
129:
121:
111:
105:
103:
102:
97:
95:
93:
82:
68:
67:
59:
21:
4289:
4288:
4284:
4283:
4282:
4280:
4279:
4278:
4264:
4263:
4262:
4253:
4232:
4201:
4185:
4159:
4133:
4096:
4092:Ladder operator
4075:
4059:
4026:
4021:
3991:
3990:
3983:
3970:
3969:
3965:
3927:
3926:
3922:
3905:
3860:
3859:
3855:
3845:Wayback Machine
3835:
3831:
3816:
3812:
3768:
3767:
3763:
3726:
3715:
3692:
3691:
3689:
3685:
3670:
3666:
3651:
3647:
3642:
3615:
3589:
3556:
3540:
3536:
3478:
3474:
3452:
3447:
3446:
3442:
3384:
3374:
3346:
3336:
3331:
3330:
3288:
3281:
3274:
3268:
3247:
3217:
3201:
3197:
3142:
3138:
3116:
3111:
3110:
3068:
3064:
3051:
3046:
3045:
3038:
3027:energy operator
3023:
2995:
2961:
2960:
2898:
2897:
2857:
2842:
2838:
2806:
2772:
2771:
2767:
2742:
2734:
2688:
2687:
2683:
2663:
2515:
2514:
2423:
2422:
2418:
2407:
2400:
2390:
2377:
2352:
2330:
2302:
2269:
2268:
2244:
2222:
2197:
2164:
2163:
2127:
2074:
2073:
2035:
1979:
1978:
1940:
1839:
1838:
1763:
1762:
1655:
1654:
1647:
1602:
1601:
1486:
1482:
1477:
1476:
1469:
1463:
1443:
1429:
1424:
1359:
1358:
1352:
1344:
1330:gauge invariant
1308:
1300:
1266:
1265:
1242:
1212:
1176:
1156:
1155:
1142:
1132:
1124:
1081:
1080:
1073:electric charge
1069:
1063:
1013:
1012:
1005:
997:
994:
986:
983:
975:
961:
960:
933:
932:
912:
902:
887:
877:
862:
852:
851:
847:
828:
827:
817:
809:
795:
782:
774:
760:
747:
739:
725:
718:
703:
702:
646:
635:
634:
627:
614:linear operator
585:
551:
550:
518:
475:
462:
447:
424:
417:
416:
412:
411:-direction and
408:
404:
352:
326:
325:
314:
306:
267:gauge invariant
265:, which is not
253:, i.e. it is a
250:
219:
211:
177:
176:
166:
132:
131:
127:
117:
107:
86:
52:
51:
44:linear momentum
28:
23:
22:
15:
12:
11:
5:
4287:
4285:
4277:
4276:
4266:
4265:
4259:
4258:
4255:
4254:
4252:
4251:
4246:
4240:
4238:
4234:
4233:
4231:
4230:
4225:
4220:
4215:
4209:
4207:
4203:
4202:
4200:
4199:
4193:
4191:
4187:
4186:
4184:
4183:
4178:
4173:
4167:
4165:
4161:
4160:
4158:
4157:
4155:Kinetic energy
4152:
4147:
4141:
4139:
4135:
4134:
4132:
4131:
4126:
4121:
4115:
4113:
4106:
4102:
4101:
4098:
4097:
4095:
4094:
4089:
4083:
4081:
4077:
4076:
4074:
4073:
4067:
4065:
4061:
4060:
4058:
4057:
4052:
4047:
4041:
4039:
4038:Space and time
4032:
4028:
4027:
4022:
4020:
4019:
4012:
4005:
3997:
3989:
3988:
3981:
3963:
3936:(1): 858â888.
3920:
3869:(3): 322â331.
3853:
3829:
3810:
3761:
3749:
3745:
3741:
3738:
3733:
3729:
3724:
3721:
3718:
3714:
3711:
3708:
3705:
3702:
3699:
3683:
3664:
3644:
3643:
3641:
3638:
3637:
3636:
3631:
3626:
3621:
3614:
3611:
3596:
3592:
3588:
3585:
3582:
3579:
3575:
3571:
3568:
3562:
3559:
3555:
3548:
3545:
3539:
3535:
3532:
3529:
3526:
3522:
3515:
3512:
3506:
3500:
3497:
3489:
3486:
3481:
3477:
3473:
3468:
3461:
3458:
3427:
3420:
3417:
3414:
3411:
3405:
3402:
3396:
3391:
3387:
3381:
3377:
3373:
3370:
3367:
3362:
3355:
3352:
3343:
3339:
3327:gamma matrices
3319:Dirac operator
3300:Dirac equation
3298:, such as the
3270:
3254:
3250:
3246:
3243:
3240:
3236:
3232:
3229:
3223:
3220:
3216:
3209:
3206:
3200:
3196:
3193:
3190:
3186:
3179:
3176:
3170:
3167:
3161:
3158:
3150:
3147:
3141:
3137:
3132:
3125:
3122:
3093:
3088:
3084:
3081:
3076:
3073:
3067:
3063:
3058:
3054:
3022:
3019:
3007:
3001:
2998:
2994:
2989:
2986:
2983:
2980:
2974:
2971:
2945:
2942:
2936:
2931:
2928:
2923:
2920:
2917:
2914:
2911:
2908:
2905:
2870:
2863:
2860:
2856:
2851:
2848:
2845:
2841:
2837:
2832:
2829:
2824:
2821:
2818:
2812:
2809:
2805:
2800:
2797:
2794:
2791:
2788:
2785:
2782:
2779:
2748:
2745:
2740:
2737:
2731:
2728:
2725:
2722:
2719:
2716:
2713:
2710:
2707:
2704:
2701:
2698:
2695:
2672:differentiable
2649:
2646:
2643:
2640:
2637:
2634:
2631:
2628:
2624:
2620:
2617:
2614:
2611:
2608:
2605:
2601:
2597:
2594:
2591:
2588:
2585:
2582:
2578:
2574:
2571:
2568:
2565:
2562:
2559:
2555:
2551:
2548:
2545:
2542:
2539:
2536:
2532:
2528:
2525:
2522:
2502:
2499:
2495:
2491:
2488:
2485:
2482:
2478:
2474:
2471:
2468:
2465:
2462:
2459:
2456:
2453:
2450:
2447:
2443:
2439:
2436:
2433:
2430:
2389:
2386:
2365:
2362:
2358:
2355:
2351:
2348:
2345:
2342:
2336:
2333:
2329:
2324:
2321:
2318:
2315:
2312:
2308:
2305:
2300:
2293:
2290:
2283:
2279:
2276:
2257:
2254:
2250:
2247:
2243:
2240:
2237:
2234:
2228:
2225:
2221:
2216:
2213:
2210:
2207:
2203:
2200:
2195:
2188:
2185:
2178:
2174:
2171:
2151:
2148:
2145:
2142:
2139:
2133:
2130:
2126:
2121:
2118:
2115:
2112:
2109:
2105:
2098:
2095:
2088:
2084:
2081:
2059:
2056:
2053:
2050:
2047:
2041:
2038:
2034:
2029:
2026:
2023:
2020:
2017:
2014:
2010:
2003:
2000:
1993:
1989:
1986:
1966:
1959:
1955:
1952:
1946:
1943:
1939:
1934:
1931:
1927:
1920:
1917:
1912:
1909:
1906:
1903:
1900:
1896:
1892:
1889:
1886:
1883:
1880:
1876:
1869:
1866:
1861:
1858:
1852:
1849:
1826:
1819:
1816:
1813:
1807:
1804:
1801:
1795:
1792:
1784:
1780:
1776:
1773:
1770:
1766:
1756:
1753:
1748:
1745:
1742:
1739:
1735:
1731:
1728:
1725:
1722:
1718:
1714:
1711:
1705:
1702:
1697:
1694:
1691:
1688:
1684:
1680:
1677:
1674:
1671:
1668:
1665:
1662:
1646:
1643:
1610:
1589:
1585:
1581:
1578:
1575:
1569:
1566:
1557:
1554:
1548:
1542:
1539:
1530:
1527:
1521:
1517:
1510:
1507:
1501:
1495:
1492:
1485:
1462:
1459:
1428:
1425:
1423:
1420:
1400:
1396:
1393:
1390:
1387:
1384:
1381:
1378:
1372:
1369:
1314:
1311:
1306:
1303:
1297:
1294:
1291:
1288:
1285:
1279:
1276:
1224:
1218:
1215:
1211:
1206:
1203:
1200:
1197:
1192:
1185:
1182:
1175:
1169:
1166:
1149:imaginary unit
1112:
1109:
1106:
1103:
1100:
1094:
1091:
1062:
1059:
1044:
1041:
1038:
1035:
1032:
1026:
1023:
1001:
990:
979:
959:
955:
949:
946:
941:
938:
936:
934:
931:
927:
921:
916:
909:
905:
901:
896:
891:
884:
880:
876:
871:
866:
859:
855:
850:
844:
841:
836:
833:
831:
829:
823:
820:
815:
812:
804:
799:
794:
788:
785:
780:
777:
769:
764:
759:
753:
750:
745:
742:
734:
729:
724:
721:
719:
717:
714:
711:
710:
688:
685:
682:
679:
675:
671:
667:
663:
658:
655:
649:
645:
642:
626:
623:
591:
588:
584:
579:
576:
573:
570:
564:
561:
536:
533:
528:
524:
521:
515:
510:
507:
504:
501:
498:
495:
492:
487:
484:
478:
472:
468:
465:
459:
453:
450:
445:
442:
439:
436:
433:
430:
427:
392:
387:
384:
381:
378:
375:
372:
369:
364:
361:
355:
351:
348:
345:
342:
339:
336:
333:
313:
310:
305:
302:
257:, just as the
241:In a basis of
225:
222:
217:
214:
208:
205:
202:
199:
196:
190:
187:
150:
147:
143:
139:
124:imaginary unit
92:
89:
85:
80:
77:
74:
71:
65:
62:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
4286:
4275:
4272:
4271:
4269:
4250:
4247:
4245:
4242:
4241:
4239:
4235:
4229:
4226:
4224:
4221:
4219:
4216:
4214:
4211:
4210:
4208:
4204:
4198:
4195:
4194:
4192:
4188:
4182:
4179:
4177:
4174:
4172:
4169:
4168:
4166:
4162:
4156:
4153:
4151:
4148:
4146:
4143:
4142:
4140:
4136:
4130:
4127:
4125:
4122:
4120:
4117:
4116:
4114:
4110:
4107:
4103:
4093:
4090:
4088:
4085:
4084:
4082:
4078:
4072:
4069:
4068:
4066:
4062:
4056:
4053:
4051:
4048:
4046:
4045:d'Alembertian
4043:
4042:
4040:
4036:
4033:
4029:
4025:
4018:
4013:
4011:
4006:
4004:
3999:
3998:
3995:
3984:
3978:
3974:
3967:
3964:
3959:
3955:
3951:
3947:
3943:
3939:
3935:
3932:(in German).
3931:
3924:
3921:
3916:
3910:
3902:
3898:
3894:
3890:
3886:
3882:
3877:
3872:
3868:
3864:
3857:
3854:
3850:
3846:
3842:
3839:
3833:
3830:
3827:
3823:
3819:
3814:
3811:
3806:
3802:
3797:
3792:
3788:
3784:
3780:
3776:
3772:
3765:
3762:
3747:
3739:
3731:
3722:
3719:
3716:
3712:
3709:
3706:
3700:
3697:
3687:
3684:
3681:
3680:0-07-145546-9
3677:
3673:
3668:
3665:
3662:
3658:
3654:
3649:
3646:
3639:
3635:
3632:
3630:
3627:
3625:
3622:
3620:
3617:
3616:
3612:
3610:
3594:
3583:
3580:
3577:
3573:
3566:
3560:
3546:
3543:
3537:
3530:
3527:
3524:
3520:
3504:
3495:
3487:
3484:
3479:
3475:
3471:
3466:
3456:
3439:
3425:
3412:
3409:
3400:
3394:
3389:
3379:
3375:
3368:
3365:
3360:
3350:
3341:
3337:
3328:
3324:
3320:
3315:
3313:
3309:
3305:
3301:
3297:
3292:
3285:
3279:
3273:
3252:
3241:
3238:
3234:
3227:
3221:
3207:
3204:
3198:
3191:
3188:
3184:
3168:
3165:
3156:
3148:
3145:
3139:
3135:
3130:
3120:
3108:
3091:
3082:
3079:
3074:
3071:
3065:
3061:
3056:
3052:
3043:
3036:
3032:
3028:
3020:
3018:
3005:
2999:
2996:
2992:
2984:
2981:
2978:
2969:
2940:
2934:
2926:
2921:
2918:
2915:
2909:
2903:
2895:
2891:
2887:
2882:
2868:
2861:
2858:
2854:
2846:
2843:
2839:
2835:
2827:
2822:
2819:
2816:
2810:
2807:
2803:
2798:
2795:
2792:
2789:
2783:
2777:
2765:
2764:infinitesimal
2746:
2743:
2738:
2735:
2729:
2726:
2720:
2714:
2711:
2705:
2702:
2699:
2693:
2681:
2680:Taylor series
2677:
2676:complex plane
2673:
2669:
2660:
2644:
2641:
2638:
2632:
2626:
2618:
2615:
2612:
2609:
2603:
2595:
2592:
2589:
2580:
2572:
2569:
2566:
2563:
2557:
2549:
2540:
2537:
2534:
2526:
2523:
2520:
2513:that becomes
2497:
2489:
2480:
2469:
2463:
2460:
2457:
2454:
2451:
2445:
2434:
2428:
2414:
2410:
2405:
2399:
2395:
2387:
2385:
2383:
2363:
2356:
2353:
2349:
2346:
2340:
2334:
2331:
2327:
2319:
2316:
2313:
2306:
2303:
2288:
2277:
2255:
2248:
2245:
2241:
2238:
2232:
2226:
2223:
2219:
2211:
2208:
2201:
2198:
2183:
2172:
2149:
2143:
2137:
2131:
2128:
2124:
2116:
2113:
2107:
2093:
2082:
2070:
2057:
2051:
2045:
2039:
2036:
2032:
2024:
2021:
2018:
2012:
1998:
1987:
1964:
1953:
1944:
1941:
1937:
1929:
1918:
1915:
1910:
1904:
1901:
1898:
1890:
1884:
1878:
1867:
1864:
1859:
1856:
1847:
1824:
1814:
1811:
1802:
1790:
1778:
1774:
1771:
1768:
1764:
1754:
1751:
1746:
1743:
1737:
1729:
1720:
1712:
1703:
1700:
1695:
1692:
1686:
1678:
1672:
1666:
1660:
1652:
1644:
1642:
1640:
1636:
1632:
1629:
1625:
1624:unit operator
1587:
1576:
1573:
1564:
1552:
1546:
1537:
1525:
1519:
1515:
1505:
1499:
1490:
1483:
1474:
1468:
1460:
1458:
1456:
1452:
1449:
1440:
1438:
1434:
1426:
1421:
1419:
1417:
1412:
1394:
1391:
1382:
1379:
1376:
1355:
1350:
1342:
1338:
1333:
1331:
1312:
1304:
1292:
1289:
1286:
1283:
1274:
1263:
1260:
1256:
1255:wave function
1252:
1248:
1240:
1235:
1222:
1216:
1201:
1198:
1195:
1190:
1180:
1173:
1164:
1152:
1150:
1145:
1140:
1135:
1130:
1104:
1101:
1098:
1078:
1074:
1068:
1060:
1058:
1055:
1036:
1033:
1030:
1010:
1004:
1000:
993:
989:
982:
978:
957:
944:
939:
937:
929:
925:
919:
907:
903:
899:
894:
882:
878:
874:
869:
857:
853:
848:
839:
834:
832:
821:
813:
802:
792:
786:
778:
767:
757:
751:
743:
732:
722:
720:
715:
683:
680:
677:
669:
653:
647:
643:
640:
632:
624:
622:
619:
618:superposition
615:
610:
608:
589:
574:
571:
568:
559:
547:
534:
531:
522:
519:
513:
505:
502:
499:
496:
493:
482:
476:
466:
463:
457:
451:
440:
437:
434:
428:
390:
382:
379:
376:
373:
370:
359:
353:
349:
343:
340:
337:
331:
323:
319:
312:One dimension
311:
309:
303:
301:
299:
298:Eugene Wigner
295:
291:
287:
282:
280:
276:
272:
268:
264:
260:
256:
248:
244:
243:Hilbert space
239:
223:
215:
203:
200:
197:
194:
185:
173:
169:
164:
148:
141:
125:
120:
115:
110:
90:
75:
72:
69:
60:
49:
45:
41:
37:
33:
19:
4213:Displacement
4145:Total energy
4118:
3972:
3966:
3933:
3929:
3923:
3909:cite journal
3866:
3862:
3856:
3832:
3817:
3813:
3781:(12): 8287.
3778:
3775:Scholarpedia
3774:
3764:
3686:
3671:
3667:
3652:
3648:
3440:
3316:
3290:
3283:
3271:
3106:
3105:obtains the
3024:
2883:
2661:
2412:
2408:
2401:
2071:
1648:
1470:
1441:
1437:normalizable
1430:
1413:
1353:
1334:
1257:undergoes a
1236:
1153:
1143:
1133:
1070:
1056:
1009:unit vectors
1002:
998:
991:
987:
980:
976:
628:
611:
548:
320:solution to
315:
307:
283:
240:
171:
167:
118:
108:
35:
29:
4150:Hamiltonian
4112:Fundamental
3323:Dirac slash
2894:translation
2406:is denoted
2380:stands for
1427:Hermiticity
1249:, during a
247:eigenstates
4071:C-symmetry
3640:References
3302:and other
3280:, and the
3278:4-gradient
3031:4-momentum
2766:values of
2392:See also:
1628:Heisenberg
1473:commutator
1444:[0, â)
1422:Properties
1131:operator,
1065:See also:
607:eigenvalue
318:plane wave
286:Niels Bohr
4064:Particles
3958:1434-6001
3805:1941-6016
3737:⟨
3728:∂
3707:∫
3704:ℏ
3698:−
3609:instead.
3595:μ
3591:∂
3587:ℏ
3581:−
3570:∇
3558:∂
3554:∂
3534:ℏ
3528:−
3514:^
3499:^
3480:−
3467:μ
3460:^
3419:∂
3416:ℏ
3404:^
3390:μ
3386:∂
3380:μ
3376:γ
3372:ℏ
3361:μ
3354:^
3342:μ
3338:γ
3253:μ
3249:∂
3245:ℏ
3231:∇
3219:∂
3215:∂
3195:ℏ
3178:^
3169:−
3160:^
3131:μ
3124:^
3083:−
3057:μ
3029:into the
2988:ℏ
2982:−
2973:^
2944:^
2935:ε
2930:ℏ
2922:−
2910:ε
2850:ℏ
2844:−
2836:ε
2831:ℏ
2823:−
2799:ε
2796:−
2784:ε
2739:ψ
2730:ε
2727:−
2715:ψ
2706:ε
2703:−
2694:ψ
2645:ε
2642:−
2633:ψ
2630:⟩
2613:∫
2607:⟩
2604:ψ
2596:ε
2593:−
2587:⟨
2584:⟩
2567:∫
2561:⟩
2558:ψ
2547:⟨
2544:⟩
2541:ε
2521:∫
2501:⟩
2498:ψ
2487:⟨
2484:⟩
2470:ε
2455:∫
2449:⟩
2446:ψ
2435:ε
2350:−
2341:δ
2323:ℏ
2317:−
2311:⟩
2292:^
2275:⟨
2242:−
2233:δ
2215:ℏ
2206:⟩
2187:^
2170:⟨
2138:ψ
2120:ℏ
2111:⟩
2108:ψ
2097:^
2080:⟨
2046:ψ
2028:ℏ
2022:−
2016:⟩
2013:ψ
2002:^
1985:⟨
1951:⟨
1933:⟩
1911:∫
1908:ℏ
1902:−
1888:⟨
1882:⟩
1860:∫
1851:^
1818:ℏ
1815:π
1794:~
1791:ψ
1783:ℏ
1747:∫
1741:⟩
1738:ψ
1727:⟨
1724:⟩
1710:⟨
1696:∫
1690:⟩
1687:ψ
1676:⟨
1661:ψ
1580:ℏ
1568:^
1556:^
1547:−
1541:^
1529:^
1509:^
1494:^
1392:−
1389:∇
1386:ℏ
1380:−
1371:^
1310:∂
1305:ψ
1302:∂
1296:ℏ
1290:−
1284:ψ
1278:^
1214:∂
1210:∂
1205:ℏ
1199:−
1184:^
1168:^
1111:∇
1108:ℏ
1102:−
1093:^
1043:∇
1040:ℏ
1034:−
1025:^
958:ψ
948:ℏ
930:ψ
843:ℏ
819:∂
814:ψ
811:∂
784:∂
779:ψ
776:∂
749:∂
744:ψ
741:∂
716:ψ
713:∇
678:−
670:⋅
657:ℏ
641:ψ
587:∂
583:∂
578:ℏ
572:−
563:^
532:ψ
527:ℏ
500:−
486:ℏ
471:ℏ
449:∂
429:ψ
426:∂
377:−
363:ℏ
332:ψ
221:∂
216:ψ
213:∂
207:ℏ
201:−
195:ψ
189:^
146:∂
138:∂
88:∂
84:∂
79:ℏ
73:−
64:^
4268:Category
4129:Rotation
4124:Position
4119:Momentum
3901:16949018
3841:Archived
3723:⟩
3613:See also
3287:becomes
2890:momentum
2668:analytic
2417:, where
2357:′
2307:′
2249:′
2202:′
1635:position
1129:gradient
1007:are the
40:operator
4228:Squeeze
4176:Orbital
4105:Quantum
4031:General
3938:Bibcode
3881:Bibcode
3783:Bibcode
3276:is the
2762:so for
1622:is the
1448:unitary
1147:is the
1137:is the
1127:is the
1075:and no
112:is the
38:is the
4206:Optics
4138:Energy
4050:Parity
3979:
3956:
3899:
3824:
3803:
3678:
3659:
3267:where
3035:1-form
3033:(as a
2888:, the
2682:about
2670:(i.e.
2666:to be
2376:where
1962:
1922:
1871:
1758:
1707:
1626:. The
1600:where
1453:. See
1351:
1343:
1245:in an
1141:, and
1123:where
996:, and
974:where
403:where
296:, and
106:where
34:, the
4171:Total
3897:S2CID
3871:arXiv
3037:with
2959:thus
1455:below
1259:local
4181:Spin
4055:Time
3977:ISBN
3954:ISSN
3915:link
3836:See
3822:ISBN
3801:ISSN
3676:ISBN
3657:ISBN
3321:and
3317:The
3310:for
2402:The
2396:and
1347:and
1335:The
1262:U(1)
1077:spin
122:the
3946:doi
3889:doi
3791:doi
3044:):
1457:.)
631:del
30:In
4270::
3952:.
3944:.
3934:34
3911:}}
3907:{{
3895:.
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3879:.
3867:69
3865:.
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3284:iħ
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292:,
288:,
281:.
172:dx
126:,
116:,
4016:e
4009:t
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3985:.
3960:.
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3903:.
3891::
3883::
3873::
3807:.
3793::
3785::
3779:3
3748:.
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3732:x
3720:x
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3574:)
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3544:1
3538:(
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3505:,
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2907:(
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2778:T
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2236:(
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2177:|
2173:p
2150:,
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2141:(
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2125:d
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2114:=
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2058:.
2055:)
2052:x
2049:(
2040:x
2037:d
2033:d
2025:i
2019:=
2009:|
1999:p
1992:|
1988:x
1965:,
1958:|
1954:x
1945:x
1942:d
1938:d
1930:x
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1879:p
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1800:(
1779:/
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1673:=
1670:)
1667:x
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1516:]
1506:p
1500:,
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1484:[
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1377:=
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