961:
693:
956:{\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}}}
3596:
3254:
1824:
2647:
3437:
534:
3101:
1645:
2869:
3426:
609:
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.
1964:
2505:
2500:
2363:
2255:
1587:
2057:
2149:
688:
2749:
1222:
407:
226:
2946:
3747:
1315:
3092:
592:
93:
2762:
1399:
3321:
1829:
390:
1435:, 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 symmetry—more specifically, it does not have
3005:
1110:
1042:
698:
3591:{\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 }}
3249:{\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 }}
2413:
1819:{\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 }}},}
148:
3295:, 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
2259:
1609:
2154:
1467:
164:) 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:
1969:
2064:
625:
2678:
1146:
2642:{\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 )}
167:
2888:
3607:
4211:
3036:
541:
42:
2386:
1349:
3903:
3829:
316:
3682:
1256:
4159:
2951:
1071:
1003:
4003:
529:{\displaystyle {\frac {\partial \psi (x,t)}{\partial x}}={\frac {ip}{\hbar }}e^{{\frac {i}{\hbar }}(px-Et)}={\frac {ip}{\hbar }}\psi .}
1622:
defines limits on how accurately the momentum and position of a single observable system can be known at once. In quantum mechanics,
273:
At the time quantum mechanics was developed in the 1920s, the momentum operator was found by many theoretical physicists, including
3612:
2392:
1439:
1046:
This momentum operator is in position space because the partial derivatives were taken with respect to the spatial variables.
4117:
3969:
3814:
3649:
1826:
so the tilde represents the
Fourier transform, in converting from coordinate space to momentum space. It then holds that
3850:
Bonneau, G., Faraut, J., Valent, G. (2001). "Self-adjoint extensions of operators and the teaching of quantum mechanics".
2864:{\displaystyle T(\varepsilon )=1-\varepsilon {d \over dx}=1-{i \over \hbar }\varepsilon \left(-i\hbar {d \over dx}\right)}
4237:
3421:{\displaystyle \gamma ^{\mu }{\hat {P}}_{\mu }=i\hbar \gamma ^{\mu }\partial _{\mu }={\hat {P}}=i\hbar \partial \!\!\!/}
1959:{\displaystyle {\hat {p}}=\int \!\!dp~|p\rangle p\langle p|=-i\hbar \int \!\!dx~|x\rangle {\frac {d}{dx}}\langle x|~,}
622:
is used instead of one partial derivative. In three dimensions, the plane wave solution to Schrödinger's equation is:
3668:
4138:
3622:
4206:
3996:
1455:
594:
so the momentum of the particle and the value that is measured when a particle is in a plane wave state is the
3836:
for a specific mathematical discussion and proof for the case of a single, uncharged, spin-zero particle. See
1627:
1424:(more technically, in math terminology a "self-adjoint operator") when it acts on physical (in particular,
4169:
310:
122:
3837:
3617:
3292:
1055:
289:. Its existence and form is sometimes taken as one of the foundational postulates of quantum mechanics.
4262:
1425:
3826:
4164:
3989:
2495:{\displaystyle T(\varepsilon )|\psi \rangle =\int dxT(\varepsilon )|x\rangle \langle x|\psi \rangle }
2370:
4185:
250:
is a multiplication operator in the position representation. Note that the definition above is the
1592:
4075:
2660:
1127:
243:
238:
expressed in the momentum representation, the action of the operator is simply multiplication by
102:
1328:, a gauge invariant physical quantity, can be expressed in terms of the canonical momentum, the
4033:
1407:. For electrically neutral particles, the canonical momentum is equal to the kinetic momentum.
4201:
3897:
2882:
1619:
1235:
606:
259:
36:
1431:(In certain artificial situations, such as the quantum states on the semi-infinite interval
605:, the momentum operator is also linear, and because any wave function can be expressed as a
3926:
3869:
3771:
3284:
2382:
1639:
1239:
1464:
to an arbitrary state in either the position or momentum basis, one can easily show that:
282:
8:
2874:
3930:
3873:
3775:
4012:
3885:
3859:
3300:
3296:
1421:
263:
251:
32:
28:
4112:
3965:
3942:
3810:
3789:
3664:
3645:
2656:
2358:{\displaystyle \langle x|{\hat {p}}|x'\rangle =-i\hbar {\frac {d}{dx}}\delta (x-x'),}
1623:
1616:
1612:
1247:
1227:
278:
247:
20:
3889:
2250:{\displaystyle \langle p|{\hat {x}}|p'\rangle =i\hbar {\frac {d}{dp}}\delta (p-p'),}
1582:{\displaystyle \left={\hat {x}}{\hat {p}}-{\hat {p}}{\hat {x}}=i\hbar \mathbb {I} ,}
404:
is the particle energy. The first order partial derivative with respect to space is
4216:
4038:
3934:
3877:
3779:
3030:
2052:{\displaystyle \langle x|{\hat {p}}|\psi \rangle =-i\hbar {\frac {d}{dx}}\psi (x).}
1966:
that is, the momentum acting in coordinate space corresponds to spatial frequency,
1461:
1436:
1404:
1337:
1329:
1325:
1318:
267:
151:
2144:{\displaystyle \langle p|{\hat {x}}|\psi \rangle =i\hbar {\frac {d}{dp}}\psi (p),}
4232:
4133:
4080:
3833:
3015:
1061:
602:
255:
683:{\displaystyle \psi =e^{{\frac {i}{\hbar }}(\mathbf {p} \cdot \mathbf {r} -Et)}}
4143:
4043:
3315:
3307:
3288:
2744:{\displaystyle \psi (x-\varepsilon )=\psi (x)-\varepsilon {\frac {d\psi }{dx}}}
2410:
represents the length of the translation. It satisfies the following identity:
1217:{\displaystyle {\hat {p}}={\hat {p}}_{x}=-i\hbar {\partial \over \partial x}.}
1137:
1065:
112:
3784:
3759:
2061:
An analogous result applies for the position operator in the momentum basis,
4256:
3946:
3793:
2752:
2668:
2664:
1243:
618:
The derivation in three dimensions is the same, except the gradient operator
286:
231:
221:{\displaystyle {\hat {p}}\psi =-i\hbar {\frac {\partial \psi }{\partial x}}}
39:. For the case of one particle in one spatial dimension, the definition is:
35:. The momentum operator is, in the position representation, an example of a
2941:{\displaystyle T(\varepsilon )=1-{\frac {i}{\hbar }}\varepsilon {\hat {p}}}
297:
The momentum and energy operators can be constructed in the following way.
3864:
3742:{\textstyle -i\hbar \int dx\left|x\right\rangle \partial _{x}\langle x|.}
3311:
1310:{\textstyle {\hat {p}}\psi =-i\hbar {\frac {\partial \psi }{\partial x}}}
997:
3283:
preceding the 3-momentum operator. This operator occurs in relativistic
4059:
3981:
3938:
3266:
3019:
595:
306:
274:
235:
3881:
3809:(2nd Edition), R. Resnick, R. Eisberg, John Wiley & Sons, 1985,
3644:(2nd Edition), R. Resnick, R. Eisberg, John Wiley & Sons, 1985,
3849:
2878:
1117:
258:
and not a measurable physical quantity for charged particles in an
3087:{\displaystyle P_{\mu }=\left({\frac {E}{c}},-\mathbf {p} \right)}
587:{\displaystyle {\hat {p}}=-i\hbar {\frac {\partial }{\partial x}}}
88:{\displaystyle {\hat {p}}=-i\hbar {\frac {\partial }{\partial x}}}
3807:
Quantum
Physics of Atoms, Molecules, Solids, Nuclei and Particles
3642:
Quantum
Physics of Atoms, Molecules, Solids, Nuclei and Particles
2885:, so the relation between translation and momentum operators is:
1394:{\displaystyle \mathbf {\hat {P}} =-i\hbar \nabla -q\mathbf {A} }
1068:, the momentum operator can be written in the position basis as:
1317:
will change its value. Therefore, the canonical momentum is not
119:
is the spatial coordinate, and a partial derivative (denoted by
3023:
1443:
2376:
1250:
385:{\displaystyle \psi (x,t)=e^{{\frac {i}{\hbar }}(px-Et)},}
619:
3964:(3rd ed.). Cambridge: Cambridge University Press.
3685:
3608:
Mathematical descriptions of the electromagnetic field
1259:
3679:
In the position coordinate representation, that is,
3440:
3324:
3104:
3039:
2954:
2891:
2765:
2681:
2508:
2416:
2262:
2157:
2067:
1972:
1832:
1648:
1595:
1470:
1352:
1149:
1074:
1006:
696:
628:
544:
410:
319:
292:
170:
125:
45:
3917:
3314:of the 4-momentum is given by contracting with the
3000:{\displaystyle {\hat {p}}=-i\hbar {\frac {d}{dx}}.}
1105:{\displaystyle \mathbf {\hat {p}} =-i\hbar \nabla }
1037:{\displaystyle \mathbf {\hat {p}} =-i\hbar \nabla }
3741:
3590:
3420:
3248:
3086:
2999:
2940:
2863:
2743:
2641:
2494:
2357:
2249:
2143:
2051:
1958:
1818:
1603:
1581:
1393:
1309:
1216:
1104:
1036:
955:
682:
586:
528:
384:
220:
142:
87:
3959:
3758:Zinn-Justin, Jean; Guida, Riccardo (2008-12-04).
3502:
3412:
3411:
3410:
3166:
3014:Inserting the 3d momentum operator above and the
1903:
1902:
1852:
1851:
1739:
1738:
1688:
1687:
1449:
1359:
1081:
1013:
4254:
3757:
1321:, and hence not a measurable physical quantity.
2387:Stone's theorem on one-parameter unitary groups
1049:
3997:
3902:: CS1 maint: multiple names: authors list (
3725:
2618:
2595:
2575:
2572:
2549:
2535:
2532:
2489:
2475:
2472:
2437:
2299:
2263:
2194:
2158:
2099:
2068:
2004:
1973:
1939:
1921:
1876:
1870:
1729:
1715:
1712:
1698:
1678:
1664:
3960:Sakurai, Jun John; Napolitano, Jim (2021).
4004:
3990:
3916:
2377:Derivation from infinitesimal translations
3863:
3783:
1597:
1572:
4011:
3663:, D. McMahon, Mc Graw Hill (USA), 2006,
3613:Translation operator (quantum mechanics)
1000:for the three spatial dimensions, hence
1143:In one spatial dimension, this becomes
538:This suggests the operator equivalence
4255:
3009:
2151:leading to further useful relations,
3985:
305:Starting in one dimension, using the
3838:Lecture notes 4 by Robert Littlejohn
3827:Lecture notes 1 by Robert Littlejohn
1633:
143:{\displaystyle \partial /\partial x}
613:
13:
3716:
3579:
3558:
3546:
3542:
3407:
3374:
3237:
3219:
3207:
3203:
1638:The following discussion uses the
1420:The momentum operator is always a
1377:
1298:
1290:
1202:
1198:
1099:
1031:
807:
799:
772:
764:
737:
729:
701:
601:Since the partial derivative is a
575:
571:
437:
414:
396:is interpreted as momentum in the
293:Origin from de Broglie plane waves
209:
201:
134:
126:
76:
72:
14:
4274:
3692:
3575:
3522:
3404:
3360:
3233:
3183:
2976:
2918:
2838:
2819:
2311:
2203:
2108:
2016:
1896:
1806:
1771:
1568:
1374:
1284:
1193:
1096:
1028:
936:
831:
645:
566:
515:
474:
459:
351:
195:
67:
3499:
3163:
3075:
1387:
1356:
1078:
1010:
942:
903:
878:
853:
786:
751:
716:
662:
654:
300:
1403:The expression above is called
1226:This is the expression for the
4207:Hanbury Brown and Twiss effect
3953:
3910:
3843:
3819:
3800:
3751:
3732:
3673:
3654:
3635:
3487:
3448:
3392:
3342:
3148:
3112:
2961:
2932:
2901:
2895:
2775:
2769:
2712:
2706:
2697:
2685:
2636:
2624:
2611:
2588:
2565:
2542:
2519:
2482:
2465:
2461:
2455:
2430:
2426:
2420:
2349:
2332:
2287:
2280:
2270:
2241:
2224:
2182:
2175:
2165:
2135:
2129:
2092:
2085:
2075:
2043:
2037:
1997:
1990:
1980:
1946:
1914:
1883:
1863:
1839:
1794:
1788:
1782:
1722:
1705:
1671:
1658:
1652:
1556:
1544:
1529:
1517:
1497:
1482:
1456:Canonical commutation relation
1450:Canonical commutation relation
1415:
1266:
1172:
1156:
1060:For a single particle with no
675:
650:
551:
497:
479:
432:
420:
374:
356:
335:
323:
177:
52:
1:
3661:Quantum Mechanics Demystified
3628:
1410:
16:Operator in quantum mechanics
1604:{\displaystyle \mathbb {I} }
7:
3852:American Journal of Physics
3623:Pauli–Lubanski pseudovector
3618:Relativistic wave equations
3601:
3432:(− + + +)
3293:relativistic wave equations
3028:(+ − − −)
1056:Position and momentum space
1050:Definition (position space)
313:of a single free particle,
10:
4279:
2380:
1453:
1253:group transformation, and
1053:
4238:Creation and annihilation
4225:
4194:
4178:
4152:
4126:
4100:
4093:
4068:
4052:
4026:
4019:
3785:10.4249/scholarpedia.8287
1230:. For a charged particle
4186:Transition dipole moment
3962:Modern quantum mechanics
3434:, the operator would be
4076:Anti-symmetric operator
4069:Operators for operators
2667:), one may expand in a
1128:reduced Planck constant
598:of the above operator.
244:multiplication operator
234:consisting of momentum
150:) is used instead of a
103:reduced Planck constant
3919:Zeitschrift für Physik
3743:
3592:
3422:
3250:
3088:
3001:
2942:
2865:
2745:
2663:in some domain of the
2651:Assuming the function
2643:
2496:
2371:Dirac's delta function
2359:
2251:
2145:
2053:
1960:
1820:
1605:
1583:
1395:
1311:
1218:
1106:
1038:
957:
684:
588:
530:
386:
311:Schrödinger's equation
222:
144:
89:
3840:for the general case.
3744:
3593:
3430:If the signature was
3423:
3251:
3089:
3002:
2943:
2866:
2746:
2644:
2497:
2360:
2252:
2146:
2054:
1961:
1821:
1620:uncertainty principle
1606:
1584:
1454:Further information:
1440:translation operators
1396:
1312:
1242:, the position space
1236:electromagnetic field
1219:
1107:
1039:
958:
685:
589:
531:
387:
260:electromagnetic field
223:
145:
90:
37:differential operator
4013:Operators in physics
3683:
3438:
3322:
3285:quantum field theory
3102:
3037:
2952:
2889:
2881:is the generator of
2873:As it is known from
2763:
2679:
2506:
2414:
2393:translation operator
2260:
2155:
2065:
1970:
1830:
1646:
1593:
1468:
1350:
1257:
1240:gauge transformation
1147:
1072:
1004:
694:
690:and the gradient is
626:
542:
408:
317:
266:is not equal to the
262:. In that case, the
168:
123:
43:
31:associated with the
3931:1925ZPhy...34..858B
3874:2001AmJPh..69..322B
3776:2008SchpJ...3.8287Z
3297:partial derivatives
3096:4-momentum operator
3010:4-momentum operator
2875:classical mechanics
1628:conjugate variables
4212:Quantum correlator
3939:10.1007/BF01328531
3832:2012-06-17 at the
3760:"Gauge invariance"
3739:
3588:
3418:
3301:Lorentz covariance
3246:
3084:
2997:
2938:
2861:
2741:
2639:
2492:
2355:
2247:
2141:
2049:
1956:
1816:
1642:. One may write
1601:
1579:
1428:) quantum states.
1422:Hermitian operator
1391:
1307:
1228:canonical momentum
1214:
1102:
1034:
953:
951:
680:
584:
526:
382:
264:canonical momentum
252:canonical momentum
218:
140:
85:
4263:Quantum mechanics
4250:
4249:
4246:
4245:
4233:Casimir invariant
4089:
4088:
3971:978-1-108-47322-4
3882:10.1119/1.1328351
3815:978-0-471-87373-0
3650:978-0-471-87373-0
3553:
3538:
3505:
3490:
3479:
3451:
3395:
3345:
3214:
3199:
3169:
3151:
3140:
3115:
3066:
2992:
2964:
2935:
2921:
2854:
2822:
2803:
2739:
2383:Noether's theorem
2327:
2283:
2219:
2178:
2124:
2088:
2032:
1993:
1952:
1937:
1912:
1861:
1842:
1811:
1809:
1785:
1748:
1697:
1634:Fourier transform
1626:and momentum are
1559:
1547:
1532:
1520:
1500:
1485:
1362:
1305:
1269:
1209:
1175:
1159:
1084:
1016:
939:
834:
814:
779:
744:
648:
582:
554:
518:
477:
462:
444:
354:
283:Erwin Schrödinger
279:Arnold Sommerfeld
248:position operator
216:
180:
83:
55:
25:momentum operator
21:quantum mechanics
4270:
4226:Particle physics
4179:Electromagnetism
4153:Angular momentum
4098:
4097:
4024:
4023:
4006:
3999:
3992:
3983:
3982:
3976:
3975:
3957:
3951:
3950:
3914:
3908:
3907:
3901:
3893:
3867:
3865:quant-ph/0103153
3847:
3841:
3823:
3817:
3804:
3798:
3797:
3787:
3755:
3749:
3748:
3746:
3745:
3740:
3735:
3724:
3723:
3714:
3677:
3671:
3658:
3652:
3639:
3597:
3595:
3594:
3589:
3587:
3586:
3565:
3561:
3554:
3552:
3541:
3539:
3531:
3512:
3508:
3507:
3506:
3498:
3492:
3491:
3483:
3480:
3472:
3459:
3458:
3453:
3452:
3444:
3433:
3427:
3425:
3424:
3419:
3417:
3397:
3396:
3388:
3382:
3381:
3372:
3371:
3353:
3352:
3347:
3346:
3338:
3334:
3333:
3282:
3275:
3264:
3255:
3253:
3252:
3247:
3245:
3244:
3226:
3222:
3215:
3213:
3202:
3200:
3192:
3176:
3172:
3171:
3170:
3162:
3153:
3152:
3144:
3141:
3133:
3123:
3122:
3117:
3116:
3108:
3093:
3091:
3090:
3085:
3083:
3079:
3078:
3067:
3059:
3049:
3048:
3031:metric signature
3029:
3006:
3004:
3003:
2998:
2993:
2991:
2980:
2966:
2965:
2957:
2947:
2945:
2944:
2939:
2937:
2936:
2928:
2922:
2914:
2870:
2868:
2867:
2862:
2860:
2856:
2855:
2853:
2842:
2823:
2815:
2804:
2802:
2791:
2758:
2750:
2748:
2747:
2742:
2740:
2738:
2730:
2722:
2674:
2654:
2648:
2646:
2645:
2640:
2614:
2591:
2568:
2545:
2522:
2501:
2499:
2498:
2493:
2485:
2468:
2433:
2409:
2405:
2368:
2364:
2362:
2361:
2356:
2348:
2328:
2326:
2315:
2298:
2290:
2285:
2284:
2276:
2273:
2256:
2254:
2253:
2248:
2240:
2220:
2218:
2207:
2193:
2185:
2180:
2179:
2171:
2168:
2150:
2148:
2147:
2142:
2125:
2123:
2112:
2095:
2090:
2089:
2081:
2078:
2058:
2056:
2055:
2050:
2033:
2031:
2020:
2000:
1995:
1994:
1986:
1983:
1965:
1963:
1962:
1957:
1950:
1949:
1938:
1936:
1925:
1917:
1910:
1886:
1866:
1859:
1844:
1843:
1835:
1825:
1823:
1822:
1817:
1812:
1810:
1799:
1797:
1787:
1786:
1778:
1775:
1774:
1770:
1750:
1746:
1725:
1708:
1695:
1674:
1640:bra–ket notation
1610:
1608:
1607:
1602:
1600:
1588:
1586:
1585:
1580:
1575:
1561:
1560:
1552:
1549:
1548:
1540:
1534:
1533:
1525:
1522:
1521:
1513:
1507:
1503:
1502:
1501:
1493:
1487:
1486:
1478:
1460:By applying the
1434:
1405:minimal coupling
1400:
1398:
1397:
1392:
1390:
1364:
1363:
1355:
1345:
1338:vector potential
1335:
1330:scalar potential
1326:kinetic momentum
1316:
1314:
1313:
1308:
1306:
1304:
1296:
1288:
1271:
1270:
1262:
1233:
1223:
1221:
1220:
1215:
1210:
1208:
1197:
1183:
1182:
1177:
1176:
1168:
1161:
1160:
1152:
1135:
1125:
1115:
1111:
1109:
1108:
1103:
1086:
1085:
1077:
1043:
1041:
1040:
1035:
1018:
1017:
1009:
995:
984:
973:
962:
960:
959:
954:
952:
945:
940:
932:
924:
917:
913:
912:
911:
906:
900:
899:
887:
886:
881:
875:
874:
862:
861:
856:
850:
849:
835:
827:
819:
815:
813:
805:
797:
795:
794:
789:
780:
778:
770:
762:
760:
759:
754:
745:
743:
735:
727:
725:
724:
719:
689:
687:
686:
681:
679:
678:
665:
657:
649:
641:
614:Three dimensions
593:
591:
590:
585:
583:
581:
570:
556:
555:
547:
535:
533:
532:
527:
519:
514:
506:
501:
500:
478:
470:
463:
458:
450:
445:
443:
435:
412:
403:
399:
395:
391:
389:
388:
383:
378:
377:
355:
347:
268:kinetic momentum
241:
227:
225:
224:
219:
217:
215:
207:
199:
182:
181:
173:
163:
152:total derivative
149:
147:
146:
141:
133:
118:
110:
100:
94:
92:
91:
86:
84:
82:
71:
57:
56:
48:
4278:
4277:
4273:
4272:
4271:
4269:
4268:
4267:
4253:
4252:
4251:
4242:
4221:
4190:
4174:
4148:
4122:
4085:
4081:Ladder operator
4064:
4048:
4015:
4010:
3980:
3979:
3972:
3958:
3954:
3915:
3911:
3895:
3894:
3848:
3844:
3834:Wayback Machine
3824:
3820:
3805:
3801:
3756:
3752:
3731:
3719:
3715:
3704:
3684:
3681:
3680:
3678:
3674:
3659:
3655:
3640:
3636:
3631:
3604:
3582:
3578:
3545:
3540:
3530:
3529:
3525:
3497:
3496:
3482:
3481:
3471:
3467:
3463:
3454:
3443:
3442:
3441:
3439:
3436:
3435:
3431:
3413:
3387:
3386:
3377:
3373:
3367:
3363:
3348:
3337:
3336:
3335:
3329:
3325:
3323:
3320:
3319:
3277:
3270:
3263:
3257:
3240:
3236:
3206:
3201:
3191:
3190:
3186:
3161:
3160:
3143:
3142:
3132:
3131:
3127:
3118:
3107:
3106:
3105:
3103:
3100:
3099:
3074:
3058:
3057:
3053:
3044:
3040:
3038:
3035:
3034:
3027:
3016:energy operator
3012:
2984:
2979:
2956:
2955:
2953:
2950:
2949:
2927:
2926:
2913:
2890:
2887:
2886:
2846:
2841:
2831:
2827:
2814:
2795:
2790:
2764:
2761:
2760:
2756:
2731:
2723:
2721:
2680:
2677:
2676:
2672:
2652:
2610:
2587:
2564:
2541:
2518:
2507:
2504:
2503:
2481:
2464:
2429:
2415:
2412:
2411:
2407:
2396:
2389:
2379:
2366:
2341:
2319:
2314:
2291:
2286:
2275:
2274:
2269:
2261:
2258:
2257:
2233:
2211:
2206:
2186:
2181:
2170:
2169:
2164:
2156:
2153:
2152:
2116:
2111:
2091:
2080:
2079:
2074:
2066:
2063:
2062:
2024:
2019:
1996:
1985:
1984:
1979:
1971:
1968:
1967:
1945:
1929:
1924:
1913:
1882:
1862:
1834:
1833:
1831:
1828:
1827:
1798:
1777:
1776:
1766:
1756:
1752:
1751:
1749:
1721:
1704:
1670:
1647:
1644:
1643:
1636:
1596:
1594:
1591:
1590:
1571:
1551:
1550:
1539:
1538:
1524:
1523:
1512:
1511:
1492:
1491:
1477:
1476:
1475:
1471:
1469:
1466:
1465:
1458:
1452:
1432:
1418:
1413:
1386:
1354:
1353:
1351:
1348:
1347:
1341:
1333:
1319:gauge invariant
1297:
1289:
1287:
1261:
1260:
1258:
1255:
1254:
1231:
1201:
1196:
1178:
1167:
1166:
1165:
1151:
1150:
1148:
1145:
1144:
1131:
1121:
1113:
1076:
1075:
1073:
1070:
1069:
1062:electric charge
1058:
1052:
1008:
1007:
1005:
1002:
1001:
994:
986:
983:
975:
972:
964:
950:
949:
941:
931:
922:
921:
907:
902:
901:
895:
891:
882:
877:
876:
870:
866:
857:
852:
851:
845:
841:
840:
836:
826:
817:
816:
806:
798:
796:
790:
785:
784:
771:
763:
761:
755:
750:
749:
736:
728:
726:
720:
715:
714:
707:
697:
695:
692:
691:
661:
653:
640:
639:
635:
627:
624:
623:
616:
603:linear operator
574:
569:
546:
545:
543:
540:
539:
507:
505:
469:
468:
464:
451:
449:
436:
413:
411:
409:
406:
405:
401:
400:-direction and
397:
393:
346:
345:
341:
318:
315:
314:
303:
295:
256:gauge invariant
254:, which is not
242:, i.e. it is a
239:
208:
200:
198:
172:
171:
169:
166:
165:
155:
129:
124:
121:
120:
116:
106:
96:
75:
70:
47:
46:
44:
41:
40:
33:linear momentum
17:
12:
11:
5:
4276:
4266:
4265:
4248:
4247:
4244:
4243:
4241:
4240:
4235:
4229:
4227:
4223:
4222:
4220:
4219:
4214:
4209:
4204:
4198:
4196:
4192:
4191:
4189:
4188:
4182:
4180:
4176:
4175:
4173:
4172:
4167:
4162:
4156:
4154:
4150:
4149:
4147:
4146:
4144:Kinetic energy
4141:
4136:
4130:
4128:
4124:
4123:
4121:
4120:
4115:
4110:
4104:
4102:
4095:
4091:
4090:
4087:
4086:
4084:
4083:
4078:
4072:
4070:
4066:
4065:
4063:
4062:
4056:
4054:
4050:
4049:
4047:
4046:
4041:
4036:
4030:
4028:
4027:Space and time
4021:
4017:
4016:
4009:
4008:
4001:
3994:
3986:
3978:
3977:
3970:
3952:
3925:(1): 858–888.
3909:
3858:(3): 322–331.
3842:
3818:
3799:
3750:
3738:
3734:
3730:
3727:
3722:
3718:
3713:
3710:
3707:
3703:
3700:
3697:
3694:
3691:
3688:
3672:
3653:
3633:
3632:
3630:
3627:
3626:
3625:
3620:
3615:
3610:
3603:
3600:
3585:
3581:
3577:
3574:
3571:
3568:
3564:
3560:
3557:
3551:
3548:
3544:
3537:
3534:
3528:
3524:
3521:
3518:
3515:
3511:
3504:
3501:
3495:
3489:
3486:
3478:
3475:
3470:
3466:
3462:
3457:
3450:
3447:
3416:
3409:
3406:
3403:
3400:
3394:
3391:
3385:
3380:
3376:
3370:
3366:
3362:
3359:
3356:
3351:
3344:
3341:
3332:
3328:
3316:gamma matrices
3308:Dirac operator
3289:Dirac equation
3287:, such as the
3259:
3243:
3239:
3235:
3232:
3229:
3225:
3221:
3218:
3212:
3209:
3205:
3198:
3195:
3189:
3185:
3182:
3179:
3175:
3168:
3165:
3159:
3156:
3150:
3147:
3139:
3136:
3130:
3126:
3121:
3114:
3111:
3082:
3077:
3073:
3070:
3065:
3062:
3056:
3052:
3047:
3043:
3011:
3008:
2996:
2990:
2987:
2983:
2978:
2975:
2972:
2969:
2963:
2960:
2934:
2931:
2925:
2920:
2917:
2912:
2909:
2906:
2903:
2900:
2897:
2894:
2859:
2852:
2849:
2845:
2840:
2837:
2834:
2830:
2826:
2821:
2818:
2813:
2810:
2807:
2801:
2798:
2794:
2789:
2786:
2783:
2780:
2777:
2774:
2771:
2768:
2737:
2734:
2729:
2726:
2720:
2717:
2714:
2711:
2708:
2705:
2702:
2699:
2696:
2693:
2690:
2687:
2684:
2661:differentiable
2638:
2635:
2632:
2629:
2626:
2623:
2620:
2617:
2613:
2609:
2606:
2603:
2600:
2597:
2594:
2590:
2586:
2583:
2580:
2577:
2574:
2571:
2567:
2563:
2560:
2557:
2554:
2551:
2548:
2544:
2540:
2537:
2534:
2531:
2528:
2525:
2521:
2517:
2514:
2511:
2491:
2488:
2484:
2480:
2477:
2474:
2471:
2467:
2463:
2460:
2457:
2454:
2451:
2448:
2445:
2442:
2439:
2436:
2432:
2428:
2425:
2422:
2419:
2378:
2375:
2354:
2351:
2347:
2344:
2340:
2337:
2334:
2331:
2325:
2322:
2318:
2313:
2310:
2307:
2304:
2301:
2297:
2294:
2289:
2282:
2279:
2272:
2268:
2265:
2246:
2243:
2239:
2236:
2232:
2229:
2226:
2223:
2217:
2214:
2210:
2205:
2202:
2199:
2196:
2192:
2189:
2184:
2177:
2174:
2167:
2163:
2160:
2140:
2137:
2134:
2131:
2128:
2122:
2119:
2115:
2110:
2107:
2104:
2101:
2098:
2094:
2087:
2084:
2077:
2073:
2070:
2048:
2045:
2042:
2039:
2036:
2030:
2027:
2023:
2018:
2015:
2012:
2009:
2006:
2003:
1999:
1992:
1989:
1982:
1978:
1975:
1955:
1948:
1944:
1941:
1935:
1932:
1928:
1923:
1920:
1916:
1909:
1906:
1901:
1898:
1895:
1892:
1889:
1885:
1881:
1878:
1875:
1872:
1869:
1865:
1858:
1855:
1850:
1847:
1841:
1838:
1815:
1808:
1805:
1802:
1796:
1793:
1790:
1784:
1781:
1773:
1769:
1765:
1762:
1759:
1755:
1745:
1742:
1737:
1734:
1731:
1728:
1724:
1720:
1717:
1714:
1711:
1707:
1703:
1700:
1694:
1691:
1686:
1683:
1680:
1677:
1673:
1669:
1666:
1663:
1660:
1657:
1654:
1651:
1635:
1632:
1599:
1578:
1574:
1570:
1567:
1564:
1558:
1555:
1546:
1543:
1537:
1531:
1528:
1519:
1516:
1510:
1506:
1499:
1496:
1490:
1484:
1481:
1474:
1451:
1448:
1417:
1414:
1412:
1409:
1389:
1385:
1382:
1379:
1376:
1373:
1370:
1367:
1361:
1358:
1303:
1300:
1295:
1292:
1286:
1283:
1280:
1277:
1274:
1268:
1265:
1213:
1207:
1204:
1200:
1195:
1192:
1189:
1186:
1181:
1174:
1171:
1164:
1158:
1155:
1138:imaginary unit
1101:
1098:
1095:
1092:
1089:
1083:
1080:
1051:
1048:
1033:
1030:
1027:
1024:
1021:
1015:
1012:
990:
979:
968:
948:
944:
938:
935:
930:
927:
925:
923:
920:
916:
910:
905:
898:
894:
890:
885:
880:
873:
869:
865:
860:
855:
848:
844:
839:
833:
830:
825:
822:
820:
818:
812:
809:
804:
801:
793:
788:
783:
777:
774:
769:
766:
758:
753:
748:
742:
739:
734:
731:
723:
718:
713:
710:
708:
706:
703:
700:
699:
677:
674:
671:
668:
664:
660:
656:
652:
647:
644:
638:
634:
631:
615:
612:
580:
577:
573:
568:
565:
562:
559:
553:
550:
525:
522:
517:
513:
510:
504:
499:
496:
493:
490:
487:
484:
481:
476:
473:
467:
461:
457:
454:
448:
442:
439:
434:
431:
428:
425:
422:
419:
416:
381:
376:
373:
370:
367:
364:
361:
358:
353:
350:
344:
340:
337:
334:
331:
328:
325:
322:
302:
299:
294:
291:
246:, just as the
230:In a basis of
214:
211:
206:
203:
197:
194:
191:
188:
185:
179:
176:
139:
136:
132:
128:
113:imaginary unit
81:
78:
74:
69:
66:
63:
60:
54:
51:
15:
9:
6:
4:
3:
2:
4275:
4264:
4261:
4260:
4258:
4239:
4236:
4234:
4231:
4230:
4228:
4224:
4218:
4215:
4213:
4210:
4208:
4205:
4203:
4200:
4199:
4197:
4193:
4187:
4184:
4183:
4181:
4177:
4171:
4168:
4166:
4163:
4161:
4158:
4157:
4155:
4151:
4145:
4142:
4140:
4137:
4135:
4132:
4131:
4129:
4125:
4119:
4116:
4114:
4111:
4109:
4106:
4105:
4103:
4099:
4096:
4092:
4082:
4079:
4077:
4074:
4073:
4071:
4067:
4061:
4058:
4057:
4055:
4051:
4045:
4042:
4040:
4037:
4035:
4034:d'Alembertian
4032:
4031:
4029:
4025:
4022:
4018:
4014:
4007:
4002:
4000:
3995:
3993:
3988:
3987:
3984:
3973:
3967:
3963:
3956:
3948:
3944:
3940:
3936:
3932:
3928:
3924:
3921:(in German).
3920:
3913:
3905:
3899:
3891:
3887:
3883:
3879:
3875:
3871:
3866:
3861:
3857:
3853:
3846:
3839:
3835:
3831:
3828:
3822:
3816:
3812:
3808:
3803:
3795:
3791:
3786:
3781:
3777:
3773:
3769:
3765:
3761:
3754:
3736:
3728:
3720:
3711:
3708:
3705:
3701:
3698:
3695:
3689:
3686:
3676:
3670:
3669:0-07-145546-9
3666:
3662:
3657:
3651:
3647:
3643:
3638:
3634:
3624:
3621:
3619:
3616:
3614:
3611:
3609:
3606:
3605:
3599:
3583:
3572:
3569:
3566:
3562:
3555:
3549:
3535:
3532:
3526:
3519:
3516:
3513:
3509:
3493:
3484:
3476:
3473:
3468:
3464:
3460:
3455:
3445:
3428:
3414:
3401:
3398:
3389:
3383:
3378:
3368:
3364:
3357:
3354:
3349:
3339:
3330:
3326:
3317:
3313:
3309:
3304:
3302:
3298:
3294:
3290:
3286:
3281:
3274:
3268:
3262:
3241:
3230:
3227:
3223:
3216:
3210:
3196:
3193:
3187:
3180:
3177:
3173:
3157:
3154:
3145:
3137:
3134:
3128:
3124:
3119:
3109:
3097:
3080:
3071:
3068:
3063:
3060:
3054:
3050:
3045:
3041:
3032:
3025:
3021:
3017:
3007:
2994:
2988:
2985:
2981:
2973:
2970:
2967:
2958:
2929:
2923:
2915:
2910:
2907:
2904:
2898:
2892:
2884:
2880:
2876:
2871:
2857:
2850:
2847:
2843:
2835:
2832:
2828:
2824:
2816:
2811:
2808:
2805:
2799:
2796:
2792:
2787:
2784:
2781:
2778:
2772:
2766:
2754:
2753:infinitesimal
2735:
2732:
2727:
2724:
2718:
2715:
2709:
2703:
2700:
2694:
2691:
2688:
2682:
2670:
2669:Taylor series
2666:
2665:complex plane
2662:
2658:
2649:
2633:
2630:
2627:
2621:
2615:
2607:
2604:
2601:
2598:
2592:
2584:
2581:
2578:
2569:
2561:
2558:
2555:
2552:
2546:
2538:
2529:
2526:
2523:
2515:
2512:
2509:
2502:that becomes
2486:
2478:
2469:
2458:
2452:
2449:
2446:
2443:
2440:
2434:
2423:
2417:
2403:
2399:
2394:
2388:
2384:
2374:
2372:
2352:
2345:
2342:
2338:
2335:
2329:
2323:
2320:
2316:
2308:
2305:
2302:
2295:
2292:
2277:
2266:
2244:
2237:
2234:
2230:
2227:
2221:
2215:
2212:
2208:
2200:
2197:
2190:
2187:
2172:
2161:
2138:
2132:
2126:
2120:
2117:
2113:
2105:
2102:
2096:
2082:
2071:
2059:
2046:
2040:
2034:
2028:
2025:
2021:
2013:
2010:
2007:
2001:
1987:
1976:
1953:
1942:
1933:
1930:
1926:
1918:
1907:
1904:
1899:
1893:
1890:
1887:
1879:
1873:
1867:
1856:
1853:
1848:
1845:
1836:
1813:
1803:
1800:
1791:
1779:
1767:
1763:
1760:
1757:
1753:
1743:
1740:
1735:
1732:
1726:
1718:
1709:
1701:
1692:
1689:
1684:
1681:
1675:
1667:
1661:
1655:
1649:
1641:
1631:
1629:
1625:
1621:
1618:
1614:
1613:unit operator
1576:
1565:
1562:
1553:
1541:
1535:
1526:
1514:
1508:
1504:
1494:
1488:
1479:
1472:
1463:
1457:
1447:
1445:
1441:
1438:
1429:
1427:
1423:
1408:
1406:
1401:
1383:
1380:
1371:
1368:
1365:
1344:
1339:
1331:
1327:
1322:
1320:
1301:
1293:
1281:
1278:
1275:
1272:
1263:
1252:
1249:
1245:
1244:wave function
1241:
1237:
1229:
1224:
1211:
1205:
1190:
1187:
1184:
1179:
1169:
1162:
1153:
1141:
1139:
1134:
1129:
1124:
1119:
1093:
1090:
1087:
1067:
1063:
1057:
1047:
1044:
1025:
1022:
1019:
999:
993:
989:
982:
978:
971:
967:
946:
933:
928:
926:
918:
914:
908:
896:
892:
888:
883:
871:
867:
863:
858:
846:
842:
837:
828:
823:
821:
810:
802:
791:
781:
775:
767:
756:
746:
740:
732:
721:
711:
709:
704:
672:
669:
666:
658:
642:
636:
632:
629:
621:
611:
608:
607:superposition
604:
599:
597:
578:
563:
560:
557:
548:
536:
523:
520:
511:
508:
502:
494:
491:
488:
485:
482:
471:
465:
455:
452:
446:
440:
429:
426:
423:
417:
379:
371:
368:
365:
362:
359:
348:
342:
338:
332:
329:
326:
320:
312:
308:
301:One dimension
298:
290:
288:
287:Eugene Wigner
284:
280:
276:
271:
269:
265:
261:
257:
253:
249:
245:
237:
233:
232:Hilbert space
228:
212:
204:
192:
189:
186:
183:
174:
162:
158:
153:
137:
130:
114:
109:
104:
99:
79:
64:
61:
58:
49:
38:
34:
30:
26:
22:
4202:Displacement
4134:Total energy
4107:
3961:
3955:
3922:
3918:
3912:
3898:cite journal
3855:
3851:
3845:
3821:
3806:
3802:
3770:(12): 8287.
3767:
3764:Scholarpedia
3763:
3753:
3675:
3660:
3656:
3641:
3637:
3429:
3305:
3279:
3272:
3260:
3095:
3094:obtains the
3013:
2872:
2650:
2401:
2397:
2390:
2060:
1637:
1459:
1430:
1426:normalizable
1419:
1402:
1342:
1323:
1246:undergoes a
1225:
1142:
1132:
1122:
1059:
1045:
998:unit vectors
991:
987:
980:
976:
969:
965:
617:
600:
537:
309:solution to
304:
296:
272:
229:
160:
156:
107:
97:
24:
18:
4139:Hamiltonian
4101:Fundamental
3312:Dirac slash
2883:translation
2395:is denoted
2369:stands for
1416:Hermiticity
1238:, during a
236:eigenstates
4060:C-symmetry
3629:References
3291:and other
3269:, and the
3267:4-gradient
3020:4-momentum
2755:values of
2381:See also:
1617:Heisenberg
1462:commutator
1433:[0, ∞)
1411:Properties
1120:operator,
1054:See also:
596:eigenvalue
307:plane wave
275:Niels Bohr
4053:Particles
3947:1434-6001
3794:1941-6016
3726:⟨
3717:∂
3696:∫
3693:ℏ
3687:−
3598:instead.
3584:μ
3580:∂
3576:ℏ
3570:−
3559:∇
3547:∂
3543:∂
3523:ℏ
3517:−
3503:^
3488:^
3469:−
3456:μ
3449:^
3408:∂
3405:ℏ
3393:^
3379:μ
3375:∂
3369:μ
3365:γ
3361:ℏ
3350:μ
3343:^
3331:μ
3327:γ
3242:μ
3238:∂
3234:ℏ
3220:∇
3208:∂
3204:∂
3184:ℏ
3167:^
3158:−
3149:^
3120:μ
3113:^
3072:−
3046:μ
3018:into the
2977:ℏ
2971:−
2962:^
2933:^
2924:ε
2919:ℏ
2911:−
2899:ε
2839:ℏ
2833:−
2825:ε
2820:ℏ
2812:−
2788:ε
2785:−
2773:ε
2728:ψ
2719:ε
2716:−
2704:ψ
2695:ε
2692:−
2683:ψ
2634:ε
2631:−
2622:ψ
2619:⟩
2602:∫
2596:⟩
2593:ψ
2585:ε
2582:−
2576:⟨
2573:⟩
2556:∫
2550:⟩
2547:ψ
2536:⟨
2533:⟩
2530:ε
2510:∫
2490:⟩
2487:ψ
2476:⟨
2473:⟩
2459:ε
2444:∫
2438:⟩
2435:ψ
2424:ε
2339:−
2330:δ
2312:ℏ
2306:−
2300:⟩
2281:^
2264:⟨
2231:−
2222:δ
2204:ℏ
2195:⟩
2176:^
2159:⟨
2127:ψ
2109:ℏ
2100:⟩
2097:ψ
2086:^
2069:⟨
2035:ψ
2017:ℏ
2011:−
2005:⟩
2002:ψ
1991:^
1974:⟨
1940:⟨
1922:⟩
1900:∫
1897:ℏ
1891:−
1877:⟨
1871:⟩
1849:∫
1840:^
1807:ℏ
1804:π
1783:~
1780:ψ
1772:ℏ
1736:∫
1730:⟩
1727:ψ
1716:⟨
1713:⟩
1699:⟨
1685:∫
1679:⟩
1676:ψ
1665:⟨
1650:ψ
1569:ℏ
1557:^
1545:^
1536:−
1530:^
1518:^
1498:^
1483:^
1381:−
1378:∇
1375:ℏ
1369:−
1360:^
1299:∂
1294:ψ
1291:∂
1285:ℏ
1279:−
1273:ψ
1267:^
1203:∂
1199:∂
1194:ℏ
1188:−
1173:^
1157:^
1100:∇
1097:ℏ
1091:−
1082:^
1032:∇
1029:ℏ
1023:−
1014:^
947:ψ
937:ℏ
919:ψ
832:ℏ
808:∂
803:ψ
800:∂
773:∂
768:ψ
765:∂
738:∂
733:ψ
730:∂
705:ψ
702:∇
667:−
659:⋅
646:ℏ
630:ψ
576:∂
572:∂
567:ℏ
561:−
552:^
521:ψ
516:ℏ
489:−
475:ℏ
460:ℏ
438:∂
418:ψ
415:∂
366:−
352:ℏ
321:ψ
210:∂
205:ψ
202:∂
196:ℏ
190:−
184:ψ
178:^
135:∂
127:∂
77:∂
73:∂
68:ℏ
62:−
53:^
4257:Category
4118:Rotation
4113:Position
4108:Momentum
3890:16949018
3830:Archived
3712:⟩
3602:See also
3276:becomes
2879:momentum
2657:analytic
2406:, where
2346:′
2296:′
2238:′
2191:′
1624:position
1118:gradient
996:are the
29:operator
4217:Squeeze
4165:Orbital
4094:Quantum
4020:General
3927:Bibcode
3870:Bibcode
3772:Bibcode
3265:is the
2751:so for
1611:is the
1437:unitary
1136:is the
1126:is the
1116:is the
1064:and no
101:is the
27:is the
4195:Optics
4127:Energy
4039:Parity
3968:
3945:
3888:
3813:
3792:
3667:
3648:
3256:where
3024:1-form
3022:(as a
2877:, the
2671:about
2659:(i.e.
2655:to be
2365:where
1951:
1911:
1860:
1747:
1696:
1615:. The
1589:where
1442:. See
1340:
1332:
1234:in an
1130:, and
1112:where
985:, and
963:where
392:where
285:, and
95:where
23:, the
4160:Total
3886:S2CID
3860:arXiv
3026:with
2948:thus
1444:below
1248:local
4170:Spin
4044:Time
3966:ISBN
3943:ISSN
3904:link
3825:See
3811:ISBN
3790:ISSN
3665:ISBN
3646:ISBN
3310:and
3306:The
3299:for
2391:The
2385:and
1336:and
1324:The
1251:U(1)
1066:spin
111:the
3935:doi
3878:doi
3780:doi
3033:):
1446:.)
620:del
19:In
4259::
3941:.
3933:.
3923:34
3900:}}
3896:{{
3884:.
3876:.
3868:.
3856:69
3854:.
3788:.
3778:.
3766:.
3762:.
3318::
3303:.
3280:iħ
3273:iħ
3098::
2759::
2675::
2373:.
1630:.
1346::
1140:.
974:,
281:,
277:,
270:.
161:dx
115:,
105:,
4005:e
3998:t
3991:v
3974:.
3949:.
3937::
3929::
3906:)
3892:.
3880::
3872::
3862::
3796:.
3782::
3774::
3768:3
3737:.
3733:|
3729:x
3721:x
3709:x
3706:|
3702:x
3699:d
3690:i
3573:i
3567:=
3563:)
3556:,
3550:t
3536:c
3533:1
3527:(
3520:i
3514:=
3510:)
3500:p
3494:,
3485:E
3477:c
3474:1
3465:(
3461:=
3446:P
3415:/
3402:i
3399:=
3390:P
3384:=
3358:i
3355:=
3340:P
3278:+
3271:−
3261:μ
3258:∂
3231:i
3228:=
3224:)
3217:,
3211:t
3197:c
3194:1
3188:(
3181:i
3178:=
3174:)
3164:p
3155:,
3146:E
3138:c
3135:1
3129:(
3125:=
3110:P
3081:)
3076:p
3069:,
3064:c
3061:E
3055:(
3051:=
3042:P
2995:.
2989:x
2986:d
2982:d
2974:i
2968:=
2959:p
2930:p
2916:i
2908:1
2905:=
2902:)
2896:(
2893:T
2858:)
2851:x
2848:d
2844:d
2836:i
2829:(
2817:i
2809:1
2806:=
2800:x
2797:d
2793:d
2782:1
2779:=
2776:)
2770:(
2767:T
2757:ε
2736:x
2733:d
2725:d
2713:)
2710:x
2707:(
2701:=
2698:)
2689:x
2686:(
2673:x
2653:ψ
2637:)
2628:x
2625:(
2616:x
2612:|
2608:x
2605:d
2599:=
2589:|
2579:x
2570:x
2566:|
2562:x
2559:d
2553:=
2543:|
2539:x
2527:+
2524:x
2520:|
2516:x
2513:d
2483:|
2479:x
2470:x
2466:|
2462:)
2456:(
2453:T
2450:x
2447:d
2441:=
2431:|
2427:)
2421:(
2418:T
2408:ε
2404:)
2402:ε
2400:(
2398:T
2367:δ
2353:,
2350:)
2343:x
2336:x
2333:(
2324:x
2321:d
2317:d
2309:i
2303:=
2293:x
2288:|
2278:p
2271:|
2267:x
2245:,
2242:)
2235:p
2228:p
2225:(
2216:p
2213:d
2209:d
2201:i
2198:=
2188:p
2183:|
2173:x
2166:|
2162:p
2139:,
2136:)
2133:p
2130:(
2121:p
2118:d
2114:d
2106:i
2103:=
2093:|
2083:x
2076:|
2072:p
2047:.
2044:)
2041:x
2038:(
2029:x
2026:d
2022:d
2014:i
2008:=
1998:|
1988:p
1981:|
1977:x
1954:,
1947:|
1943:x
1934:x
1931:d
1927:d
1919:x
1915:|
1908:x
1905:d
1894:i
1888:=
1884:|
1880:p
1874:p
1868:p
1864:|
1857:p
1854:d
1846:=
1837:p
1814:,
1801:2
1795:)
1792:p
1789:(
1768:/
1764:p
1761:x
1758:i
1754:e
1744:p
1741:d
1733:=
1723:|
1719:p
1710:p
1706:|
1702:x
1693:p
1690:d
1682:=
1672:|
1668:x
1662:=
1659:)
1656:x
1653:(
1598:I
1577:,
1573:I
1566:i
1563:=
1554:x
1542:p
1527:p
1515:x
1509:=
1505:]
1495:p
1489:,
1480:x
1473:[
1388:A
1384:q
1372:i
1366:=
1357:P
1343:A
1334:φ
1302:x
1282:i
1276:=
1264:p
1232:q
1212:.
1206:x
1191:i
1185:=
1180:x
1170:p
1163:=
1154:p
1133:i
1123:ħ
1114:∇
1094:i
1088:=
1079:p
1026:i
1020:=
1011:p
992:z
988:e
981:y
977:e
970:x
966:e
943:p
934:i
929:=
915:)
909:z
904:e
897:z
893:p
889:+
884:y
879:e
872:y
868:p
864:+
859:x
854:e
847:x
843:p
838:(
829:i
824:=
811:z
792:z
787:e
782:+
776:y
757:y
752:e
747:+
741:x
722:x
717:e
712:=
676:)
673:t
670:E
663:r
655:p
651:(
643:i
637:e
633:=
579:x
564:i
558:=
549:p
524:.
512:p
509:i
503:=
498:)
495:t
492:E
486:x
483:p
480:(
472:i
466:e
456:p
453:i
447:=
441:x
433:)
430:t
427:,
424:x
421:(
402:E
398:x
394:p
380:,
375:)
372:t
369:E
363:x
360:p
357:(
349:i
343:e
339:=
336:)
333:t
330:,
327:x
324:(
240:p
213:x
193:i
187:=
175:p
159:/
157:d
154:(
138:x
131:/
117:x
108:i
98:ħ
80:x
65:i
59:=
50:p
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.