2384:
momentum, whose possible values are taken from a locally compact
Abelian group and its dual. A quantum mechanical state can be fully represented in terms of either variables, and the transformation used to go between position and momentum spaces is, in each of the three cases, a variant of the Fourier transform. The table uses bra-ket notation as well as mathematical terminology describing Canonical commutation relations (CCR).
992:
2401:. Since a quantum mechanical particle has a frequency proportional to the momentum (de Broglie's equation given above), describing the particle as a sum of its momentum components is equivalent to describing it as a sum of frequency components (i.e. a Fourier transform). This becomes clear when we ask ourselves how we can transform from one representation to another.
680:
2380:
1977:
1719:
3674:
3045:
463:
137:
These quantities and ideas transcend all of classical and quantum physics, and a physical system can be described using either the positions of the constituent particles, or their momenta, both formulations equivalently provide the same information about the system in consideration. Another quantity
2095:
The advantage of the
Legendre transformation is that the relation between the new and old functions and their variables are obtained in the process. Both the coordinate and momentum forms of the equation are equivalent and contain the same information about the dynamics of the system. This form may
2383:
Comparison and summary of relations between conjugate variables in discrete-variable (DV), rotor (ROT), and continuous-variable (CV) phase spaces (taken from arXiv:1709.04460). Most physically relevant phase spaces consist of combinations of these three. Each phase space consists of position and
3308:
2621:
1732:
987:{\displaystyle dL=\sum _{i=1}^{n}\left({\frac {\partial L}{\partial q_{i}}}dq_{i}+{\frac {\partial L}{\partial {\dot {q}}_{i}}}d{\dot {q}}_{i}\right)+{\frac {\partial L}{\partial t}}dt=\sum _{i=1}^{n}({\dot {p}}_{i}dq_{i}+p_{i}d{\dot {q}}_{i})+{\frac {\partial L}{\partial t}}dt\,,}
3536:
3493:
1517:
2253:
2913:
3392:
2091:
2898:
1506:
312:
3205:
2521:
2788:
1249:
1125:
3200:
2513:
2375:
A feature of quantum mechanics is that phase spaces can come in different types: discrete-variable, rotor, and continuous-variable. The table below summarizes some relations involved in the three types of phase spaces.
3397:
2131:
3319:
609:
539:
2805:
1254:
1984:
1972:{\displaystyle L'=L-\sum _{i=1}^{n}(q_{i}{\dot {p}}_{i}+{\dot {q}}_{i}p_{i})\,,\quad -{\dot {q}}_{i}={\frac {\partial L'}{\partial p_{i}}}\,,\quad -q_{i}={\frac {\partial L'}{\partial {\dot {p}}_{i}}}\,.}
2337:
By choosing the eigenfunctions of a different operator as a set of basis functions, one can arrive at a number of different representations of the same state. If one picks the eigenfunctions of the
2661:
3669:{\displaystyle \phi (\mathbf {k} )={\frac {1}{({\sqrt {2\pi }})^{3}}}\int _{\mathbf {r} {\text{-space}}}\psi (\mathbf {r} )e^{-i\mathbf {k} \cdot \mathbf {r} }\mathrm {d} ^{3}\mathbf {r} .}
1714:{\displaystyle dL'=\sum _{i=1}^{n}\left({\frac {\partial L'}{\partial p_{i}}}dp_{i}+{\frac {\partial L'}{\partial {\dot {p}}_{i}}}d{\dot {p}}_{i}\right)+{\frac {\partial L'}{\partial t}}dt}
3040:{\displaystyle \psi (\mathbf {r} )={\frac {1}{({\sqrt {2\pi }})^{3}}}\int _{\mathbf {k} {\text{-space}}}\phi (\mathbf {k} )e^{i\mathbf {k} \cdot \mathbf {r} }\mathrm {d} ^{3}\mathbf {k} }
3122:
3531:
3084:
2692:
2370:
1002:
for differentials allows the exchange of differentials in the generalized coordinates and velocities for the differentials in generalized momenta and their time derivatives,
2735:
1129:
1005:
458:{\displaystyle {\frac {d}{dt}}{\frac {\partial L}{\partial {\dot {q}}_{i}}}={\frac {\partial L}{\partial q_{i}}}\,,\quad {\dot {q}}_{i}\equiv {\frac {dq_{i}}{dt}}\,.}
3127:
2723:
3699:
being given explicitly by the
Fourier transform, namely a quarter-cycle rotation in phase space, generated by the oscillator Hamiltonian. Thus, they have the same
2443:
544:
474:
3303:{\displaystyle \phi (\mathbf {k} )=\int _{\mathbf {r} {\text{-space}}}\psi (\mathbf {r} )\phi _{\mathbf {r} }(\mathbf {k} )\mathrm {d} ^{3}\mathbf {r} .}
2616:{\displaystyle \psi (\mathbf {r} )=\int _{\mathbf {k} {\text{-space}}}\phi (\mathbf {k} )\psi _{\mathbf {k} }(\mathbf {r} )\mathrm {d} ^{3}\mathbf {k} }
220:
which states the momentum and wavevector of a free particle are proportional to each other. In this context, when it is unambiguous, the terms "
3998:
Albert, Victor V; Pascazio, Saverio; Devoret, Michel H (2017). "General phase spaces: from discrete variables to rotor and continuum limits".
4082:
4052:
3982:
3954:
3927:
3488:{\displaystyle \phi _{\mathbf {r} }(\mathbf {k} )={\frac {1}{\left({\sqrt {2\pi }}\right)^{3}}}e^{-i\mathbf {k} \cdot \mathbf {r} }}
2248:{\displaystyle {\dot {q}}_{i}={\frac {\partial H}{\partial p_{i}}}\,,\quad {\dot {p}}_{i}=-{\frac {\partial H}{\partial q_{i}}}\,.}
3387:{\displaystyle \mathbf {\hat {r}} =i\hbar {\frac {\partial }{\partial \mathbf {p} }}=i{\frac {\partial }{\partial \mathbf {k} }}}
2112:
the momenta, the
Hamiltonian equations of motion place coordinates and momenta on equal footing. For a system with Hamiltonian
3700:
1721:
so by comparison of differentials of the
Lagrangians, the momenta, and their time derivatives, the momentum space Lagrangian
3839:
3754:
265:
3844:
2893:{\displaystyle \psi _{\mathbf {k} }(\mathbf {r} )={\frac {1}{({\sqrt {2\pi }})^{3}}}e^{i\mathbf {k} \cdot \mathbf {r} }}
2626:
1501:{\displaystyle d\left=-\sum _{i=1}^{n}({\dot {q}}_{i}dp_{i}+q_{i}d{\dot {p}}_{i})+{\frac {\partial L}{\partial t}}dt\,.}
306:
2086:{\displaystyle {\frac {d}{dt}}{\frac {\partial L'}{\partial {\dot {p}}_{i}}}={\frac {\partial L'}{\partial p_{i}}}\,.}
4109:
3758:
994:
where the definition of generalized momentum and Euler–Lagrange equations have replaced the partial derivatives of
3533:
can be made in terms of the eigenfunctions of this operator, which turns out to be the inverse
Fourier transform,
3047:
and we see that the momentum representation is related to the position representation by a
Fourier transform.
3089:
2327:
670:
41:
3692:
2289:
302:
111:
3505:
3058:
2666:
2344:
101:
a physical system can have; the momentum vector of a particle corresponds to its motion, with units of .
2277:
2105:
224:" and "wavevector" are used interchangeably. However, the de Broglie relation is not true in a crystal.
191:
134:). Conversely, the inverse Fourier transform of a momentum space function is a position space function.
3809:
3784:-space is still meaningful and extremely useful, but it differs in several ways from the non-crystal
2795:
238:
206:/2 stating that position and momentum cannot be simultaneously known to arbitrary precision, and the
73:
4073:
3712:
207:
181:
4025:
4007:
3793:
3724:
2281:
674:
106:
4104:
4078:
4048:
3978:
3950:
3923:
3313:
2729:
2394:
2338:
2301:
2269:
2263:
187:
158:
154:
123:
54:
4017:
3834:
3762:
3742:
3696:
2398:
2708:
2791:
468:
94:
69:
61:
614:
81:
20:
2783:{\displaystyle \mathbf {\hat {p}} =-i\hbar {\frac {\partial }{\partial \mathbf {r} }}}
4098:
4029:
3916:
2799:
2410:
2305:
2297:
2273:
2292:
states. In principle one is free to choose the set of basis states, as long as they
471:). Introducing the definition of canonical momentum for each generalized coordinate
190:
provides two fundamental examples of the duality between position and momentum, the
2424:, then we can write this functions as a weighted sum of orthogonal basis functions
2285:
1981:
Combining the last two equations gives the momentum space Euler–Lagrange equations
999:
37:
3944:
3829:
3496:
2901:
2293:
1244:{\displaystyle p_{i}d{\dot {q}}_{i}=d({\dot {q}}_{i}p_{i})-{\dot {q}}_{i}dp_{i}}
1120:{\displaystyle {\dot {p}}_{i}dq_{i}=d(q_{i}{\dot {p}}_{i})-q_{i}d{\dot {p}}_{i}}
143:
4021:
3195:{\displaystyle \phi (\mathbf {k} )=\sum _{j}\psi _{j}\phi _{j}(\mathbf {k} ),}
2393:
The momentum representation of a wave function is very closely related to the
85:
2508:{\displaystyle \psi (\mathbf {r} )=\sum _{j}\phi _{j}\psi _{j}(\mathbf {r} )}
3766:
104:
Mathematically, the duality between position and momentum is an example of
80:; a position vector defines a point in space. (If the position vector of a
16:
Physical spaces representing position and momentum, Fourier-transform duals
2663:, say as the set of eigenfunctions of the momentum operator, the function
2379:
3805:
3734:
3730:
2516:
221:
33:
2096:
be more useful when momentum or angular momentum enters the
Lagrangian.
29:
3773:
give any information about how the wave varies within each unit cell.
604:{\displaystyle {\dot {p}}_{i}={\frac {\partial L}{\partial q_{i}}}\,.}
534:{\displaystyle p_{i}={\frac {\partial L}{\partial {\dot {q}}_{i}}}\,,}
3780:
relates to crystal momentum instead of true momentum, the concept of
2323:
77:
3086:
can be expressed as a weighted sum of orthogonal basis functions
2108:, unlike Lagrangian mechanics which uses either all the coordinates
4012:
2334:
is an example of quantum mechanics in the position representation.
3918:
Quantum
Physics of Atoms, Molecules, Solids, Nuclei, and Particles
2378:
2319:
298:
3055:
Conversely, a three-dimensional wave function in momentum space
165:
139:
1510:
Now, the total differential of the momentum space
Lagrangian
3679:
Unitary equivalence between position and momentum operator
2705:
and is therefore an alternative description for the state
2341:
as a set of basis functions, the resulting wave function
19:"Momentum space" redirects here. For the jazz album, see
2304:
as a set of basis functions, one speaks of a state as a
3707:
acting on momentum space wave functions is the same as
3792:-space, there is an infinite set of points called the
1251:
which after substitution simplifies and rearranges to
3820:
is "equivalent" to exactly one point in this region.
3539:
3508:
3400:
3322:
3208:
3130:
3092:
3061:
2916:
2808:
2738:
2711:
2669:
2629:
2524:
2446:
2347:
2134:
1987:
1735:
1520:
1257:
1132:
1008:
683:
547:
477:
315:
3973:
Peleg, Y.; Pnini, R.; Zaarur, E.; Hecht, E. (2010).
3788:-space discussed above. For example, in a crystal's
2623:
It is clear that if we specify the set of functions
3711:acting on position space wave functions (under the
2694:holds all the information necessary to reconstruct
2372:is said to be the wave function in momentum space.
228:
Position and momentum spaces in classical mechanics
180:, though the converse can also be true, such as in
4000:Journal of Physics A: Mathematical and Theoretical
3915:
3668:
3525:
3487:
3386:
3302:
3194:
3116:
3078:
3039:
2892:
2782:
2717:
2686:
2656:{\displaystyle \psi _{\mathbf {k} }(\mathbf {r} )}
2655:
2615:
2507:
2364:
2247:
2085:
1971:
1713:
1500:
1243:
1119:
986:
603:
533:
457:
3329:
2745:
2258:Position and momentum spaces in quantum mechanics
677:of the generalized coordinate space Lagrangian;
84:varies with time, it will trace out a path, the
2794:for the denominator notation) with appropriate
1725:′ and the generalized coordinates derived from
2276:. This quantum state can be represented as a
8:
3765:that describes how the wave varies from one
673:is performed to change the variables in the
3975:Quantum Mechanics (Schaum's Outline Series)
3761:for an example. Crystal momentum is like a
2389:Relation between space and reciprocal space
541:the Euler–Lagrange equations take the form
2318:in position space (our ordinary notion of
4066:
4064:
4011:
3658:
3652:
3647:
3639:
3631:
3624:
3612:
3599:
3594:
3593:
3580:
3566:
3557:
3546:
3538:
3515:
3507:
3478:
3470:
3463:
3451:
3437:
3427:
3416:
3406:
3405:
3399:
3376:
3367:
3353:
3344:
3324:
3323:
3321:
3292:
3286:
3281:
3272:
3262:
3261:
3249:
3236:
3231:
3230:
3215:
3207:
3181:
3172:
3162:
3152:
3137:
3129:
3106:
3097:
3091:
3068:
3060:
3051:Functions and operators in momentum space
3032:
3026:
3021:
3013:
3005:
3001:
2989:
2976:
2971:
2970:
2957:
2943:
2934:
2923:
2915:
2883:
2875:
2871:
2858:
2844:
2835:
2824:
2814:
2813:
2807:
2772:
2763:
2740:
2739:
2737:
2710:
2676:
2668:
2645:
2635:
2634:
2628:
2608:
2602:
2597:
2588:
2578:
2577:
2565:
2552:
2547:
2546:
2531:
2523:
2497:
2488:
2478:
2468:
2453:
2445:
2405:Functions and operators in position space
2354:
2346:
2241:
2232:
2214:
2202:
2191:
2190:
2184:
2175:
2157:
2148:
2137:
2136:
2133:
2079:
2070:
2047:
2035:
2024:
2023:
2003:
1988:
1986:
1965:
1956:
1945:
1944:
1924:
1915:
1903:
1894:
1871:
1862:
1851:
1850:
1841:
1832:
1822:
1811:
1810:
1800:
1789:
1788:
1781:
1768:
1757:
1734:
1680:
1666:
1655:
1654:
1641:
1630:
1629:
1609:
1600:
1584:
1561:
1550:
1539:
1519:
1494:
1468:
1456:
1445:
1444:
1434:
1421:
1408:
1397:
1396:
1386:
1375:
1351:
1341:
1330:
1329:
1319:
1308:
1307:
1300:
1287:
1276:
1256:
1235:
1222:
1211:
1210:
1197:
1187:
1176:
1175:
1159:
1148:
1147:
1137:
1131:
1111:
1100:
1099:
1089:
1073:
1062:
1061:
1054:
1035:
1022:
1011:
1010:
1007:
980:
954:
942:
931:
930:
920:
907:
894:
883:
882:
872:
861:
828:
814:
803:
802:
789:
778:
777:
762:
753:
737:
719:
708:
697:
682:
597:
588:
570:
561:
550:
549:
546:
527:
518:
507:
506:
491:
482:
476:
451:
434:
424:
415:
404:
403:
397:
388:
370:
358:
347:
346:
331:
316:
314:
126:obtains the function in momentum space,
44:but in general of any finite dimension.
3968:
3966:
3922:(2nd ed.). John Wiley & Sons.
3906:
3856:
3341:
3117:{\displaystyle \phi _{j}(\mathbf {k} )}
2760:
3943:Hand, Louis N; Finch, Janet D (1998).
3745:, not its normal momentum. Therefore,
138:is useful to define in the context of
4047:. Addison Wesley, Prentice Hall Inc.
3875:, the differential of the product is
669:-tuple of the generalized momenta. A
7:
2409:Suppose we have a three-dimensional
3526:{\displaystyle \phi (\mathbf {k} )}
3079:{\displaystyle \phi (\mathbf {k} )}
2687:{\displaystyle \phi (\mathbf {k} )}
2365:{\displaystyle \phi (\mathbf {k} )}
613:The Lagrangian can be expressed in
176:is more intuitive and simpler than
164:which has dimensions of reciprocal
3648:
3373:
3369:
3350:
3346:
3282:
3022:
2769:
2765:
2598:
2515:or, in the continuous case, as an
2225:
2217:
2168:
2160:
2063:
2050:
2019:
2006:
1940:
1927:
1887:
1874:
1696:
1683:
1625:
1612:
1577:
1564:
1479:
1471:
965:
957:
839:
831:
773:
765:
730:
722:
581:
573:
502:
494:
381:
373:
342:
334:
14:
3914:Eisberg, R.; Resnick, R. (1985).
3816:-space, such that every possible
168:. The set of all wave vectors is
3659:
3640:
3632:
3613:
3595:
3547:
3516:
3502:. So a similar decomposition of
3479:
3471:
3417:
3407:
3377:
3354:
3326:
3293:
3273:
3263:
3250:
3232:
3216:
3182:
3138:
3107:
3069:
3033:
3014:
3006:
2990:
2972:
2924:
2884:
2876:
2825:
2815:
2773:
2742:
2677:
2646:
2636:
2609:
2589:
2579:
2566:
2548:
2532:
2498:
2454:
2355:
192:Heisenberg uncertainty principle
36:, there are two closely related
2272:, a particle is described by a
2188:
1907:
1845:
401:
3757:but play different roles. See
3617:
3609:
3577:
3563:
3551:
3543:
3520:
3512:
3421:
3413:
3277:
3269:
3254:
3246:
3220:
3212:
3186:
3178:
3142:
3134:
3111:
3103:
3073:
3065:
2994:
2986:
2954:
2940:
2928:
2920:
2855:
2841:
2829:
2821:
2681:
2673:
2650:
2642:
2593:
2585:
2570:
2562:
2536:
2528:
2502:
2494:
2458:
2450:
2359:
2351:
2296:the space. If one chooses the
1838:
1774:
1462:
1392:
1357:
1293:
1203:
1171:
1079:
1047:
948:
878:
1:
3977:(2nd ed.). McGraw Hill.
3741:relates almost always to its
3737:) in a crystal, its value of
3719:Reciprocal space and crystals
3845:Fractional Fourier transform
153:-vector") has dimensions of
114:is given in position space,
3715:of the Fourier transform).
467:(One overdot indicates one
157:, making it an analogue of
4126:
3796:which are "equivalent" to
3722:
2728:In quantum mechanics, the
2261:
18:
2330:in terms of the position
4022:10.1088/1751-8121/aa9314
3812:" is a finite volume of
3703:. In physical language,
307:Euler–Lagrange equations
3759:k·p perturbation theory
671:Legendre transformation
303:generalized coordinates
3804:(this is analogous to
3769:to the next, but does
3670:
3527:
3489:
3388:
3304:
3196:
3118:
3080:
3041:
2894:
2784:
2719:
2688:
2657:
2617:
2509:
2385:
2366:
2249:
2087:
1973:
1773:
1715:
1555:
1502:
1391:
1292:
1245:
1121:
988:
877:
713:
605:
535:
459:
110:. In particular, if a
3671:
3528:
3490:
3389:
3305:
3197:
3119:
3081:
3042:
2895:
2785:
2720:
2718:{\displaystyle \psi }
2689:
2658:
2618:
2510:
2382:
2367:
2262:Further information:
2250:
2128:), the equations are
2106:Hamiltonian mechanics
2100:Hamiltonian mechanics
2088:
1974:
1753:
1716:
1535:
1503:
1371:
1272:
1246:
1122:
989:
857:
693:
606:
536:
460:
3946:Analytical Mechanics
3810:first Brillouin zone
3693:unitarily equivalent
3537:
3506:
3398:
3394:with eigenfunctions
3320:
3206:
3128:
3090:
3059:
2914:
2806:
2736:
2709:
2667:
2627:
2522:
2444:
2345:
2328:Schrödinger equation
2132:
1985:
1733:
1518:
1255:
1130:
1006:
681:
545:
475:
313:
239:Lagrangian mechanics
233:Lagrangian mechanics
59:) is the set of all
4074:The Road to Reality
4071:R. Penrose (2007).
3840:Configuration space
3202:or as an integral,
2397:and the concept of
1729:′ are respectively
266:configuration space
208:de Broglie relation
182:solid-state physics
4043:Abers, E. (2004).
3863:For two functions
3808:). Likewise, the "
3794:reciprocal lattice
3725:Reciprocal lattice
3666:
3523:
3485:
3384:
3300:
3192:
3157:
3114:
3076:
3037:
2890:
2780:
2715:
2684:
2653:
2613:
2505:
2473:
2413:in position space
2386:
2362:
2282:linear combination
2245:
2083:
1969:
1711:
1498:
1241:
1117:
984:
675:total differential
601:
531:
455:
107:Pontryagin duality
92:is the set of all
4110:Quantum mechanics
4084:978-0-679-77631-4
4077:. Vintage books.
4054:978-0-13-146100-0
4045:Quantum Mechanics
3984:978-0-07-162358-2
3956:978-0-521-57572-0
3929:978-0-471-87373-0
3602:
3587:
3574:
3457:
3445:
3382:
3359:
3332:
3314:position operator
3239:
3148:
2979:
2964:
2951:
2865:
2852:
2778:
2748:
2730:momentum operator
2555:
2464:
2395:Fourier transform
2339:momentum operator
2302:position operator
2270:quantum mechanics
2264:Momentum operator
2239:
2199:
2182:
2145:
2077:
2042:
2032:
2001:
1963:
1953:
1901:
1859:
1819:
1797:
1703:
1663:
1648:
1638:
1591:
1486:
1453:
1405:
1338:
1316:
1219:
1184:
1156:
1108:
1070:
1019:
972:
939:
891:
846:
811:
796:
786:
744:
595:
558:
525:
515:
449:
412:
395:
365:
355:
329:
241:, the Lagrangian
188:Quantum mechanics
159:angular frequency
155:reciprocal length
124:Fourier transform
42:three-dimensional
4117:
4089:
4088:
4068:
4059:
4058:
4040:
4034:
4033:
4015:
3995:
3989:
3988:
3970:
3961:
3960:
3940:
3934:
3933:
3921:
3911:
3894:
3892:
3874:
3868:
3861:
3835:Reciprocal space
3743:crystal momentum
3697:unitary operator
3675:
3673:
3672:
3667:
3662:
3657:
3656:
3651:
3645:
3644:
3643:
3635:
3616:
3605:
3604:
3603:
3600:
3598:
3588:
3586:
3585:
3584:
3575:
3567:
3558:
3550:
3532:
3530:
3529:
3524:
3519:
3494:
3492:
3491:
3486:
3484:
3483:
3482:
3474:
3458:
3456:
3455:
3450:
3446:
3438:
3428:
3420:
3412:
3411:
3410:
3393:
3391:
3390:
3385:
3383:
3381:
3380:
3368:
3360:
3358:
3357:
3345:
3334:
3333:
3325:
3309:
3307:
3306:
3301:
3296:
3291:
3290:
3285:
3276:
3268:
3267:
3266:
3253:
3242:
3241:
3240:
3237:
3235:
3219:
3201:
3199:
3198:
3193:
3185:
3177:
3176:
3167:
3166:
3156:
3141:
3123:
3121:
3120:
3115:
3110:
3102:
3101:
3085:
3083:
3082:
3077:
3072:
3046:
3044:
3043:
3038:
3036:
3031:
3030:
3025:
3019:
3018:
3017:
3009:
2993:
2982:
2981:
2980:
2977:
2975:
2965:
2963:
2962:
2961:
2952:
2944:
2935:
2927:
2899:
2897:
2896:
2891:
2889:
2888:
2887:
2879:
2866:
2864:
2863:
2862:
2853:
2845:
2836:
2828:
2820:
2819:
2818:
2789:
2787:
2786:
2781:
2779:
2777:
2776:
2764:
2750:
2749:
2741:
2724:
2722:
2721:
2716:
2704:
2693:
2691:
2690:
2685:
2680:
2662:
2660:
2659:
2654:
2649:
2641:
2640:
2639:
2622:
2620:
2619:
2614:
2612:
2607:
2606:
2601:
2592:
2584:
2583:
2582:
2569:
2558:
2557:
2556:
2553:
2551:
2535:
2514:
2512:
2511:
2506:
2501:
2493:
2492:
2483:
2482:
2472:
2457:
2439:
2423:
2399:frequency domain
2371:
2369:
2368:
2363:
2358:
2326:). The familiar
2317:
2254:
2252:
2251:
2246:
2240:
2238:
2237:
2236:
2223:
2215:
2207:
2206:
2201:
2200:
2192:
2183:
2181:
2180:
2179:
2166:
2158:
2153:
2152:
2147:
2146:
2138:
2092:
2090:
2089:
2084:
2078:
2076:
2075:
2074:
2061:
2060:
2048:
2043:
2041:
2040:
2039:
2034:
2033:
2025:
2017:
2016:
2004:
2002:
2000:
1989:
1978:
1976:
1975:
1970:
1964:
1962:
1961:
1960:
1955:
1954:
1946:
1938:
1937:
1925:
1920:
1919:
1902:
1900:
1899:
1898:
1885:
1884:
1872:
1867:
1866:
1861:
1860:
1852:
1837:
1836:
1827:
1826:
1821:
1820:
1812:
1805:
1804:
1799:
1798:
1790:
1786:
1785:
1772:
1767:
1743:
1720:
1718:
1717:
1712:
1704:
1702:
1694:
1693:
1681:
1676:
1672:
1671:
1670:
1665:
1664:
1656:
1649:
1647:
1646:
1645:
1640:
1639:
1631:
1623:
1622:
1610:
1605:
1604:
1592:
1590:
1589:
1588:
1575:
1574:
1562:
1554:
1549:
1531:
1507:
1505:
1504:
1499:
1487:
1485:
1477:
1469:
1461:
1460:
1455:
1454:
1446:
1439:
1438:
1426:
1425:
1413:
1412:
1407:
1406:
1398:
1390:
1385:
1364:
1360:
1356:
1355:
1346:
1345:
1340:
1339:
1331:
1324:
1323:
1318:
1317:
1309:
1305:
1304:
1291:
1286:
1250:
1248:
1247:
1242:
1240:
1239:
1227:
1226:
1221:
1220:
1212:
1202:
1201:
1192:
1191:
1186:
1185:
1177:
1164:
1163:
1158:
1157:
1149:
1142:
1141:
1126:
1124:
1123:
1118:
1116:
1115:
1110:
1109:
1101:
1094:
1093:
1078:
1077:
1072:
1071:
1063:
1059:
1058:
1040:
1039:
1027:
1026:
1021:
1020:
1012:
993:
991:
990:
985:
973:
971:
963:
955:
947:
946:
941:
940:
932:
925:
924:
912:
911:
899:
898:
893:
892:
884:
876:
871:
847:
845:
837:
829:
824:
820:
819:
818:
813:
812:
804:
797:
795:
794:
793:
788:
787:
779:
771:
763:
758:
757:
745:
743:
742:
741:
728:
720:
712:
707:
610:
608:
607:
602:
596:
594:
593:
592:
579:
571:
566:
565:
560:
559:
551:
540:
538:
537:
532:
526:
524:
523:
522:
517:
516:
508:
500:
492:
487:
486:
464:
462:
461:
456:
450:
448:
440:
439:
438:
425:
420:
419:
414:
413:
405:
396:
394:
393:
392:
379:
371:
366:
364:
363:
362:
357:
356:
348:
340:
332:
330:
328:
317:
95:momentum vectors
88:of a particle.)
62:position vectors
4125:
4124:
4120:
4119:
4118:
4116:
4115:
4114:
4095:
4094:
4093:
4092:
4085:
4070:
4069:
4062:
4055:
4042:
4041:
4037:
3997:
3996:
3992:
3985:
3972:
3971:
3964:
3957:
3949:. p. 190.
3942:
3941:
3937:
3930:
3913:
3912:
3908:
3903:
3898:
3897:
3876:
3870:
3864:
3862:
3858:
3853:
3826:
3753:are not simply
3727:
3721:
3681:
3646:
3620:
3589:
3576:
3562:
3535:
3534:
3504:
3503:
3459:
3433:
3432:
3401:
3396:
3395:
3372:
3349:
3318:
3317:
3280:
3257:
3226:
3204:
3203:
3168:
3158:
3126:
3125:
3093:
3088:
3087:
3057:
3056:
3053:
3020:
2997:
2966:
2953:
2939:
2912:
2911:
2867:
2854:
2840:
2809:
2804:
2803:
2792:matrix calculus
2768:
2734:
2733:
2707:
2706:
2695:
2665:
2664:
2630:
2625:
2624:
2596:
2573:
2542:
2520:
2519:
2484:
2474:
2442:
2441:
2433:
2425:
2414:
2407:
2391:
2343:
2342:
2308:
2266:
2260:
2228:
2224:
2216:
2189:
2171:
2167:
2159:
2135:
2130:
2129:
2102:
2066:
2062:
2053:
2049:
2022:
2018:
2009:
2005:
1993:
1983:
1982:
1943:
1939:
1930:
1926:
1911:
1890:
1886:
1877:
1873:
1849:
1828:
1809:
1787:
1777:
1736:
1731:
1730:
1695:
1686:
1682:
1653:
1628:
1624:
1615:
1611:
1596:
1580:
1576:
1567:
1563:
1560:
1556:
1524:
1516:
1515:
1478:
1470:
1443:
1430:
1417:
1395:
1347:
1328:
1306:
1296:
1265:
1261:
1253:
1252:
1231:
1209:
1193:
1174:
1146:
1133:
1128:
1127:
1098:
1085:
1060:
1050:
1031:
1009:
1004:
1003:
964:
956:
929:
916:
903:
881:
838:
830:
801:
776:
772:
764:
749:
733:
729:
721:
718:
714:
679:
678:
663:
657:
650:
584:
580:
572:
548:
543:
542:
505:
501:
493:
478:
473:
472:
469:time derivative
441:
430:
426:
402:
384:
380:
372:
345:
341:
333:
321:
311:
310:
291:
285:
278:
235:
230:
70:Euclidean space
24:
17:
12:
11:
5:
4123:
4121:
4113:
4112:
4107:
4097:
4096:
4091:
4090:
4083:
4060:
4053:
4035:
4006:(50): 504002.
3990:
3983:
3962:
3955:
3935:
3928:
3905:
3904:
3902:
3899:
3896:
3895:
3855:
3854:
3852:
3849:
3848:
3847:
3842:
3837:
3832:
3825:
3822:
3723:Main article:
3720:
3717:
3691:operators are
3680:
3677:
3665:
3661:
3655:
3650:
3642:
3638:
3634:
3630:
3627:
3623:
3619:
3615:
3611:
3608:
3597:
3592:
3583:
3579:
3573:
3570:
3565:
3561:
3556:
3553:
3549:
3545:
3542:
3522:
3518:
3514:
3511:
3481:
3477:
3473:
3469:
3466:
3462:
3454:
3449:
3444:
3441:
3436:
3431:
3426:
3423:
3419:
3415:
3409:
3404:
3379:
3375:
3371:
3366:
3363:
3356:
3352:
3348:
3343:
3340:
3337:
3331:
3328:
3299:
3295:
3289:
3284:
3279:
3275:
3271:
3265:
3260:
3256:
3252:
3248:
3245:
3234:
3229:
3225:
3222:
3218:
3214:
3211:
3191:
3188:
3184:
3180:
3175:
3171:
3165:
3161:
3155:
3151:
3147:
3144:
3140:
3136:
3133:
3113:
3109:
3105:
3100:
3096:
3075:
3071:
3067:
3064:
3052:
3049:
3035:
3029:
3024:
3016:
3012:
3008:
3004:
3000:
2996:
2992:
2988:
2985:
2974:
2969:
2960:
2956:
2950:
2947:
2942:
2938:
2933:
2930:
2926:
2922:
2919:
2886:
2882:
2878:
2874:
2870:
2861:
2857:
2851:
2848:
2843:
2839:
2834:
2831:
2827:
2823:
2817:
2812:
2800:eigenfunctions
2775:
2771:
2767:
2762:
2759:
2756:
2753:
2747:
2744:
2714:
2683:
2679:
2675:
2672:
2652:
2648:
2644:
2638:
2633:
2611:
2605:
2600:
2595:
2591:
2587:
2581:
2576:
2572:
2568:
2564:
2561:
2550:
2545:
2541:
2538:
2534:
2530:
2527:
2504:
2500:
2496:
2491:
2487:
2481:
2477:
2471:
2467:
2463:
2460:
2456:
2452:
2449:
2429:
2406:
2403:
2390:
2387:
2361:
2357:
2353:
2350:
2298:eigenfunctions
2259:
2256:
2244:
2235:
2231:
2227:
2222:
2219:
2213:
2210:
2205:
2198:
2195:
2187:
2178:
2174:
2170:
2165:
2162:
2156:
2151:
2144:
2141:
2101:
2098:
2082:
2073:
2069:
2065:
2059:
2056:
2052:
2046:
2038:
2031:
2028:
2021:
2015:
2012:
2008:
1999:
1996:
1992:
1968:
1959:
1952:
1949:
1942:
1936:
1933:
1929:
1923:
1918:
1914:
1910:
1906:
1897:
1893:
1889:
1883:
1880:
1876:
1870:
1865:
1858:
1855:
1848:
1844:
1840:
1835:
1831:
1825:
1818:
1815:
1808:
1803:
1796:
1793:
1784:
1780:
1776:
1771:
1766:
1763:
1760:
1756:
1752:
1749:
1746:
1742:
1739:
1710:
1707:
1701:
1698:
1692:
1689:
1685:
1679:
1675:
1669:
1662:
1659:
1652:
1644:
1637:
1634:
1627:
1621:
1618:
1614:
1608:
1603:
1599:
1595:
1587:
1583:
1579:
1573:
1570:
1566:
1559:
1553:
1548:
1545:
1542:
1538:
1534:
1530:
1527:
1523:
1497:
1493:
1490:
1484:
1481:
1476:
1473:
1467:
1464:
1459:
1452:
1449:
1442:
1437:
1433:
1429:
1424:
1420:
1416:
1411:
1404:
1401:
1394:
1389:
1384:
1381:
1378:
1374:
1370:
1367:
1363:
1359:
1354:
1350:
1344:
1337:
1334:
1327:
1322:
1315:
1312:
1303:
1299:
1295:
1290:
1285:
1282:
1279:
1275:
1271:
1268:
1264:
1260:
1238:
1234:
1230:
1225:
1218:
1215:
1208:
1205:
1200:
1196:
1190:
1183:
1180:
1173:
1170:
1167:
1162:
1155:
1152:
1145:
1140:
1136:
1114:
1107:
1104:
1097:
1092:
1088:
1084:
1081:
1076:
1069:
1066:
1057:
1053:
1049:
1046:
1043:
1038:
1034:
1030:
1025:
1018:
1015:
983:
979:
976:
970:
967:
962:
959:
953:
950:
945:
938:
935:
928:
923:
919:
915:
910:
906:
902:
897:
890:
887:
880:
875:
870:
867:
864:
860:
856:
853:
850:
844:
841:
836:
833:
827:
823:
817:
810:
807:
800:
792:
785:
782:
775:
770:
767:
761:
756:
752:
748:
740:
736:
732:
727:
724:
717:
711:
706:
703:
700:
696:
692:
689:
686:
661:
655:
648:
615:momentum space
600:
591:
587:
583:
578:
575:
569:
564:
557:
554:
530:
521:
514:
511:
504:
499:
496:
490:
485:
481:
454:
447:
444:
437:
433:
429:
423:
418:
411:
408:
400:
391:
387:
383:
378:
375:
369:
361:
354:
351:
344:
339:
336:
327:
324:
320:
309:of motion are
289:
283:
276:
237:Most often in
234:
231:
229:
226:
90:Momentum space
82:point particle
46:Position space
21:Momentum Space
15:
13:
10:
9:
6:
4:
3:
2:
4122:
4111:
4108:
4106:
4103:
4102:
4100:
4086:
4080:
4076:
4075:
4067:
4065:
4061:
4056:
4050:
4046:
4039:
4036:
4031:
4027:
4023:
4019:
4014:
4009:
4005:
4001:
3994:
3991:
3986:
3980:
3976:
3969:
3967:
3963:
3958:
3952:
3948:
3947:
3939:
3936:
3931:
3925:
3920:
3919:
3910:
3907:
3900:
3891:
3887:
3883:
3879:
3873:
3867:
3860:
3857:
3850:
3846:
3843:
3841:
3838:
3836:
3833:
3831:
3828:
3827:
3823:
3821:
3819:
3815:
3811:
3807:
3803:
3799:
3795:
3791:
3787:
3783:
3779:
3774:
3772:
3768:
3764:
3763:wave envelope
3760:
3756:
3752:
3748:
3744:
3740:
3736:
3732:
3726:
3718:
3716:
3714:
3710:
3706:
3702:
3698:
3694:
3690:
3686:
3678:
3676:
3663:
3653:
3636:
3628:
3625:
3621:
3606:
3590:
3581:
3571:
3568:
3559:
3554:
3540:
3509:
3501:
3498:
3475:
3467:
3464:
3460:
3452:
3447:
3442:
3439:
3434:
3429:
3424:
3402:
3364:
3361:
3338:
3335:
3315:
3310:
3297:
3287:
3258:
3243:
3227:
3223:
3209:
3189:
3173:
3169:
3163:
3159:
3153:
3149:
3145:
3131:
3098:
3094:
3062:
3050:
3048:
3027:
3010:
3002:
2998:
2983:
2967:
2958:
2948:
2945:
2936:
2931:
2917:
2909:
2906:
2903:
2880:
2872:
2868:
2859:
2849:
2846:
2837:
2832:
2810:
2801:
2797:
2793:
2757:
2754:
2751:
2731:
2726:
2712:
2702:
2698:
2670:
2631:
2603:
2574:
2559:
2543:
2539:
2525:
2518:
2489:
2485:
2479:
2475:
2469:
2465:
2461:
2447:
2437:
2432:
2428:
2421:
2417:
2412:
2411:wave function
2404:
2402:
2400:
2396:
2388:
2381:
2377:
2373:
2348:
2340:
2335:
2333:
2329:
2325:
2321:
2315:
2311:
2307:
2306:wave function
2303:
2299:
2295:
2291:
2287:
2283:
2279:
2278:superposition
2275:
2274:quantum state
2271:
2265:
2257:
2255:
2242:
2233:
2229:
2220:
2211:
2208:
2203:
2196:
2193:
2185:
2176:
2172:
2163:
2154:
2149:
2142:
2139:
2127:
2123:
2119:
2115:
2111:
2107:
2099:
2097:
2093:
2080:
2071:
2067:
2057:
2054:
2044:
2036:
2029:
2026:
2013:
2010:
1997:
1994:
1990:
1979:
1966:
1957:
1950:
1947:
1934:
1931:
1921:
1916:
1912:
1908:
1904:
1895:
1891:
1881:
1878:
1868:
1863:
1856:
1853:
1846:
1842:
1833:
1829:
1823:
1816:
1813:
1806:
1801:
1794:
1791:
1782:
1778:
1769:
1764:
1761:
1758:
1754:
1750:
1747:
1744:
1740:
1737:
1728:
1724:
1708:
1705:
1699:
1690:
1687:
1677:
1673:
1667:
1660:
1657:
1650:
1642:
1635:
1632:
1619:
1616:
1606:
1601:
1597:
1593:
1585:
1581:
1571:
1568:
1557:
1551:
1546:
1543:
1540:
1536:
1532:
1528:
1525:
1521:
1513:
1508:
1495:
1491:
1488:
1482:
1474:
1465:
1457:
1450:
1447:
1440:
1435:
1431:
1427:
1422:
1418:
1414:
1409:
1402:
1399:
1387:
1382:
1379:
1376:
1372:
1368:
1365:
1361:
1352:
1348:
1342:
1335:
1332:
1325:
1320:
1313:
1310:
1301:
1297:
1288:
1283:
1280:
1277:
1273:
1269:
1266:
1262:
1258:
1236:
1232:
1228:
1223:
1216:
1213:
1206:
1198:
1194:
1188:
1181:
1178:
1168:
1165:
1160:
1153:
1150:
1143:
1138:
1134:
1112:
1105:
1102:
1095:
1090:
1086:
1082:
1074:
1067:
1064:
1055:
1051:
1044:
1041:
1036:
1032:
1028:
1023:
1016:
1013:
1001:
997:
981:
977:
974:
968:
960:
951:
943:
936:
933:
926:
921:
917:
913:
908:
904:
900:
895:
888:
885:
873:
868:
865:
862:
858:
854:
851:
848:
842:
834:
825:
821:
815:
808:
805:
798:
790:
783:
780:
768:
759:
754:
750:
746:
738:
734:
725:
715:
709:
704:
701:
698:
694:
690:
687:
684:
676:
672:
668:
664:
654:
647:
643:
639:
635:
631:
628:
624:
620:
616:
611:
598:
589:
585:
576:
567:
562:
555:
552:
528:
519:
512:
509:
497:
488:
483:
479:
470:
465:
452:
445:
442:
435:
431:
427:
421:
416:
409:
406:
398:
389:
385:
376:
367:
359:
352:
349:
337:
325:
322:
318:
308:
304:
300:
296:
292:
282:
275:
271:
267:
263:
259:
255:
252:
248:
244:
240:
232:
227:
225:
223:
219:
216:
212:
209:
205:
201:
197:
193:
189:
185:
183:
179:
175:
171:
167:
163:
160:
156:
152:
148:
145:
141:
135:
133:
129:
125:
121:
117:
113:
109:
108:
102:
100:
97:
96:
91:
87:
83:
79:
75:
71:
67:
64:
63:
58:
56:
51:
47:
43:
39:
38:vector spaces
35:
31:
26:
22:
4072:
4044:
4038:
4003:
3999:
3993:
3974:
3945:
3938:
3917:
3909:
3889:
3885:
3881:
3877:
3871:
3865:
3859:
3817:
3813:
3801:
3797:
3789:
3785:
3781:
3777:
3775:
3770:
3755:proportional
3750:
3746:
3738:
3728:
3708:
3704:
3688:
3684:
3682:
3499:
3316:is given by
3311:
3054:
2907:
2904:
2732:is given by
2727:
2700:
2696:
2435:
2430:
2426:
2419:
2415:
2408:
2392:
2374:
2336:
2331:
2322:in terms of
2313:
2309:
2286:weighted sum
2267:
2125:
2121:
2117:
2113:
2109:
2103:
2094:
1980:
1726:
1722:
1511:
1509:
1000:product rule
995:
666:
659:
652:
645:
641:
637:
633:
629:
626:
622:
618:
612:
466:
294:
287:
280:
273:
269:
261:
257:
253:
250:
246:
242:
236:
217:
214:
210:
203:
199:
195:
186:
177:
173:
169:
161:
150:
149:(or simply "
146:
136:
131:
127:
122:), then its
119:
115:
105:
103:
98:
93:
89:
65:
60:
53:
49:
45:
27:
25:
3830:Phase space
3695:, with the
3497:eigenvalues
2902:eigenvalues
144:wave vector
4099:Categories
4013:1709.04460
3901:References
3733:(or other
172:. Usually
86:trajectory
74:dimensions
72:, and has
55:coordinate
50:real space
40:, usually
4030:119290497
3851:Footnotes
3767:unit cell
3637:⋅
3626:−
3607:ψ
3591:∫
3572:π
3541:ϕ
3510:ϕ
3476:⋅
3465:−
3443:π
3403:ϕ
3374:∂
3370:∂
3351:∂
3347:∂
3342:ℏ
3330:^
3259:ϕ
3244:ψ
3228:∫
3210:ϕ
3170:ϕ
3160:ψ
3150:∑
3132:ϕ
3095:ϕ
3063:ϕ
3011:⋅
2984:ϕ
2968:∫
2949:π
2918:ψ
2881:⋅
2850:π
2811:ψ
2770:∂
2766:∂
2761:ℏ
2755:−
2746:^
2713:ψ
2671:ϕ
2632:ψ
2575:ψ
2560:ϕ
2544:∫
2526:ψ
2486:ψ
2476:ϕ
2466:∑
2448:ψ
2349:ϕ
2226:∂
2218:∂
2212:−
2197:˙
2169:∂
2161:∂
2143:˙
2064:∂
2051:∂
2030:˙
2020:∂
2007:∂
1951:˙
1941:∂
1928:∂
1909:−
1888:∂
1875:∂
1857:˙
1847:−
1817:˙
1795:˙
1755:∑
1751:−
1697:∂
1684:∂
1661:˙
1636:˙
1626:∂
1613:∂
1578:∂
1565:∂
1537:∑
1480:∂
1472:∂
1451:˙
1403:˙
1373:∑
1369:−
1336:˙
1314:˙
1274:∑
1270:−
1217:˙
1207:−
1182:˙
1154:˙
1106:˙
1083:−
1068:˙
1017:˙
966:∂
958:∂
937:˙
889:˙
859:∑
840:∂
832:∂
809:˙
784:˙
774:∂
766:∂
731:∂
723:∂
695:∑
640:), where
582:∂
574:∂
556:˙
513:˙
503:∂
495:∂
422:≡
410:˙
382:∂
374:∂
353:˙
343:∂
335:∂
4105:Momentum
3824:See also
3806:aliasing
3735:particle
3731:electron
3701:spectrum
2517:integral
2280:(i.e. a
2058:′
2014:′
1935:′
1882:′
1741:′
1691:′
1620:′
1572:′
1529:′
665:) is an
293:) is an
268:, where
264:) is in
222:momentum
112:function
34:geometry
3729:For an
2300:of the
658:, ...,
301:of the
170:k-space
30:physics
4081:
4051:
4028:
3981:
3953:
3926:
3601:-space
3238:-space
2978:-space
2798:. The
2796:domain
2554:-space
2324:length
998:. The
617:also,
305:. The
286:,...,
142:. The
78:length
48:(also
4026:S2CID
4008:arXiv
3776:When
3713:image
2910:. So
2790:(see
2320:space
2290:basis
2288:) of
2284:as a
1514:′ is
299:tuple
140:waves
57:space
4079:ISBN
4049:ISBN
3979:ISBN
3951:ISBN
3924:ISBN
3884:) =
3869:and
3749:and
3687:and
3683:The
3495:and
3312:The
2900:and
2802:are
2294:span
166:time
32:and
4018:doi
3890:vdu
3886:udv
3771:not
2268:In
2104:In
644:= (
272:= (
76:of
68:in
52:or
28:In
4101::
4063:^
4024:.
4016:.
4004:50
4002:.
3965:^
3888:+
3882:uv
3800:=
3124:,
2725:.
2440::
2124:,
2120:,
2110:or
651:,
636:,
634:dt
625:,
621:′(
279:,
260:,
258:dt
249:,
213:=
202:≥
184:.
4087:.
4057:.
4032:.
4020::
4010::
3987:.
3959:.
3932:.
3893:.
3880:(
3878:d
3872:v
3866:u
3818:k
3814:k
3802:0
3798:k
3790:k
3786:k
3782:k
3778:k
3751:p
3747:k
3739:k
3709:r
3705:p
3689:p
3685:r
3664:.
3660:r
3654:3
3649:d
3641:r
3633:k
3629:i
3622:e
3618:)
3614:r
3610:(
3596:r
3582:3
3578:)
3569:2
3564:(
3560:1
3555:=
3552:)
3548:k
3544:(
3521:)
3517:k
3513:(
3500:r
3480:r
3472:k
3468:i
3461:e
3453:3
3448:)
3440:2
3435:(
3430:1
3425:=
3422:)
3418:k
3414:(
3408:r
3378:k
3365:i
3362:=
3355:p
3339:i
3336:=
3327:r
3298:.
3294:r
3288:3
3283:d
3278:)
3274:k
3270:(
3264:r
3255:)
3251:r
3247:(
3233:r
3224:=
3221:)
3217:k
3213:(
3190:,
3187:)
3183:k
3179:(
3174:j
3164:j
3154:j
3146:=
3143:)
3139:k
3135:(
3112:)
3108:k
3104:(
3099:j
3074:)
3070:k
3066:(
3034:k
3028:3
3023:d
3015:r
3007:k
3003:i
2999:e
2995:)
2991:k
2987:(
2973:k
2959:3
2955:)
2946:2
2941:(
2937:1
2932:=
2929:)
2925:r
2921:(
2908:k
2905:ħ
2885:r
2877:k
2873:i
2869:e
2860:3
2856:)
2847:2
2842:(
2838:1
2833:=
2830:)
2826:r
2822:(
2816:k
2774:r
2758:i
2752:=
2743:p
2703:)
2701:r
2699:(
2697:ψ
2682:)
2678:k
2674:(
2651:)
2647:r
2643:(
2637:k
2610:k
2604:3
2599:d
2594:)
2590:r
2586:(
2580:k
2571:)
2567:k
2563:(
2549:k
2540:=
2537:)
2533:r
2529:(
2503:)
2499:r
2495:(
2490:j
2480:j
2470:j
2462:=
2459:)
2455:r
2451:(
2438:)
2436:r
2434:(
2431:j
2427:ψ
2422:)
2420:r
2418:(
2416:ψ
2360:)
2356:k
2352:(
2332:r
2316:)
2314:r
2312:(
2310:ψ
2243:.
2234:i
2230:q
2221:H
2209:=
2204:i
2194:p
2186:,
2177:i
2173:p
2164:H
2155:=
2150:i
2140:q
2126:t
2122:p
2118:q
2116:(
2114:H
2081:.
2072:i
2068:p
2055:L
2045:=
2037:i
2027:p
2011:L
1998:t
1995:d
1991:d
1967:.
1958:i
1948:p
1932:L
1922:=
1917:i
1913:q
1905:,
1896:i
1892:p
1879:L
1869:=
1864:i
1854:q
1843:,
1839:)
1834:i
1830:p
1824:i
1814:q
1807:+
1802:i
1792:p
1783:i
1779:q
1775:(
1770:n
1765:1
1762:=
1759:i
1748:L
1745:=
1738:L
1727:L
1723:L
1709:t
1706:d
1700:t
1688:L
1678:+
1674:)
1668:i
1658:p
1651:d
1643:i
1633:p
1617:L
1607:+
1602:i
1598:p
1594:d
1586:i
1582:p
1569:L
1558:(
1552:n
1547:1
1544:=
1541:i
1533:=
1526:L
1522:d
1512:L
1496:.
1492:t
1489:d
1483:t
1475:L
1466:+
1463:)
1458:i
1448:p
1441:d
1436:i
1432:q
1428:+
1423:i
1419:p
1415:d
1410:i
1400:q
1393:(
1388:n
1383:1
1380:=
1377:i
1366:=
1362:]
1358:)
1353:i
1349:p
1343:i
1333:q
1326:+
1321:i
1311:p
1302:i
1298:q
1294:(
1289:n
1284:1
1281:=
1278:i
1267:L
1263:[
1259:d
1237:i
1233:p
1229:d
1224:i
1214:q
1204:)
1199:i
1195:p
1189:i
1179:q
1172:(
1169:d
1166:=
1161:i
1151:q
1144:d
1139:i
1135:p
1113:i
1103:p
1096:d
1091:i
1087:q
1080:)
1075:i
1065:p
1056:i
1052:q
1048:(
1045:d
1042:=
1037:i
1033:q
1029:d
1024:i
1014:p
996:L
982:,
978:t
975:d
969:t
961:L
952:+
949:)
944:i
934:q
927:d
922:i
918:p
914:+
909:i
905:q
901:d
896:i
886:p
879:(
874:n
869:1
866:=
863:i
855:=
852:t
849:d
843:t
835:L
826:+
822:)
816:i
806:q
799:d
791:i
781:q
769:L
760:+
755:i
751:q
747:d
739:i
735:q
726:L
716:(
710:n
705:1
702:=
699:i
691:=
688:L
685:d
667:n
662:n
660:p
656:2
653:p
649:1
646:p
642:p
638:t
632:/
630:p
627:d
623:p
619:L
599:.
590:i
586:q
577:L
568:=
563:i
553:p
529:,
520:i
510:q
498:L
489:=
484:i
480:p
453:.
446:t
443:d
436:i
432:q
428:d
417:i
407:q
399:,
390:i
386:q
377:L
368:=
360:i
350:q
338:L
326:t
323:d
319:d
297:-
295:n
290:n
288:q
284:2
281:q
277:1
274:q
270:q
262:t
256:/
254:q
251:d
247:q
245:(
243:L
218:k
215:ħ
211:p
204:ħ
200:p
198:Δ
196:x
194:Δ
178:k
174:r
162:ω
151:k
147:k
132:p
130:(
128:φ
120:r
118:(
116:f
99:p
66:r
23:.
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