2395:
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).
1003:
2412:. 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.
691:
2391:
1988:
1730:
3685:
3056:
474:
148:
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
2106:
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
2394:
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
3319:
2632:
1743:
998:{\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\,,}
3547:
3504:
1528:
2264:
2924:
3403:
2102:
2909:
1517:
323:
3216:
2532:
2799:
1260:
1136:
3211:
2524:
2386:
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.
3408:
2142:
3330:
620:
550:
2816:
1265:
1995:
1983:{\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}}}\,.}
2348:
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
2672:
3680:{\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} .}
1725:{\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}
3051:{\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} }
3133:
3542:
3095:
2703:
2381:
1013:
for differentials allows the exchange of differentials in the generalized coordinates and velocities for the differentials in generalized momenta and their time derivatives,
2746:
1140:
1016:
469:{\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}}\,.}
3138:
2734:
3710:
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
2454:
555:
485:
3314:{\displaystyle \phi (\mathbf {k} )=\int _{\mathbf {r} {\text{-space}}}\psi (\mathbf {r} )\phi _{\mathbf {r} }(\mathbf {k} )\mathrm {d} ^{3}\mathbf {r} .}
2627:{\displaystyle \psi (\mathbf {r} )=\int _{\mathbf {k} {\text{-space}}}\phi (\mathbf {k} )\psi _{\mathbf {k} }(\mathbf {r} )\mathrm {d} ^{3}\mathbf {k} }
231:
which states the momentum and wavevector of a free particle are proportional to each other. In this context, when it is unambiguous, the terms "
4009:
Albert, Victor V; Pascazio, Saverio; Devoret, Michel H (2017). "General phase spaces: from discrete variables to rotor and continuum limits".
4093:
4063:
3993:
3965:
3938:
3499:{\displaystyle \phi _{\mathbf {r} }(\mathbf {k} )={\frac {1}{\left({\sqrt {2\pi }}\right)^{3}}}e^{-i\mathbf {k} \cdot \mathbf {r} }}
2259:{\displaystyle {\dot {q}}_{i}={\frac {\partial H}{\partial p_{i}}}\,,\quad {\dot {p}}_{i}=-{\frac {\partial H}{\partial q_{i}}}\,.}
3398:{\displaystyle \mathbf {\hat {r}} =i\hbar {\frac {\partial }{\partial \mathbf {p} }}=i{\frac {\partial }{\partial \mathbf {k} }}}
2123:
the momenta, the
Hamiltonian equations of motion place coordinates and momenta on equal footing. For a system with Hamiltonian
3711:
1732:
so by comparison of differentials of the
Lagrangians, the momenta, and their time derivatives, the momentum space Lagrangian
3850:
3765:
276:
3855:
2904:{\displaystyle \psi _{\mathbf {k} }(\mathbf {r} )={\frac {1}{({\sqrt {2\pi }})^{3}}}e^{i\mathbf {k} \cdot \mathbf {r} }}
2637:
1512:{\displaystyle d\left=-\sum _{i=1}^{n}({\dot {q}}_{i}dp_{i}+q_{i}d{\dot {p}}_{i})+{\frac {\partial L}{\partial t}}dt\,.}
317:
2097:{\displaystyle {\frac {d}{dt}}{\frac {\partial L'}{\partial {\dot {p}}_{i}}}={\frac {\partial L'}{\partial p_{i}}}\,.}
4120:
3769:
1005:
where the definition of generalized momentum and Euler–Lagrange equations have replaced the partial derivatives of
3544:
can be made in terms of the eigenfunctions of this operator, which turns out to be the inverse
Fourier transform,
3058:
and we see that the momentum representation is related to the position representation by a
Fourier transform.
3100:
2338:
681:
52:
3703:
2300:
313:
122:
3516:
3069:
2677:
2355:
112:
a physical system can have; the momentum vector of a particle corresponds to its motion, with units of .
2288:
2116:
235:" and "wavevector" are used interchangeably. However, the de Broglie relation is not true in a crystal.
202:
145:). Conversely, the inverse Fourier transform of a momentum space function is a position space function.
3820:
3795:-space is still meaningful and extremely useful, but it differs in several ways from the non-crystal
2806:
249:
217:/2 stating that position and momentum cannot be simultaneously known to arbitrary precision, and the
84:
4084:
3723:
218:
192:
4036:
4018:
3804:
3735:
2292:
685:
117:
4115:
4089:
4059:
3989:
3961:
3934:
3324:
2740:
2405:
2349:
2312:
2280:
2274:
198:
169:
165:
134:
65:
4028:
3845:
3773:
3753:
3707:
2409:
2719:
2802:
479:
105:
80:
72:
625:
92:
31:
2794:{\displaystyle \mathbf {\hat {p}} =-i\hbar {\frac {\partial }{\partial \mathbf {r} }}}
4109:
4040:
3927:
2810:
2421:
2316:
2308:
2284:
2303:
states. In principle one is free to choose the set of basis states, as long as they
482:). Introducing the definition of canonical momentum for each generalized coordinate
201:
provides two fundamental examples of the duality between position and momentum, the
2435:, then we can write this functions as a weighted sum of orthogonal basis functions
2296:
1992:
Combining the last two equations gives the momentum space Euler–Lagrange equations
1010:
48:
3955:
3840:
3507:
2912:
2304:
1255:{\displaystyle p_{i}d{\dot {q}}_{i}=d({\dot {q}}_{i}p_{i})-{\dot {q}}_{i}dp_{i}}
1131:{\displaystyle {\dot {p}}_{i}dq_{i}=d(q_{i}{\dot {p}}_{i})-q_{i}d{\dot {p}}_{i}}
154:
4032:
3206:{\displaystyle \phi (\mathbf {k} )=\sum _{j}\psi _{j}\phi _{j}(\mathbf {k} ),}
2404:
The momentum representation of a wave function is very closely related to the
96:
17:
2519:{\displaystyle \psi (\mathbf {r} )=\sum _{j}\phi _{j}\psi _{j}(\mathbf {r} )}
3777:
115:
Mathematically, the duality between position and momentum is an example of
91:; a position vector defines a point in space. (If the position vector of a
27:
Physical spaces representing position and momentum, Fourier-transform duals
2674:, say as the set of eigenfunctions of the momentum operator, the function
2390:
3816:
3745:
3741:
2527:
232:
44:
2107:
be more useful when momentum or angular momentum enters the
Lagrangian.
40:
3784:
give any information about how the wave varies within each unit cell.
615:{\displaystyle {\dot {p}}_{i}={\frac {\partial L}{\partial q_{i}}}\,.}
545:{\displaystyle p_{i}={\frac {\partial L}{\partial {\dot {q}}_{i}}}\,,}
3791:
relates to crystal momentum instead of true momentum, the concept of
2334:
88:
3097:
can be expressed as a weighted sum of orthogonal basis functions
2119:, unlike Lagrangian mechanics which uses either all the coordinates
4023:
2345:
is an example of quantum mechanics in the position representation.
3929:
Quantum
Physics of Atoms, Molecules, Solids, Nuclei, and Particles
2389:
2330:
309:
3066:
Conversely, a three-dimensional wave function in momentum space
176:
150:
1521:
Now, the total differential of the momentum space
Lagrangian
3690:
Unitary equivalence between position and momentum operator
2716:
and is therefore an alternative description for the state
2352:
as a set of basis functions, the resulting wave function
30:"Momentum space" redirects here. For the jazz album, see
2315:
as a set of basis functions, one speaks of a state as a
3718:
acting on momentum space wave functions is the same as
3803:-space, there is an infinite set of points called the
1262:
which after substitution simplifies and rearranges to
3831:
is "equivalent" to exactly one point in this region.
3550:
3519:
3411:
3333:
3219:
3141:
3103:
3072:
2927:
2819:
2749:
2722:
2680:
2640:
2535:
2457:
2358:
2145:
1998:
1746:
1531:
1268:
1143:
1019:
694:
558:
488:
326:
3984:
Peleg, Y.; Pnini, R.; Zaarur, E.; Hecht, E. (2010).
3799:-space discussed above. For example, in a crystal's
2634:
It is clear that if we specify the set of functions
3722:acting on position space wave functions (under the
2705:holds all the information necessary to reconstruct
2383:is said to be the wave function in momentum space.
239:
Position and momentum spaces in classical mechanics
191:, though the converse can also be true, such as in
4011:Journal of Physics A: Mathematical and Theoretical
3926:
3679:
3536:
3498:
3397:
3313:
3205:
3127:
3089:
3050:
2903:
2793:
2728:
2697:
2667:{\displaystyle \psi _{\mathbf {k} }(\mathbf {r} )}
2666:
2626:
2518:
2375:
2258:
2096:
1982:
1724:
1511:
1254:
1130:
997:
614:
544:
468:
3340:
2756:
2269:Position and momentum spaces in quantum mechanics
688:of the generalized coordinate space Lagrangian;
95:varies with time, it will trace out a path, the
2805:for the denominator notation) with appropriate
1736:′ and the generalized coordinates derived from
2287:. This quantum state can be represented as a
8:
3776:that describes how the wave varies from one
684:is performed to change the variables in the
3986:Quantum Mechanics (Schaum's Outline Series)
3772:for an example. Crystal momentum is like a
2400:Relation between space and reciprocal space
552:the Euler–Lagrange equations take the form
2329:in position space (our ordinary notion of
4077:
4075:
4022:
3669:
3663:
3658:
3650:
3642:
3635:
3623:
3610:
3605:
3604:
3591:
3577:
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3526:
3518:
3489:
3481:
3474:
3462:
3448:
3438:
3427:
3417:
3416:
3410:
3387:
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3364:
3355:
3335:
3334:
3332:
3303:
3297:
3292:
3283:
3273:
3272:
3260:
3247:
3242:
3241:
3226:
3218:
3192:
3183:
3173:
3163:
3148:
3140:
3117:
3108:
3102:
3079:
3071:
3062:Functions and operators in momentum space
3043:
3037:
3032:
3024:
3016:
3012:
3000:
2987:
2982:
2981:
2968:
2954:
2945:
2934:
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2886:
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2783:
2774:
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2748:
2721:
2687:
2679:
2656:
2646:
2645:
2639:
2619:
2613:
2608:
2599:
2589:
2588:
2576:
2563:
2558:
2557:
2542:
2534:
2508:
2499:
2489:
2479:
2464:
2456:
2416:Functions and operators in position space
2365:
2357:
2252:
2243:
2225:
2213:
2202:
2201:
2195:
2186:
2168:
2159:
2148:
2147:
2144:
2090:
2081:
2058:
2046:
2035:
2034:
2014:
1999:
1997:
1976:
1967:
1956:
1955:
1935:
1926:
1914:
1905:
1882:
1873:
1862:
1861:
1852:
1843:
1833:
1822:
1821:
1811:
1800:
1799:
1792:
1779:
1768:
1745:
1691:
1677:
1666:
1665:
1652:
1641:
1640:
1620:
1611:
1595:
1572:
1561:
1550:
1530:
1505:
1479:
1467:
1456:
1455:
1445:
1432:
1419:
1408:
1407:
1397:
1386:
1362:
1352:
1341:
1340:
1330:
1319:
1318:
1311:
1298:
1287:
1267:
1246:
1233:
1222:
1221:
1208:
1198:
1187:
1186:
1170:
1159:
1158:
1148:
1142:
1122:
1111:
1110:
1100:
1084:
1073:
1072:
1065:
1046:
1033:
1022:
1021:
1018:
991:
965:
953:
942:
941:
931:
918:
905:
894:
893:
883:
872:
839:
825:
814:
813:
800:
789:
788:
773:
764:
748:
730:
719:
708:
693:
608:
599:
581:
572:
561:
560:
557:
538:
529:
518:
517:
502:
493:
487:
462:
445:
435:
426:
415:
414:
408:
399:
381:
369:
358:
357:
342:
327:
325:
137:obtains the function in momentum space,
55:but in general of any finite dimension.
3979:
3977:
3933:(2nd ed.). John Wiley & Sons.
3917:
3867:
3352:
3128:{\displaystyle \phi _{j}(\mathbf {k} )}
2771:
3954:Hand, Louis N; Finch, Janet D (1998).
3756:, not its normal momentum. Therefore,
149:is useful to define in the context of
4058:. Addison Wesley, Prentice Hall Inc.
3886:, the differential of the product is
680:-tuple of the generalized momenta. A
7:
2420:Suppose we have a three-dimensional
3537:{\displaystyle \phi (\mathbf {k} )}
3090:{\displaystyle \phi (\mathbf {k} )}
2698:{\displaystyle \phi (\mathbf {k} )}
2376:{\displaystyle \phi (\mathbf {k} )}
624:The Lagrangian can be expressed in
187:is more intuitive and simpler than
175:which has dimensions of reciprocal
3659:
3384:
3380:
3361:
3357:
3293:
3033:
2780:
2776:
2609:
2526:or, in the continuous case, as an
2236:
2228:
2179:
2171:
2074:
2061:
2030:
2017:
1951:
1938:
1898:
1885:
1707:
1694:
1636:
1623:
1588:
1575:
1490:
1482:
976:
968:
850:
842:
784:
776:
741:
733:
592:
584:
513:
505:
392:
384:
353:
345:
25:
3925:Eisberg, R.; Resnick, R. (1985).
3827:-space, such that every possible
179:. The set of all wave vectors is
3670:
3651:
3643:
3624:
3606:
3558:
3527:
3513:. So a similar decomposition of
3490:
3482:
3428:
3418:
3388:
3365:
3337:
3304:
3284:
3274:
3261:
3243:
3227:
3193:
3149:
3118:
3080:
3044:
3025:
3017:
3001:
2983:
2935:
2895:
2887:
2836:
2826:
2784:
2753:
2688:
2657:
2647:
2620:
2600:
2590:
2577:
2559:
2543:
2509:
2465:
2366:
203:Heisenberg uncertainty principle
47:, there are two closely related
2283:, a particle is described by a
2199:
1918:
1856:
412:
3768:but play different roles. See
3628:
3620:
3588:
3574:
3562:
3554:
3531:
3523:
3432:
3424:
3288:
3280:
3265:
3257:
3231:
3223:
3197:
3189:
3153:
3145:
3122:
3114:
3084:
3076:
3005:
2997:
2965:
2951:
2939:
2931:
2866:
2852:
2840:
2832:
2692:
2684:
2661:
2653:
2604:
2596:
2581:
2573:
2547:
2539:
2513:
2505:
2469:
2461:
2370:
2362:
2307:the space. If one chooses the
1849:
1785:
1473:
1403:
1368:
1304:
1214:
1182:
1090:
1058:
959:
889:
1:
3988:(2nd ed.). McGraw Hill.
3752:relates almost always to its
3748:) in a crystal, its value of
3730:Reciprocal space and crystals
3856:Fractional Fourier transform
164:-vector") has dimensions of
125:is given in position space,
3726:of the Fourier transform).
478:(One overdot indicates one
168:, making it an analogue of
4137:
3807:which are "equivalent" to
3733:
2739:In quantum mechanics, the
2272:
29:
2341:in terms of the position
4033:10.1088/1751-8121/aa9314
3823:" is a finite volume of
3714:. In physical language,
318:Euler–Lagrange equations
3770:k·p perturbation theory
682:Legendre transformation
314:generalized coordinates
3815:(this is analogous to
3780:to the next, but does
3681:
3538:
3500:
3399:
3315:
3207:
3129:
3091:
3052:
2905:
2795:
2730:
2699:
2668:
2628:
2520:
2396:
2377:
2260:
2098:
1984:
1784:
1726:
1566:
1513:
1402:
1303:
1256:
1132:
999:
888:
724:
616:
546:
470:
121:. In particular, if a
3682:
3539:
3501:
3400:
3316:
3208:
3130:
3092:
3053:
2906:
2796:
2731:
2729:{\displaystyle \psi }
2700:
2669:
2629:
2521:
2393:
2378:
2273:Further information:
2261:
2139:), the equations are
2117:Hamiltonian mechanics
2111:Hamiltonian mechanics
2099:
1985:
1764:
1727:
1546:
1514:
1382:
1283:
1257:
1133:
1000:
868:
704:
617:
547:
471:
3957:Analytical Mechanics
3821:first Brillouin zone
3704:unitarily equivalent
3548:
3517:
3409:
3405:with eigenfunctions
3331:
3217:
3139:
3101:
3070:
2925:
2817:
2747:
2720:
2678:
2638:
2533:
2455:
2356:
2339:Schrödinger equation
2143:
1996:
1744:
1529:
1266:
1141:
1017:
692:
556:
486:
324:
250:Lagrangian mechanics
244:Lagrangian mechanics
70:) is the set of all
4085:The Road to Reality
4082:R. Penrose (2007).
3851:Configuration space
3213:or as an integral,
2408:and the concept of
1740:′ are respectively
277:configuration space
219:de Broglie relation
193:solid-state physics
4054:Abers, E. (2004).
3874:For two functions
3819:). Likewise, the "
3805:reciprocal lattice
3736:Reciprocal lattice
3677:
3534:
3496:
3395:
3311:
3203:
3168:
3125:
3087:
3048:
2901:
2791:
2726:
2695:
2664:
2624:
2516:
2484:
2424:in position space
2397:
2373:
2293:linear combination
2256:
2094:
1980:
1722:
1509:
1252:
1128:
995:
686:total differential
612:
542:
466:
118:Pontryagin duality
103:is the set of all
4121:Quantum mechanics
4095:978-0-679-77631-4
4088:. Vintage books.
4065:978-0-13-146100-0
4056:Quantum Mechanics
3995:978-0-07-162358-2
3967:978-0-521-57572-0
3940:978-0-471-87373-0
3613:
3598:
3585:
3468:
3456:
3393:
3370:
3343:
3325:position operator
3250:
3159:
2990:
2975:
2962:
2876:
2863:
2789:
2759:
2741:momentum operator
2566:
2475:
2406:Fourier transform
2350:momentum operator
2313:position operator
2281:quantum mechanics
2275:Momentum operator
2250:
2210:
2193:
2156:
2088:
2053:
2043:
2012:
1974:
1964:
1912:
1870:
1830:
1808:
1714:
1674:
1659:
1649:
1602:
1497:
1464:
1416:
1349:
1327:
1230:
1195:
1167:
1119:
1081:
1030:
983:
950:
902:
857:
822:
807:
797:
755:
606:
569:
536:
526:
460:
423:
406:
376:
366:
340:
252:, the Lagrangian
199:Quantum mechanics
170:angular frequency
166:reciprocal length
135:Fourier transform
53:three-dimensional
16:(Redirected from
4128:
4100:
4099:
4079:
4070:
4069:
4051:
4045:
4044:
4026:
4006:
4000:
3999:
3981:
3972:
3971:
3951:
3945:
3944:
3932:
3922:
3905:
3903:
3885:
3879:
3872:
3846:Reciprocal space
3754:crystal momentum
3708:unitary operator
3686:
3684:
3683:
3678:
3673:
3668:
3667:
3662:
3656:
3655:
3654:
3646:
3627:
3616:
3615:
3614:
3611:
3609:
3599:
3597:
3596:
3595:
3586:
3578:
3569:
3561:
3543:
3541:
3540:
3535:
3530:
3505:
3503:
3502:
3497:
3495:
3494:
3493:
3485:
3469:
3467:
3466:
3461:
3457:
3449:
3439:
3431:
3423:
3422:
3421:
3404:
3402:
3401:
3396:
3394:
3392:
3391:
3379:
3371:
3369:
3368:
3356:
3345:
3344:
3336:
3320:
3318:
3317:
3312:
3307:
3302:
3301:
3296:
3287:
3279:
3278:
3277:
3264:
3253:
3252:
3251:
3248:
3246:
3230:
3212:
3210:
3209:
3204:
3196:
3188:
3187:
3178:
3177:
3167:
3152:
3134:
3132:
3131:
3126:
3121:
3113:
3112:
3096:
3094:
3093:
3088:
3083:
3057:
3055:
3054:
3049:
3047:
3042:
3041:
3036:
3030:
3029:
3028:
3020:
3004:
2993:
2992:
2991:
2988:
2986:
2976:
2974:
2973:
2972:
2963:
2955:
2946:
2938:
2910:
2908:
2907:
2902:
2900:
2899:
2898:
2890:
2877:
2875:
2874:
2873:
2864:
2856:
2847:
2839:
2831:
2830:
2829:
2800:
2798:
2797:
2792:
2790:
2788:
2787:
2775:
2761:
2760:
2752:
2735:
2733:
2732:
2727:
2715:
2704:
2702:
2701:
2696:
2691:
2673:
2671:
2670:
2665:
2660:
2652:
2651:
2650:
2633:
2631:
2630:
2625:
2623:
2618:
2617:
2612:
2603:
2595:
2594:
2593:
2580:
2569:
2568:
2567:
2564:
2562:
2546:
2525:
2523:
2522:
2517:
2512:
2504:
2503:
2494:
2493:
2483:
2468:
2450:
2434:
2410:frequency domain
2382:
2380:
2379:
2374:
2369:
2337:). The familiar
2328:
2265:
2263:
2262:
2257:
2251:
2249:
2248:
2247:
2234:
2226:
2218:
2217:
2212:
2211:
2203:
2194:
2192:
2191:
2190:
2177:
2169:
2164:
2163:
2158:
2157:
2149:
2103:
2101:
2100:
2095:
2089:
2087:
2086:
2085:
2072:
2071:
2059:
2054:
2052:
2051:
2050:
2045:
2044:
2036:
2028:
2027:
2015:
2013:
2011:
2000:
1989:
1987:
1986:
1981:
1975:
1973:
1972:
1971:
1966:
1965:
1957:
1949:
1948:
1936:
1931:
1930:
1913:
1911:
1910:
1909:
1896:
1895:
1883:
1878:
1877:
1872:
1871:
1863:
1848:
1847:
1838:
1837:
1832:
1831:
1823:
1816:
1815:
1810:
1809:
1801:
1797:
1796:
1783:
1778:
1754:
1731:
1729:
1728:
1723:
1715:
1713:
1705:
1704:
1692:
1687:
1683:
1682:
1681:
1676:
1675:
1667:
1660:
1658:
1657:
1656:
1651:
1650:
1642:
1634:
1633:
1621:
1616:
1615:
1603:
1601:
1600:
1599:
1586:
1585:
1573:
1565:
1560:
1542:
1518:
1516:
1515:
1510:
1498:
1496:
1488:
1480:
1472:
1471:
1466:
1465:
1457:
1450:
1449:
1437:
1436:
1424:
1423:
1418:
1417:
1409:
1401:
1396:
1375:
1371:
1367:
1366:
1357:
1356:
1351:
1350:
1342:
1335:
1334:
1329:
1328:
1320:
1316:
1315:
1302:
1297:
1261:
1259:
1258:
1253:
1251:
1250:
1238:
1237:
1232:
1231:
1223:
1213:
1212:
1203:
1202:
1197:
1196:
1188:
1175:
1174:
1169:
1168:
1160:
1153:
1152:
1137:
1135:
1134:
1129:
1127:
1126:
1121:
1120:
1112:
1105:
1104:
1089:
1088:
1083:
1082:
1074:
1070:
1069:
1051:
1050:
1038:
1037:
1032:
1031:
1023:
1004:
1002:
1001:
996:
984:
982:
974:
966:
958:
957:
952:
951:
943:
936:
935:
923:
922:
910:
909:
904:
903:
895:
887:
882:
858:
856:
848:
840:
835:
831:
830:
829:
824:
823:
815:
808:
806:
805:
804:
799:
798:
790:
782:
774:
769:
768:
756:
754:
753:
752:
739:
731:
723:
718:
621:
619:
618:
613:
607:
605:
604:
603:
590:
582:
577:
576:
571:
570:
562:
551:
549:
548:
543:
537:
535:
534:
533:
528:
527:
519:
511:
503:
498:
497:
475:
473:
472:
467:
461:
459:
451:
450:
449:
436:
431:
430:
425:
424:
416:
407:
405:
404:
403:
390:
382:
377:
375:
374:
373:
368:
367:
359:
351:
343:
341:
339:
328:
106:momentum vectors
99:of a particle.)
73:position vectors
21:
4136:
4135:
4131:
4130:
4129:
4127:
4126:
4125:
4106:
4105:
4104:
4103:
4096:
4081:
4080:
4073:
4066:
4053:
4052:
4048:
4008:
4007:
4003:
3996:
3983:
3982:
3975:
3968:
3960:. p. 190.
3953:
3952:
3948:
3941:
3924:
3923:
3919:
3914:
3909:
3908:
3887:
3881:
3875:
3873:
3869:
3864:
3837:
3764:are not simply
3738:
3732:
3692:
3657:
3631:
3600:
3587:
3573:
3546:
3545:
3515:
3514:
3470:
3444:
3443:
3412:
3407:
3406:
3383:
3360:
3329:
3328:
3291:
3268:
3237:
3215:
3214:
3179:
3169:
3137:
3136:
3104:
3099:
3098:
3068:
3067:
3064:
3031:
3008:
2977:
2964:
2950:
2923:
2922:
2878:
2865:
2851:
2820:
2815:
2814:
2803:matrix calculus
2779:
2745:
2744:
2718:
2717:
2706:
2676:
2675:
2641:
2636:
2635:
2607:
2584:
2553:
2531:
2530:
2495:
2485:
2453:
2452:
2444:
2436:
2425:
2418:
2402:
2354:
2353:
2319:
2277:
2271:
2239:
2235:
2227:
2200:
2182:
2178:
2170:
2146:
2141:
2140:
2113:
2077:
2073:
2064:
2060:
2033:
2029:
2020:
2016:
2004:
1994:
1993:
1954:
1950:
1941:
1937:
1922:
1901:
1897:
1888:
1884:
1860:
1839:
1820:
1798:
1788:
1747:
1742:
1741:
1706:
1697:
1693:
1664:
1639:
1635:
1626:
1622:
1607:
1591:
1587:
1578:
1574:
1571:
1567:
1535:
1527:
1526:
1489:
1481:
1454:
1441:
1428:
1406:
1358:
1339:
1317:
1307:
1276:
1272:
1264:
1263:
1242:
1220:
1204:
1185:
1157:
1144:
1139:
1138:
1109:
1096:
1071:
1061:
1042:
1020:
1015:
1014:
975:
967:
940:
927:
914:
892:
849:
841:
812:
787:
783:
775:
760:
744:
740:
732:
729:
725:
690:
689:
674:
668:
661:
595:
591:
583:
559:
554:
553:
516:
512:
504:
489:
484:
483:
480:time derivative
452:
441:
437:
413:
395:
391:
383:
356:
352:
344:
332:
322:
321:
302:
296:
289:
246:
241:
81:Euclidean space
35:
28:
23:
22:
15:
12:
11:
5:
4134:
4132:
4124:
4123:
4118:
4108:
4107:
4102:
4101:
4094:
4071:
4064:
4046:
4017:(50): 504002.
4001:
3994:
3973:
3966:
3946:
3939:
3916:
3915:
3913:
3910:
3907:
3906:
3866:
3865:
3863:
3860:
3859:
3858:
3853:
3848:
3843:
3836:
3833:
3734:Main article:
3731:
3728:
3702:operators are
3691:
3688:
3676:
3672:
3666:
3661:
3653:
3649:
3645:
3641:
3638:
3634:
3630:
3626:
3622:
3619:
3608:
3603:
3594:
3590:
3584:
3581:
3576:
3572:
3567:
3564:
3560:
3556:
3553:
3533:
3529:
3525:
3522:
3492:
3488:
3484:
3480:
3477:
3473:
3465:
3460:
3455:
3452:
3447:
3442:
3437:
3434:
3430:
3426:
3420:
3415:
3390:
3386:
3382:
3377:
3374:
3367:
3363:
3359:
3354:
3351:
3348:
3342:
3339:
3310:
3306:
3300:
3295:
3290:
3286:
3282:
3276:
3271:
3267:
3263:
3259:
3256:
3245:
3240:
3236:
3233:
3229:
3225:
3222:
3202:
3199:
3195:
3191:
3186:
3182:
3176:
3172:
3166:
3162:
3158:
3155:
3151:
3147:
3144:
3124:
3120:
3116:
3111:
3107:
3086:
3082:
3078:
3075:
3063:
3060:
3046:
3040:
3035:
3027:
3023:
3019:
3015:
3011:
3007:
3003:
2999:
2996:
2985:
2980:
2971:
2967:
2961:
2958:
2953:
2949:
2944:
2941:
2937:
2933:
2930:
2897:
2893:
2889:
2885:
2881:
2872:
2868:
2862:
2859:
2854:
2850:
2845:
2842:
2838:
2834:
2828:
2823:
2811:eigenfunctions
2786:
2782:
2778:
2773:
2770:
2767:
2764:
2758:
2755:
2725:
2694:
2690:
2686:
2683:
2663:
2659:
2655:
2649:
2644:
2622:
2616:
2611:
2606:
2602:
2598:
2592:
2587:
2583:
2579:
2575:
2572:
2561:
2556:
2552:
2549:
2545:
2541:
2538:
2515:
2511:
2507:
2502:
2498:
2492:
2488:
2482:
2478:
2474:
2471:
2467:
2463:
2460:
2440:
2417:
2414:
2401:
2398:
2372:
2368:
2364:
2361:
2309:eigenfunctions
2270:
2267:
2255:
2246:
2242:
2238:
2233:
2230:
2224:
2221:
2216:
2209:
2206:
2198:
2189:
2185:
2181:
2176:
2173:
2167:
2162:
2155:
2152:
2112:
2109:
2093:
2084:
2080:
2076:
2070:
2067:
2063:
2057:
2049:
2042:
2039:
2032:
2026:
2023:
2019:
2010:
2007:
2003:
1979:
1970:
1963:
1960:
1953:
1947:
1944:
1940:
1934:
1929:
1925:
1921:
1917:
1908:
1904:
1900:
1894:
1891:
1887:
1881:
1876:
1869:
1866:
1859:
1855:
1851:
1846:
1842:
1836:
1829:
1826:
1819:
1814:
1807:
1804:
1795:
1791:
1787:
1782:
1777:
1774:
1771:
1767:
1763:
1760:
1757:
1753:
1750:
1721:
1718:
1712:
1709:
1703:
1700:
1696:
1690:
1686:
1680:
1673:
1670:
1663:
1655:
1648:
1645:
1638:
1632:
1629:
1625:
1619:
1614:
1610:
1606:
1598:
1594:
1590:
1584:
1581:
1577:
1570:
1564:
1559:
1556:
1553:
1549:
1545:
1541:
1538:
1534:
1508:
1504:
1501:
1495:
1492:
1487:
1484:
1478:
1475:
1470:
1463:
1460:
1453:
1448:
1444:
1440:
1435:
1431:
1427:
1422:
1415:
1412:
1405:
1400:
1395:
1392:
1389:
1385:
1381:
1378:
1374:
1370:
1365:
1361:
1355:
1348:
1345:
1338:
1333:
1326:
1323:
1314:
1310:
1306:
1301:
1296:
1293:
1290:
1286:
1282:
1279:
1275:
1271:
1249:
1245:
1241:
1236:
1229:
1226:
1219:
1216:
1211:
1207:
1201:
1194:
1191:
1184:
1181:
1178:
1173:
1166:
1163:
1156:
1151:
1147:
1125:
1118:
1115:
1108:
1103:
1099:
1095:
1092:
1087:
1080:
1077:
1068:
1064:
1060:
1057:
1054:
1049:
1045:
1041:
1036:
1029:
1026:
994:
990:
987:
981:
978:
973:
970:
964:
961:
956:
949:
946:
939:
934:
930:
926:
921:
917:
913:
908:
901:
898:
891:
886:
881:
878:
875:
871:
867:
864:
861:
855:
852:
847:
844:
838:
834:
828:
821:
818:
811:
803:
796:
793:
786:
781:
778:
772:
767:
763:
759:
751:
747:
743:
738:
735:
728:
722:
717:
714:
711:
707:
703:
700:
697:
672:
666:
659:
626:momentum space
611:
602:
598:
594:
589:
586:
580:
575:
568:
565:
541:
532:
525:
522:
515:
510:
507:
501:
496:
492:
465:
458:
455:
448:
444:
440:
434:
429:
422:
419:
411:
402:
398:
394:
389:
386:
380:
372:
365:
362:
355:
350:
347:
338:
335:
331:
320:of motion are
300:
294:
287:
248:Most often in
245:
242:
240:
237:
101:Momentum space
93:point particle
57:Position space
32:Momentum Space
26:
24:
18:Momentum space
14:
13:
10:
9:
6:
4:
3:
2:
4133:
4122:
4119:
4117:
4114:
4113:
4111:
4097:
4091:
4087:
4086:
4078:
4076:
4072:
4067:
4061:
4057:
4050:
4047:
4042:
4038:
4034:
4030:
4025:
4020:
4016:
4012:
4005:
4002:
3997:
3991:
3987:
3980:
3978:
3974:
3969:
3963:
3959:
3958:
3950:
3947:
3942:
3936:
3931:
3930:
3921:
3918:
3911:
3902:
3898:
3894:
3890:
3884:
3878:
3871:
3868:
3861:
3857:
3854:
3852:
3849:
3847:
3844:
3842:
3839:
3838:
3834:
3832:
3830:
3826:
3822:
3818:
3814:
3810:
3806:
3802:
3798:
3794:
3790:
3785:
3783:
3779:
3775:
3774:wave envelope
3771:
3767:
3763:
3759:
3755:
3751:
3747:
3743:
3737:
3729:
3727:
3725:
3721:
3717:
3713:
3709:
3705:
3701:
3697:
3689:
3687:
3674:
3664:
3647:
3639:
3636:
3632:
3617:
3601:
3592:
3582:
3579:
3570:
3565:
3551:
3520:
3512:
3509:
3486:
3478:
3475:
3471:
3463:
3458:
3453:
3450:
3445:
3440:
3435:
3413:
3375:
3372:
3349:
3346:
3326:
3321:
3308:
3298:
3269:
3254:
3238:
3234:
3220:
3200:
3184:
3180:
3174:
3170:
3164:
3160:
3156:
3142:
3109:
3105:
3073:
3061:
3059:
3038:
3021:
3013:
3009:
2994:
2978:
2969:
2959:
2956:
2947:
2942:
2928:
2920:
2917:
2914:
2891:
2883:
2879:
2870:
2860:
2857:
2848:
2843:
2821:
2812:
2808:
2804:
2768:
2765:
2762:
2742:
2737:
2723:
2713:
2709:
2681:
2642:
2614:
2585:
2570:
2554:
2550:
2536:
2529:
2500:
2496:
2490:
2486:
2480:
2476:
2472:
2458:
2448:
2443:
2439:
2432:
2428:
2423:
2422:wave function
2415:
2413:
2411:
2407:
2399:
2392:
2388:
2384:
2359:
2351:
2346:
2344:
2340:
2336:
2332:
2326:
2322:
2318:
2317:wave function
2314:
2310:
2306:
2302:
2298:
2294:
2290:
2289:superposition
2286:
2285:quantum state
2282:
2276:
2268:
2266:
2253:
2244:
2240:
2231:
2222:
2219:
2214:
2207:
2204:
2196:
2187:
2183:
2174:
2165:
2160:
2153:
2150:
2138:
2134:
2130:
2126:
2122:
2118:
2110:
2108:
2104:
2091:
2082:
2078:
2068:
2065:
2055:
2047:
2040:
2037:
2024:
2021:
2008:
2005:
2001:
1990:
1977:
1968:
1961:
1958:
1945:
1942:
1932:
1927:
1923:
1919:
1915:
1906:
1902:
1892:
1889:
1879:
1874:
1867:
1864:
1857:
1853:
1844:
1840:
1834:
1827:
1824:
1817:
1812:
1805:
1802:
1793:
1789:
1780:
1775:
1772:
1769:
1765:
1761:
1758:
1755:
1751:
1748:
1739:
1735:
1719:
1716:
1710:
1701:
1698:
1688:
1684:
1678:
1671:
1668:
1661:
1653:
1646:
1643:
1630:
1627:
1617:
1612:
1608:
1604:
1596:
1592:
1582:
1579:
1568:
1562:
1557:
1554:
1551:
1547:
1543:
1539:
1536:
1532:
1524:
1519:
1506:
1502:
1499:
1493:
1485:
1476:
1468:
1461:
1458:
1451:
1446:
1442:
1438:
1433:
1429:
1425:
1420:
1413:
1410:
1398:
1393:
1390:
1387:
1383:
1379:
1376:
1372:
1363:
1359:
1353:
1346:
1343:
1336:
1331:
1324:
1321:
1312:
1308:
1299:
1294:
1291:
1288:
1284:
1280:
1277:
1273:
1269:
1247:
1243:
1239:
1234:
1227:
1224:
1217:
1209:
1205:
1199:
1192:
1189:
1179:
1176:
1171:
1164:
1161:
1154:
1149:
1145:
1123:
1116:
1113:
1106:
1101:
1097:
1093:
1085:
1078:
1075:
1066:
1062:
1055:
1052:
1047:
1043:
1039:
1034:
1027:
1024:
1012:
1008:
992:
988:
985:
979:
971:
962:
954:
947:
944:
937:
932:
928:
924:
919:
915:
911:
906:
899:
896:
884:
879:
876:
873:
869:
865:
862:
859:
853:
845:
836:
832:
826:
819:
816:
809:
801:
794:
791:
779:
770:
765:
761:
757:
749:
745:
736:
726:
720:
715:
712:
709:
705:
701:
698:
695:
687:
683:
679:
675:
665:
658:
654:
650:
646:
642:
639:
635:
631:
627:
622:
609:
600:
596:
587:
578:
573:
566:
563:
539:
530:
523:
520:
508:
499:
494:
490:
481:
476:
463:
456:
453:
446:
442:
438:
432:
427:
420:
417:
409:
400:
396:
387:
378:
370:
363:
360:
348:
336:
333:
329:
319:
315:
311:
307:
303:
293:
286:
282:
278:
274:
270:
266:
263:
259:
255:
251:
243:
238:
236:
234:
230:
227:
223:
220:
216:
212:
208:
204:
200:
196:
194:
190:
186:
182:
178:
174:
171:
167:
163:
159:
156:
152:
146:
144:
140:
136:
132:
128:
124:
120:
119:
113:
111:
108:
107:
102:
98:
94:
90:
86:
82:
78:
75:
74:
69:
67:
62:
58:
54:
50:
49:vector spaces
46:
42:
37:
33:
19:
4083:
4055:
4049:
4014:
4010:
4004:
3985:
3956:
3949:
3928:
3920:
3900:
3896:
3892:
3888:
3882:
3876:
3870:
3828:
3824:
3812:
3808:
3800:
3796:
3792:
3788:
3786:
3781:
3766:proportional
3761:
3757:
3749:
3739:
3719:
3715:
3699:
3695:
3693:
3510:
3327:is given by
3322:
3065:
2918:
2915:
2743:is given by
2738:
2711:
2707:
2446:
2441:
2437:
2430:
2426:
2419:
2403:
2385:
2347:
2342:
2333:in terms of
2324:
2320:
2297:weighted sum
2278:
2136:
2132:
2128:
2124:
2120:
2114:
2105:
1991:
1737:
1733:
1522:
1520:
1011:product rule
1006:
677:
670:
663:
656:
652:
648:
644:
640:
637:
633:
629:
623:
477:
305:
298:
291:
284:
280:
272:
268:
264:
261:
257:
253:
247:
228:
225:
221:
214:
210:
206:
197:
188:
184:
180:
172:
161:
160:(or simply "
157:
147:
142:
138:
133:), then its
130:
126:
116:
114:
109:
104:
100:
76:
71:
64:
60:
56:
38:
36:
3841:Phase space
3706:, with the
3508:eigenvalues
2913:eigenvalues
155:wave vector
4110:Categories
4024:1709.04460
3912:References
3744:(or other
183:. Usually
97:trajectory
85:dimensions
83:, and has
66:coordinate
61:real space
51:, usually
4041:119290497
3862:Footnotes
3778:unit cell
3648:⋅
3637:−
3618:ψ
3602:∫
3583:π
3552:ϕ
3521:ϕ
3487:⋅
3476:−
3454:π
3414:ϕ
3385:∂
3381:∂
3362:∂
3358:∂
3353:ℏ
3341:^
3270:ϕ
3255:ψ
3239:∫
3221:ϕ
3181:ϕ
3171:ψ
3161:∑
3143:ϕ
3106:ϕ
3074:ϕ
3022:⋅
2995:ϕ
2979:∫
2960:π
2929:ψ
2892:⋅
2861:π
2822:ψ
2781:∂
2777:∂
2772:ℏ
2766:−
2757:^
2724:ψ
2682:ϕ
2643:ψ
2586:ψ
2571:ϕ
2555:∫
2537:ψ
2497:ψ
2487:ϕ
2477:∑
2459:ψ
2360:ϕ
2237:∂
2229:∂
2223:−
2208:˙
2180:∂
2172:∂
2154:˙
2075:∂
2062:∂
2041:˙
2031:∂
2018:∂
1962:˙
1952:∂
1939:∂
1920:−
1899:∂
1886:∂
1868:˙
1858:−
1828:˙
1806:˙
1766:∑
1762:−
1708:∂
1695:∂
1672:˙
1647:˙
1637:∂
1624:∂
1589:∂
1576:∂
1548:∑
1491:∂
1483:∂
1462:˙
1414:˙
1384:∑
1380:−
1347:˙
1325:˙
1285:∑
1281:−
1228:˙
1218:−
1193:˙
1165:˙
1117:˙
1094:−
1079:˙
1028:˙
977:∂
969:∂
948:˙
900:˙
870:∑
851:∂
843:∂
820:˙
795:˙
785:∂
777:∂
742:∂
734:∂
706:∑
651:), where
593:∂
585:∂
567:˙
524:˙
514:∂
506:∂
433:≡
421:˙
393:∂
385:∂
364:˙
354:∂
346:∂
4116:Momentum
3835:See also
3817:aliasing
3746:particle
3742:electron
3712:spectrum
2528:integral
2291:(i.e. a
2069:′
2025:′
1946:′
1893:′
1752:′
1702:′
1631:′
1583:′
1540:′
676:) is an
304:) is an
279:, where
275:) is in
233:momentum
123:function
45:geometry
3740:For an
2311:of the
669:, ...,
312:of the
181:k-space
41:physics
4092:
4062:
4039:
3992:
3964:
3937:
3612:-space
3249:-space
2989:-space
2809:. The
2807:domain
2565:-space
2335:length
1009:. The
628:also,
316:. The
297:,...,
153:. The
89:length
59:(also
4037:S2CID
4019:arXiv
3787:When
3724:image
2921:. So
2801:(see
2331:space
2301:basis
2299:) of
2295:as a
1525:′ is
310:tuple
151:waves
68:space
4090:ISBN
4060:ISBN
3990:ISBN
3962:ISBN
3935:ISBN
3895:) =
3880:and
3760:and
3698:and
3694:The
3506:and
3323:The
2911:and
2813:are
2305:span
177:time
43:and
4029:doi
3901:vdu
3897:udv
3782:not
2279:In
2115:In
655:= (
283:= (
87:of
79:in
63:or
39:In
4112::
4074:^
4035:.
4027:.
4015:50
4013:.
3976:^
3899:+
3893:uv
3811:=
3135:,
2736:.
2451::
2135:,
2131:,
2121:or
662:,
647:,
645:dt
636:,
632:′(
290:,
271:,
269:dt
260:,
224:=
213:≥
195:.
4098:.
4068:.
4043:.
4031::
4021::
3998:.
3970:.
3943:.
3904:.
3891:(
3889:d
3883:v
3877:u
3829:k
3825:k
3813:0
3809:k
3801:k
3797:k
3793:k
3789:k
3762:p
3758:k
3750:k
3720:r
3716:p
3700:p
3696:r
3675:.
3671:r
3665:3
3660:d
3652:r
3644:k
3640:i
3633:e
3629:)
3625:r
3621:(
3607:r
3593:3
3589:)
3580:2
3575:(
3571:1
3566:=
3563:)
3559:k
3555:(
3532:)
3528:k
3524:(
3511:r
3491:r
3483:k
3479:i
3472:e
3464:3
3459:)
3451:2
3446:(
3441:1
3436:=
3433:)
3429:k
3425:(
3419:r
3389:k
3376:i
3373:=
3366:p
3350:i
3347:=
3338:r
3309:.
3305:r
3299:3
3294:d
3289:)
3285:k
3281:(
3275:r
3266:)
3262:r
3258:(
3244:r
3235:=
3232:)
3228:k
3224:(
3201:,
3198:)
3194:k
3190:(
3185:j
3175:j
3165:j
3157:=
3154:)
3150:k
3146:(
3123:)
3119:k
3115:(
3110:j
3085:)
3081:k
3077:(
3045:k
3039:3
3034:d
3026:r
3018:k
3014:i
3010:e
3006:)
3002:k
2998:(
2984:k
2970:3
2966:)
2957:2
2952:(
2948:1
2943:=
2940:)
2936:r
2932:(
2919:k
2916:ħ
2896:r
2888:k
2884:i
2880:e
2871:3
2867:)
2858:2
2853:(
2849:1
2844:=
2841:)
2837:r
2833:(
2827:k
2785:r
2769:i
2763:=
2754:p
2714:)
2712:r
2710:(
2708:ψ
2693:)
2689:k
2685:(
2662:)
2658:r
2654:(
2648:k
2621:k
2615:3
2610:d
2605:)
2601:r
2597:(
2591:k
2582:)
2578:k
2574:(
2560:k
2551:=
2548:)
2544:r
2540:(
2514:)
2510:r
2506:(
2501:j
2491:j
2481:j
2473:=
2470:)
2466:r
2462:(
2449:)
2447:r
2445:(
2442:j
2438:ψ
2433:)
2431:r
2429:(
2427:ψ
2371:)
2367:k
2363:(
2343:r
2327:)
2325:r
2323:(
2321:ψ
2254:.
2245:i
2241:q
2232:H
2220:=
2215:i
2205:p
2197:,
2188:i
2184:p
2175:H
2166:=
2161:i
2151:q
2137:t
2133:p
2129:q
2127:(
2125:H
2092:.
2083:i
2079:p
2066:L
2056:=
2048:i
2038:p
2022:L
2009:t
2006:d
2002:d
1978:.
1969:i
1959:p
1943:L
1933:=
1928:i
1924:q
1916:,
1907:i
1903:p
1890:L
1880:=
1875:i
1865:q
1854:,
1850:)
1845:i
1841:p
1835:i
1825:q
1818:+
1813:i
1803:p
1794:i
1790:q
1786:(
1781:n
1776:1
1773:=
1770:i
1759:L
1756:=
1749:L
1738:L
1734:L
1720:t
1717:d
1711:t
1699:L
1689:+
1685:)
1679:i
1669:p
1662:d
1654:i
1644:p
1628:L
1618:+
1613:i
1609:p
1605:d
1597:i
1593:p
1580:L
1569:(
1563:n
1558:1
1555:=
1552:i
1544:=
1537:L
1533:d
1523:L
1507:.
1503:t
1500:d
1494:t
1486:L
1477:+
1474:)
1469:i
1459:p
1452:d
1447:i
1443:q
1439:+
1434:i
1430:p
1426:d
1421:i
1411:q
1404:(
1399:n
1394:1
1391:=
1388:i
1377:=
1373:]
1369:)
1364:i
1360:p
1354:i
1344:q
1337:+
1332:i
1322:p
1313:i
1309:q
1305:(
1300:n
1295:1
1292:=
1289:i
1278:L
1274:[
1270:d
1248:i
1244:p
1240:d
1235:i
1225:q
1215:)
1210:i
1206:p
1200:i
1190:q
1183:(
1180:d
1177:=
1172:i
1162:q
1155:d
1150:i
1146:p
1124:i
1114:p
1107:d
1102:i
1098:q
1091:)
1086:i
1076:p
1067:i
1063:q
1059:(
1056:d
1053:=
1048:i
1044:q
1040:d
1035:i
1025:p
1007:L
993:,
989:t
986:d
980:t
972:L
963:+
960:)
955:i
945:q
938:d
933:i
929:p
925:+
920:i
916:q
912:d
907:i
897:p
890:(
885:n
880:1
877:=
874:i
866:=
863:t
860:d
854:t
846:L
837:+
833:)
827:i
817:q
810:d
802:i
792:q
780:L
771:+
766:i
762:q
758:d
750:i
746:q
737:L
727:(
721:n
716:1
713:=
710:i
702:=
699:L
696:d
678:n
673:n
671:p
667:2
664:p
660:1
657:p
653:p
649:t
643:/
641:p
638:d
634:p
630:L
610:.
601:i
597:q
588:L
579:=
574:i
564:p
540:,
531:i
521:q
509:L
500:=
495:i
491:p
464:.
457:t
454:d
447:i
443:q
439:d
428:i
418:q
410:,
401:i
397:q
388:L
379:=
371:i
361:q
349:L
337:t
334:d
330:d
308:-
306:n
301:n
299:q
295:2
292:q
288:1
285:q
281:q
273:t
267:/
265:q
262:d
258:q
256:(
254:L
229:k
226:ħ
222:p
215:ħ
211:p
209:Δ
207:x
205:Δ
189:k
185:r
173:ω
162:k
158:k
143:p
141:(
139:φ
131:r
129:(
127:f
110:p
77:r
34:.
20:)
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