1575:
912:
1570:{\displaystyle {\begin{aligned}{\frac {d}{dt}}\left\langle Q\right\rangle &={\frac {d}{dt}}\left\langle \psi \right|Q\left|\psi \right\rangle \\&=\left({\frac {d}{dt}}\left\langle \psi \right|\right)Q\left|\psi \right\rangle +\left\langle \psi \right|{\frac {dQ}{dt}}\left|\psi \right\rangle +\left\langle \psi \right|Q\left({\frac {d}{dt}}\left|\psi \right\rangle \right)\\&=-{\frac {1}{i\hbar }}\left\langle H\psi \right|Q\left|\psi \right\rangle +\left\langle \psi \right|{\frac {dQ}{dt}}\left|\psi \right\rangle +{\frac {1}{i\hbar }}\left\langle \psi \right|Q\left|H\psi \right\rangle \\&=-{\frac {1}{i\hbar }}\left\langle \psi \right|HQ\left|\psi \right\rangle +\left\langle \psi \right|{\frac {dQ}{dt}}\left|\psi \right\rangle +{\frac {1}{i\hbar }}\left\langle \psi \right|QH\left|\psi \right\rangle \\&=-{\frac {1}{i\hbar }}\left\langle \psi \right|\left\left|\psi \right\rangle +\left\langle \psi \right|{\frac {dQ}{dt}}\left|\psi \right\rangle \end{aligned}}}
723:
2358:
2153:
1749:
563:
2404:, defined as any functions of only the phase-space coordinates that are constant along an orbit. Every integral of motion is a constant of motion, but the converse is not true because a constant of motion may depend on time. Examples of integrals of motion are the angular momentum vector,
2207:
120:
is the intersection of a sphere (conservation of total angular momentum) and an ellipsoid (conservation of energy), a trajectory that might be otherwise hard to derive and visualize. Therefore, the identification of constants of motion is an important objective in
1987:
1587:
2519:
917:
335:
897:
718:{\displaystyle {\frac {d}{dt}}\langle \psi |Q|\psi \rangle =-{\frac {1}{i\hbar }}\left\langle \psi \right|\left\left|\psi \right\rangle +\left\langle \psi \right|{\frac {dQ}{dt}}\left|\psi \right\rangle \,}
2440:
2212:
424:
1971:
1860:
2202:
2353:{\displaystyle {\begin{aligned}{\frac {d}{dt}}\langle Q\rangle &=-{\frac {1}{i\hbar }}\left(E\langle \psi |Q|\psi \rangle -E\langle \psi |Q|\psi \rangle \right)\\&=0\end{aligned}}}
2593:
observables or fixes a gauge. In a canonical language, this usually means either constructing functions which
Poisson-commute on the constraint surface with the gauge generating
2575:
2148:{\displaystyle {\frac {d}{dt}}\langle Q\rangle =-{\frac {1}{i\hbar }}\langle \psi |\left|\psi \rangle =-{\frac {1}{i\hbar }}\langle \psi |\left(HQ-QH\right)|\psi \rangle }
1924:
833:
774:
1809:
503:
1882:
1744:{\displaystyle {\frac {d}{dt}}\langle \psi |Q|\psi \rangle ={\frac {-1}{i\hbar }}\langle \psi |\left|\psi \rangle +\langle \psi |{\frac {dQ}{dt}}|\psi \rangle \,}
253:
471:
451:
355:
248:
2396:
coordinates (position and velocity, or position and momentum) and time that is constant throughout a trajectory. A subset of the constants of motion are the
137:
The simplest but least systematic approach is the intuitive ("psychic") derivation, in which a quantity is hypothesized to be constant (perhaps because of
847:
61:
2445:
368:
2598:
1929:
1818:
2158:
2601:. Such gauge invariant observables are thus the `constants of motion' of the gauge generators and referred to as Dirac observables.
2407:
516:
constants of motion, such that the
Poisson bracket of any pair of constants of motion vanishes, is known as a completely
2688:
2652:
2626:
77:
2384:
only if it is integrable; as of 2024, there is no known consistent method for quantizing chaotic dynamical systems.
204:
73:
1887:
791:
17:
728:
148:
provide a commonly used and straightforward method for identifying constants of motion, particularly when the
1772:
92:
Constants of motion are useful because they allow properties of the motion to be derived without solving the
2521:. An example of a function that is a constant of motion but not an integral of motion would be the function
2707:
145:
533:
101:
2524:
521:
188:
69:
43:
2594:
842:
176:
153:
109:
65:
54:
553:
476:
362:
149:
1867:
529:
216:
208:
192:
180:
172:
168:
8:
2360:
This is the reason why eigenstates of the
Hamiltonian are also called stationary states.
42:
throughout the motion, imposing in effect a constraint on the motion. However, it is a
2677:
2672:
532:
not explicitly dependent on time), the energy of the system is a constant of motion (a
456:
436:
340:
233:
175:
provides a systematic way of deriving such quantities from the symmetry. For example,
93:
81:
50:
2684:
2648:
2622:
2381:
2369:
517:
138:
36:
2380:; such systems are termed chaotic. In general, a classical mechanical system can be
2377:
160:
39:
2590:
2514:{\textstyle H(\mathbf {x} ,\mathbf {v} )={\frac {1}{2}}v^{2}+\Phi (\mathbf {x} )}
358:
2586:
2642:
2392:
A constant of motion may be defined in a given force field as any function of
2701:
838:
525:
330:{\displaystyle 0={\frac {dA}{dt}}={\frac {\partial A}{\partial t}}+\{A,H\},}
906:
2393:
549:
141:) and later shown mathematically to be conserved throughout the motion.
117:
105:
97:
560:, and it does not itself depend explicitly on time. This is because
122:
28:
2597:
or to fix the flow of the latter by singling out points within each
892:{\displaystyle i\hbar {\frac {\partial \psi }{\partial t}}=H\psi .}
212:
164:
113:
250:
is a constant of the motion if its total time derivative is zero
2373:
133:
There are several methods for identifying constants of motion.
128:
520:. Such a collection of constants of motion are said to be in
196:
2435:{\displaystyle \mathbf {L} =\mathbf {x} \times \mathbf {v} }
2372:
has constants of motion other than the energy. By contrast,
2577:
for an object moving at a constant speed in one dimension.
184:
1761:
For an arbitrary state of a
Quantum Mechanical system, if
419:{\displaystyle {\frac {\partial A}{\partial t}}=-\{A,H\}.}
365:
equals minus its partial derivative with respect to time
108:
corresponding to the constants of motion. For example,
901:
Taking the time derivative of the expectation value of
2448:
2527:
2410:
2210:
2161:
1990:
1932:
1890:
1870:
1821:
1775:
1590:
915:
850:
794:
731:
566:
479:
473:
are constants of motion, so is their
Poisson bracket
459:
439:
371:
343:
256:
236:
2442:, or a Hamiltonian without time dependence, such as
219:
corresponds to a constant of motion, often called a
215:. The converse is also true; every symmetry of the
2671:
2676:
2569:
2513:
2434:
2352:
2196:
2147:
1966:{\displaystyle {\frac {d}{dt}}\langle Q\rangle =0}
1965:
1918:
1884:is an eigenfunction of Hamiltonian, then even if
1876:
1855:{\displaystyle {\frac {d}{dt}}\langle Q\rangle =0}
1854:
1803:
1743:
1569:
891:
827:
768:
717:
497:
465:
445:
418:
349:
329:
242:
2197:{\displaystyle H|\psi \rangle =E|\psi \rangle \,}
2699:
2647:. Princeton University Press. 27 January 2008.
2644:Binney, J. and Tremaine, S.: Galactic Dynamics
2616:
2585:In order to extract physical information from
2363:
788:which depends on position, momentum and time,
2325:
2303:
2294:
2272:
2236:
2230:
2190:
2173:
2142:
2098:
2074:
2036:
2012:
2006:
1954:
1948:
1843:
1837:
1737:
1698:
1692:
1654:
1628:
1606:
604:
582:
492:
480:
410:
398:
321:
309:
2679:Introduction to Quantum Mechanics (2nd ed.)
129:Methods for identifying constants of motion
1815:is not explicitly dependent on time, then
2193:
1740:
765:
714:
159:Another approach is to recognize that a
152:adopts recognizable functional forms in
539:
14:
2700:
784:Say there is some observable quantity
433:, which states that if two quantities
2387:
2580:
2376:is the only constant of motion in a
100:of the motion can be derived as the
548:will be a constant of motion if it
207:results from the invariance of the
191:results from the invariance of the
179:results from the invariance of the
24:
2497:
868:
860:
383:
375:
297:
289:
25:
2719:
2617:Landau, L.; Lifshitz, E. (1960).
2258:
2092:
2030:
1648:
1471:
1415:
1324:
1266:
1173:
854:
622:
49:, the natural consequence of the
2504:
2464:
2456:
2428:
2420:
2412:
1581:
205:conservation of angular momentum
96:. In fortunate cases, even the
2621:. Pergamon Press. p. 135.
1980:
189:conservation of linear momentum
87:
2635:
2610:
2549:
2531:
2508:
2500:
2468:
2452:
2318:
2310:
2287:
2279:
2183:
2166:
2135:
2105:
2067:
2043:
1730:
1705:
1685:
1661:
1621:
1613:
822:
804:
744:
732:
597:
589:
195:under shifts in the origin of
183:under shifts in the origin of
13:
1:
2604:
2570:{\displaystyle C(x,v,t)=x-vt}
779:
776:is the commutator relation.
64:). Common examples include
7:
2364:Relevance for quantum chaos
1919:{\displaystyle \left\neq 0}
112:shows that the torque-free
60:(which would require extra
10:
2724:
1926:it is still the case that
1756:
837:And also, that there is a
828:{\displaystyle Q=Q(x,p,t)}
429:Another useful result is
146:Hamilton–Jacobi equations
82:inverse-square force laws
78:Laplace–Runge–Lenz vector
2589:, one either constructs
1977:is independent of time.
769:{\displaystyle =HQ-QH\,}
512:degrees of freedom, and
2595:first class constraints
1804:{\displaystyle \left=0}
544:An observable quantity
524:with each other. For a
498:{\displaystyle \{A,B\}}
2571:
2515:
2436:
2354:
2198:
2149:
1967:
1920:
1878:
1856:
1805:
1745:
1571:
893:
843:Schrödinger's equation
829:
770:
719:
499:
467:
447:
420:
351:
331:
244:
201:translational symmetry
177:conservation of energy
154:orthogonal coordinates
110:Poinsot's construction
2572:
2516:
2437:
2378:non-integrable system
2355:
2199:
2150:
1968:
1921:
1879:
1877:{\displaystyle \psi }
1857:
1806:
1746:
1572:
894:
830:
771:
720:
500:
468:
448:
421:
352:
332:
245:
2525:
2446:
2408:
2208:
2159:
1988:
1930:
1888:
1868:
1819:
1773:
1588:
913:
905:requires use of the
848:
792:
729:
564:
540:In quantum mechanics
477:
457:
437:
369:
341:
254:
234:
2708:Classical mechanics
2673:Griffiths, David J.
2398:integrals of motion
1769:commute, i.e. if
94:equations of motion
51:equations of motion
2567:
2511:
2432:
2388:Integral of motion
2350:
2348:
2194:
2145:
1963:
1916:
1874:
1852:
1801:
1741:
1567:
1565:
889:
825:
766:
715:
534:conserved quantity
495:
463:
443:
416:
347:
337:which occurs when
327:
240:
161:conserved quantity
33:constant of motion
2683:. Prentice Hall.
2581:Dirac observables
2482:
2370:integrable system
2262:
2228:
2096:
2034:
2004:
1946:
1835:
1754:
1753:
1727:
1652:
1604:
1550:
1475:
1419:
1390:
1328:
1270:
1241:
1177:
1133:
1085:
1021:
966:
933:
909:, and results in
875:
701:
626:
580:
518:integrable system
466:{\displaystyle B}
446:{\displaystyle A}
431:Poisson's theorem
390:
350:{\displaystyle A}
304:
281:
243:{\displaystyle A}
173:Noether's theorem
163:corresponds to a
139:experimental data
62:constraint forces
37:physical quantity
16:(Redirected from
2715:
2694:
2682:
2665:
2664:
2662:
2661:
2639:
2633:
2632:
2614:
2576:
2574:
2573:
2568:
2520:
2518:
2517:
2512:
2507:
2493:
2492:
2483:
2475:
2467:
2459:
2441:
2439:
2438:
2433:
2431:
2423:
2415:
2359:
2357:
2356:
2351:
2349:
2336:
2332:
2328:
2321:
2313:
2290:
2282:
2263:
2261:
2250:
2229:
2227:
2216:
2203:
2201:
2200:
2195:
2186:
2169:
2154:
2152:
2151:
2146:
2138:
2133:
2129:
2108:
2097:
2095:
2084:
2070:
2065:
2061:
2046:
2035:
2033:
2022:
2005:
2003:
1992:
1976:
1972:
1970:
1969:
1964:
1947:
1945:
1934:
1925:
1923:
1922:
1917:
1909:
1905:
1883:
1881:
1880:
1875:
1861:
1859:
1858:
1853:
1836:
1834:
1823:
1814:
1810:
1808:
1807:
1802:
1794:
1790:
1768:
1764:
1750:
1748:
1747:
1742:
1733:
1728:
1726:
1718:
1710:
1708:
1688:
1683:
1679:
1664:
1653:
1651:
1643:
1635:
1624:
1616:
1605:
1603:
1592:
1582:
1576:
1574:
1573:
1568:
1566:
1562:
1551:
1549:
1541:
1533:
1531:
1517:
1506:
1502:
1487:
1476:
1474:
1463:
1452:
1448:
1431:
1420:
1418:
1407:
1402:
1391:
1389:
1381:
1373:
1371:
1357:
1340:
1329:
1327:
1316:
1305:
1301:
1297:
1282:
1271:
1269:
1258:
1253:
1242:
1240:
1232:
1224:
1222:
1208:
1194:
1190:
1178:
1176:
1165:
1154:
1150:
1146:
1145:
1134:
1132:
1121:
1111:
1097:
1086:
1084:
1076:
1068:
1066:
1052:
1038:
1034:
1033:
1022:
1020:
1009:
996:
992:
978:
967:
965:
954:
945:
934:
932:
921:
904:
898:
896:
895:
890:
876:
874:
866:
858:
834:
832:
831:
826:
787:
775:
773:
772:
767:
724:
722:
721:
716:
713:
702:
700:
692:
684:
682:
668:
657:
653:
638:
627:
625:
614:
600:
592:
581:
579:
568:
504:
502:
501:
496:
472:
470:
469:
464:
452:
450:
449:
444:
425:
423:
422:
417:
391:
389:
381:
373:
356:
354:
353:
348:
336:
334:
333:
328:
305:
303:
295:
287:
282:
280:
272:
264:
249:
247:
246:
241:
221:conserved charge
74:angular momentum
53:, rather than a
21:
2723:
2722:
2718:
2717:
2716:
2714:
2713:
2712:
2698:
2697:
2691:
2668:
2659:
2657:
2655:
2641:
2640:
2636:
2629:
2615:
2611:
2607:
2591:gauge invariant
2583:
2526:
2523:
2522:
2503:
2488:
2484:
2474:
2463:
2455:
2447:
2444:
2443:
2427:
2419:
2411:
2409:
2406:
2405:
2402:first integrals
2390:
2368:In general, an
2366:
2347:
2346:
2334:
2333:
2317:
2309:
2286:
2278:
2268:
2264:
2254:
2249:
2239:
2220:
2215:
2211:
2209:
2206:
2205:
2182:
2165:
2160:
2157:
2156:
2134:
2113:
2109:
2104:
2088:
2083:
2066:
2051:
2047:
2042:
2026:
2021:
1996:
1991:
1989:
1986:
1985:
1983:
1974:
1938:
1933:
1931:
1928:
1927:
1895:
1891:
1889:
1886:
1885:
1869:
1866:
1865:
1827:
1822:
1820:
1817:
1816:
1812:
1780:
1776:
1774:
1771:
1770:
1766:
1762:
1759:
1729:
1719:
1711:
1709:
1704:
1684:
1669:
1665:
1660:
1644:
1636:
1634:
1620:
1612:
1596:
1591:
1589:
1586:
1585:
1564:
1563:
1552:
1542:
1534:
1532:
1521:
1507:
1492:
1488:
1477:
1467:
1462:
1450:
1449:
1438:
1421:
1411:
1406:
1392:
1382:
1374:
1372:
1361:
1347:
1330:
1320:
1315:
1303:
1302:
1290:
1286:
1272:
1262:
1257:
1243:
1233:
1225:
1223:
1212:
1198:
1183:
1179:
1169:
1164:
1152:
1151:
1135:
1125:
1120:
1119:
1115:
1101:
1087:
1077:
1069:
1067:
1056:
1042:
1023:
1013:
1008:
1007:
1003:
994:
993:
982:
968:
958:
953:
946:
935:
925:
920:
916:
914:
911:
910:
902:
867:
859:
857:
849:
846:
845:
793:
790:
789:
785:
782:
730:
727:
726:
703:
693:
685:
683:
672:
658:
643:
639:
628:
618:
613:
596:
588:
572:
567:
565:
562:
561:
542:
478:
475:
474:
458:
455:
454:
438:
435:
434:
382:
374:
372:
370:
367:
366:
359:Poisson bracket
342:
339:
338:
296:
288:
286:
273:
265:
263:
255:
252:
251:
235:
232:
231:
131:
90:
70:linear momentum
23:
22:
15:
12:
11:
5:
2721:
2711:
2710:
2696:
2695:
2689:
2667:
2666:
2653:
2634:
2627:
2608:
2606:
2603:
2587:gauge theories
2582:
2579:
2566:
2563:
2560:
2557:
2554:
2551:
2548:
2545:
2542:
2539:
2536:
2533:
2530:
2510:
2506:
2502:
2499:
2496:
2491:
2487:
2481:
2478:
2473:
2470:
2466:
2462:
2458:
2454:
2451:
2430:
2426:
2422:
2418:
2414:
2389:
2386:
2365:
2362:
2345:
2342:
2339:
2337:
2335:
2331:
2327:
2324:
2320:
2316:
2312:
2308:
2305:
2302:
2299:
2296:
2293:
2289:
2285:
2281:
2277:
2274:
2271:
2267:
2260:
2257:
2253:
2248:
2245:
2242:
2240:
2238:
2235:
2232:
2226:
2223:
2219:
2214:
2213:
2192:
2189:
2185:
2181:
2178:
2175:
2172:
2168:
2164:
2144:
2141:
2137:
2132:
2128:
2125:
2122:
2119:
2116:
2112:
2107:
2103:
2100:
2094:
2091:
2087:
2082:
2079:
2076:
2073:
2069:
2064:
2060:
2057:
2054:
2050:
2045:
2041:
2038:
2032:
2029:
2025:
2020:
2017:
2014:
2011:
2008:
2002:
1999:
1995:
1982:
1979:
1962:
1959:
1956:
1953:
1950:
1944:
1941:
1937:
1915:
1912:
1908:
1904:
1901:
1898:
1894:
1873:
1851:
1848:
1845:
1842:
1839:
1833:
1830:
1826:
1800:
1797:
1793:
1789:
1786:
1783:
1779:
1758:
1755:
1752:
1751:
1739:
1736:
1732:
1725:
1722:
1717:
1714:
1707:
1703:
1700:
1697:
1694:
1691:
1687:
1682:
1678:
1675:
1672:
1668:
1663:
1659:
1656:
1650:
1647:
1642:
1639:
1633:
1630:
1627:
1623:
1619:
1615:
1611:
1608:
1602:
1599:
1595:
1561:
1558:
1555:
1548:
1545:
1540:
1537:
1530:
1527:
1524:
1520:
1516:
1513:
1510:
1505:
1501:
1498:
1495:
1491:
1486:
1483:
1480:
1473:
1470:
1466:
1461:
1458:
1455:
1453:
1451:
1447:
1444:
1441:
1437:
1434:
1430:
1427:
1424:
1417:
1414:
1410:
1405:
1401:
1398:
1395:
1388:
1385:
1380:
1377:
1370:
1367:
1364:
1360:
1356:
1353:
1350:
1346:
1343:
1339:
1336:
1333:
1326:
1323:
1319:
1314:
1311:
1308:
1306:
1304:
1300:
1296:
1293:
1289:
1285:
1281:
1278:
1275:
1268:
1265:
1261:
1256:
1252:
1249:
1246:
1239:
1236:
1231:
1228:
1221:
1218:
1215:
1211:
1207:
1204:
1201:
1197:
1193:
1189:
1186:
1182:
1175:
1172:
1168:
1163:
1160:
1157:
1155:
1153:
1149:
1144:
1141:
1138:
1131:
1128:
1124:
1118:
1114:
1110:
1107:
1104:
1100:
1096:
1093:
1090:
1083:
1080:
1075:
1072:
1065:
1062:
1059:
1055:
1051:
1048:
1045:
1041:
1037:
1032:
1029:
1026:
1019:
1016:
1012:
1006:
1002:
999:
997:
995:
991:
988:
985:
981:
977:
974:
971:
964:
961:
957:
952:
949:
947:
944:
941:
938:
931:
928:
924:
919:
918:
888:
885:
882:
879:
873:
870:
865:
862:
856:
853:
824:
821:
818:
815:
812:
809:
806:
803:
800:
797:
781:
778:
764:
761:
758:
755:
752:
749:
746:
743:
740:
737:
734:
712:
709:
706:
699:
696:
691:
688:
681:
678:
675:
671:
667:
664:
661:
656:
652:
649:
646:
642:
637:
634:
631:
624:
621:
617:
612:
609:
606:
603:
599:
595:
591:
587:
584:
578:
575:
571:
541:
538:
508:A system with
494:
491:
488:
485:
482:
462:
442:
427:
426:
415:
412:
409:
406:
403:
400:
397:
394:
388:
385:
380:
377:
346:
326:
323:
320:
317:
314:
311:
308:
302:
299:
294:
291:
285:
279:
276:
271:
268:
262:
259:
239:
228:
157:
142:
130:
127:
89:
86:
18:First integral
9:
6:
4:
3:
2:
2720:
2709:
2706:
2705:
2703:
2692:
2690:0-13-805326-X
2686:
2681:
2680:
2674:
2670:
2669:
2656:
2654:9780691130279
2650:
2646:
2645:
2638:
2630:
2628:0-7506-2896-0
2624:
2620:
2613:
2609:
2602:
2600:
2596:
2592:
2588:
2578:
2564:
2561:
2558:
2555:
2552:
2546:
2543:
2540:
2537:
2534:
2528:
2494:
2489:
2485:
2479:
2476:
2471:
2460:
2449:
2424:
2416:
2403:
2399:
2395:
2385:
2383:
2379:
2375:
2371:
2361:
2343:
2340:
2338:
2329:
2322:
2314:
2306:
2300:
2297:
2291:
2283:
2275:
2269:
2265:
2255:
2251:
2246:
2243:
2241:
2233:
2224:
2221:
2217:
2187:
2179:
2176:
2170:
2162:
2139:
2130:
2126:
2123:
2120:
2117:
2114:
2110:
2101:
2089:
2085:
2080:
2077:
2071:
2062:
2058:
2055:
2052:
2048:
2039:
2027:
2023:
2018:
2015:
2009:
2000:
1997:
1993:
1978:
1960:
1957:
1951:
1942:
1939:
1935:
1913:
1910:
1906:
1902:
1899:
1896:
1892:
1871:
1862:
1849:
1846:
1840:
1831:
1828:
1824:
1798:
1795:
1791:
1787:
1784:
1781:
1777:
1734:
1723:
1720:
1715:
1712:
1701:
1695:
1689:
1680:
1676:
1673:
1670:
1666:
1657:
1645:
1640:
1637:
1631:
1625:
1617:
1609:
1600:
1597:
1593:
1584:
1583:
1580:
1577:
1559:
1556:
1553:
1546:
1543:
1538:
1535:
1528:
1525:
1522:
1518:
1514:
1511:
1508:
1503:
1499:
1496:
1493:
1489:
1484:
1481:
1478:
1468:
1464:
1459:
1456:
1454:
1445:
1442:
1439:
1435:
1432:
1428:
1425:
1422:
1412:
1408:
1403:
1399:
1396:
1393:
1386:
1383:
1378:
1375:
1368:
1365:
1362:
1358:
1354:
1351:
1348:
1344:
1341:
1337:
1334:
1331:
1321:
1317:
1312:
1309:
1307:
1298:
1294:
1291:
1287:
1283:
1279:
1276:
1273:
1263:
1259:
1254:
1250:
1247:
1244:
1237:
1234:
1229:
1226:
1219:
1216:
1213:
1209:
1205:
1202:
1199:
1195:
1191:
1187:
1184:
1180:
1170:
1166:
1161:
1158:
1156:
1147:
1142:
1139:
1136:
1129:
1126:
1122:
1116:
1112:
1108:
1105:
1102:
1098:
1094:
1091:
1088:
1081:
1078:
1073:
1070:
1063:
1060:
1057:
1053:
1049:
1046:
1043:
1039:
1035:
1030:
1027:
1024:
1017:
1014:
1010:
1004:
1000:
998:
989:
986:
983:
979:
975:
972:
969:
962:
959:
955:
950:
948:
942:
939:
936:
929:
926:
922:
908:
899:
886:
883:
880:
877:
871:
863:
851:
844:
840:
839:wave function
835:
819:
816:
813:
810:
807:
801:
798:
795:
777:
762:
759:
756:
753:
750:
747:
741:
738:
735:
710:
707:
704:
697:
694:
689:
686:
679:
676:
673:
669:
665:
662:
659:
654:
650:
647:
644:
640:
635:
632:
629:
619:
615:
610:
607:
601:
593:
585:
576:
573:
569:
559:
555:
551:
547:
537:
535:
531:
527:
526:closed system
523:
519:
515:
511:
506:
489:
486:
483:
460:
440:
432:
413:
407:
404:
401:
395:
392:
386:
378:
364:
360:
344:
324:
318:
315:
312:
306:
300:
292:
283:
277:
274:
269:
266:
260:
257:
237:
229:
226:
222:
218:
214:
210:
206:
202:
198:
194:
190:
186:
182:
178:
174:
170:
166:
162:
158:
155:
151:
147:
143:
140:
136:
135:
134:
126:
124:
119:
115:
111:
107:
103:
99:
95:
85:
83:
79:
75:
71:
67:
63:
59:
57:
52:
48:
46:
41:
38:
34:
30:
19:
2678:
2658:. Retrieved
2643:
2637:
2618:
2612:
2584:
2401:
2397:
2391:
2367:
1984:
1863:
1760:
1579:So finally,
1578:
907:product rule
900:
841:which obeys
836:
783:
557:
545:
543:
513:
509:
507:
430:
428:
224:
220:
200:
132:
102:intersection
91:
88:Applications
55:
45:mathematical
44:
32:
26:
2599:gauge orbit
2394:phase-space
554:Hamiltonian
363:Hamiltonian
230:A quantity
150:Hamiltonian
106:isosurfaces
2660:2011-05-05
2605:References
1981:Derivation
780:Derivation
530:Lagrangian
522:involution
217:Lagrangian
209:Lagrangian
193:Lagrangian
181:Lagrangian
169:Lagrangian
118:rigid body
98:trajectory
58:constraint
47:constraint
2619:Mechanics
2559:−
2498:Φ
2425:×
2382:quantized
2326:⟩
2323:ψ
2307:ψ
2304:⟨
2298:−
2295:⟩
2292:ψ
2276:ψ
2273:⟨
2259:ℏ
2247:−
2237:⟩
2231:⟨
2191:⟩
2188:ψ
2174:⟩
2171:ψ
2143:⟩
2140:ψ
2121:−
2102:ψ
2099:⟨
2093:ℏ
2081:−
2075:⟩
2072:ψ
2040:ψ
2037:⟨
2031:ℏ
2019:−
2013:⟩
2007:⟨
1973:provided
1955:⟩
1949:⟨
1911:≠
1872:ψ
1844:⟩
1838:⟨
1738:⟩
1735:ψ
1702:ψ
1699:⟨
1693:⟩
1690:ψ
1658:ψ
1655:⟨
1649:ℏ
1638:−
1629:⟩
1626:ψ
1610:ψ
1607:⟨
1557:ψ
1526:ψ
1512:ψ
1482:ψ
1472:ℏ
1460:−
1443:ψ
1426:ψ
1416:ℏ
1397:ψ
1366:ψ
1352:ψ
1335:ψ
1325:ℏ
1313:−
1295:ψ
1277:ψ
1267:ℏ
1248:ψ
1217:ψ
1203:ψ
1188:ψ
1174:ℏ
1162:−
1140:ψ
1106:ψ
1092:ψ
1061:ψ
1047:ψ
1028:ψ
987:ψ
973:ψ
884:ψ
869:∂
864:ψ
861:∂
855:ℏ
757:−
708:ψ
677:ψ
663:ψ
633:ψ
623:ℏ
611:−
605:⟩
602:ψ
586:ψ
583:⟨
552:with the
396:−
384:∂
376:∂
361:with the
298:∂
290:∂
213:rotations
123:mechanics
40:conserved
29:mechanics
2702:Category
2675:(2004).
1560:⟩
1523:⟨
1515:⟩
1479:⟨
1446:⟩
1423:⟨
1400:⟩
1363:⟨
1355:⟩
1332:⟨
1299:⟩
1274:⟨
1251:⟩
1214:⟨
1206:⟩
1181:⟨
1143:⟩
1103:⟨
1095:⟩
1058:⟨
1050:⟩
1025:⟨
990:⟩
970:⟨
943:⟩
937:⟨
711:⟩
674:⟨
666:⟩
630:⟨
550:commutes
165:symmetry
114:rotation
76:and the
56:physical
2155:Since
1864:But if
1757:Comment
225:current
167:of the
2687:
2651:
2625:
2374:energy
725:where
211:under
203:) and
66:energy
2400:, or
2204:then
197:space
116:of a
80:(for
35:is a
2685:ISBN
2649:ISBN
2623:ISBN
1811:and
1765:and
453:and
185:time
144:The
31:, a
536:).
357:'s
223:or
104:of
84:).
27:In
2704::
556:,
505:.
187:,
171:.
125:.
72:,
68:,
2693:.
2663:.
2631:.
2565:t
2562:v
2556:x
2553:=
2550:)
2547:t
2544:,
2541:v
2538:,
2535:x
2532:(
2529:C
2509:)
2505:x
2501:(
2495:+
2490:2
2486:v
2480:2
2477:1
2472:=
2469:)
2465:v
2461:,
2457:x
2453:(
2450:H
2429:v
2421:x
2417:=
2413:L
2344:0
2341:=
2330:)
2319:|
2315:Q
2311:|
2301:E
2288:|
2284:Q
2280:|
2270:E
2266:(
2256:i
2252:1
2244:=
2234:Q
2225:t
2222:d
2218:d
2184:|
2180:E
2177:=
2167:|
2163:H
2136:|
2131:)
2127:H
2124:Q
2118:Q
2115:H
2111:(
2106:|
2090:i
2086:1
2078:=
2068:|
2063:]
2059:Q
2056:,
2053:H
2049:[
2044:|
2028:i
2024:1
2016:=
2010:Q
2001:t
1998:d
1994:d
1975:Q
1961:0
1958:=
1952:Q
1943:t
1940:d
1936:d
1914:0
1907:]
1903:Q
1900:,
1897:H
1893:[
1850:0
1847:=
1841:Q
1832:t
1829:d
1825:d
1813:Q
1799:0
1796:=
1792:]
1788:Q
1785:,
1782:H
1778:[
1767:Q
1763:H
1731:|
1724:t
1721:d
1716:Q
1713:d
1706:|
1696:+
1686:|
1681:]
1677:Q
1674:,
1671:H
1667:[
1662:|
1646:i
1641:1
1632:=
1622:|
1618:Q
1614:|
1601:t
1598:d
1594:d
1554:|
1547:t
1544:d
1539:Q
1536:d
1529:|
1519:+
1509:|
1504:]
1500:Q
1497:,
1494:H
1490:[
1485:|
1469:i
1465:1
1457:=
1440:|
1436:H
1433:Q
1429:|
1413:i
1409:1
1404:+
1394:|
1387:t
1384:d
1379:Q
1376:d
1369:|
1359:+
1349:|
1345:Q
1342:H
1338:|
1322:i
1318:1
1310:=
1292:H
1288:|
1284:Q
1280:|
1264:i
1260:1
1255:+
1245:|
1238:t
1235:d
1230:Q
1227:d
1220:|
1210:+
1200:|
1196:Q
1192:|
1185:H
1171:i
1167:1
1159:=
1148:)
1137:|
1130:t
1127:d
1123:d
1117:(
1113:Q
1109:|
1099:+
1089:|
1082:t
1079:d
1074:Q
1071:d
1064:|
1054:+
1044:|
1040:Q
1036:)
1031:|
1018:t
1015:d
1011:d
1005:(
1001:=
984:|
980:Q
976:|
963:t
960:d
956:d
951:=
940:Q
930:t
927:d
923:d
903:Q
887:.
881:H
878:=
872:t
852:i
823:)
820:t
817:,
814:p
811:,
808:x
805:(
802:Q
799:=
796:Q
786:Q
763:H
760:Q
754:Q
751:H
748:=
745:]
742:Q
739:,
736:H
733:[
705:|
698:t
695:d
690:Q
687:d
680:|
670:+
660:|
655:]
651:Q
648:,
645:H
641:[
636:|
620:i
616:1
608:=
598:|
594:Q
590:|
577:t
574:d
570:d
558:H
546:Q
528:(
514:n
510:n
493:}
490:B
487:,
484:A
481:{
461:B
441:A
414:.
411:}
408:H
405:,
402:A
399:{
393:=
387:t
379:A
345:A
325:,
322:}
319:H
316:,
313:A
310:{
307:+
301:t
293:A
284:=
278:t
275:d
270:A
267:d
261:=
258:0
238:A
227:.
199:(
156:.
20:)
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