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Spinor spherical harmonics

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1014: 579: 1241: 753: 126: 1009:{\displaystyle Y_{l,\pm {\frac {1}{2}},j,m}={\frac {1}{\sqrt {2{\bigl (}j\mp {\frac {1}{2}}{\bigr )}+1}}}{\begin{pmatrix}\pm {\sqrt {j\mp {\frac {1}{2}}\pm m+{\frac {1}{2}}}}Y_{l}^{m-{\frac {1}{2}}}\\{\sqrt {j\mp {\frac {1}{2}}\mp m+{\frac {1}{2}}}}Y_{l}^{m+{\frac {1}{2}}}\end{pmatrix}}.} 574:{\displaystyle {\begin{aligned}\mathbf {j} ^{2}Y_{l,s,j,m}&=j(j+1)Y_{l,s,j,m}\\\mathrm {j} _{\mathrm {z} }Y_{l,s,j,m}&=mY_{l,s,j,m}\;;\;m=-j,-(j-1),\cdots ,j-1,j\\\mathbf {l} ^{2}Y_{l,s,j,m}&=l(l+1)Y_{l,s,j,m}\\\mathbf {s} ^{2}Y_{l,s,j,m}&=s(s+1)Y_{l,s,j,m}\end{aligned}}} 131: 738: 1049: 71: 1282: 1211: 1186: 1159: 1125: 1093: 20: 1316: 1306: 1148:(6 December 2012). "9.3 Separation of the Variables for the Dirac Equation with Central Potential (minimally coupled)". 645: 87: 1311: 1275: 1117: 51: 117: 59: 621: 1176: 629: 62:, the spinor spherical harmonics are a basis for the total angular momentum operator (angular momentum plus 1268: 1077: 1052: 55: 1022: 1248: 1217: 1207: 1182: 1155: 1121: 1089: 47: 27: 50:
defined over the sphere. The spinor spherical harmonics are the natural spinor analog of the
1301: 636: 1252: 1206:. Translated by J. B. Sykes; J. S. Bell (2nd ed.). Oxford: Butterworth-Heinemann. 1145: 1110: 1085: 79: 67: 63: 1295: 83: 1149: 113: 1221: 616:
are the (dimensionless) total, orbital and spin angular momentum operators,
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Berestetskii, V. B.; E. M. Lifshitz; L. P. Pitaevskii (2008).
1082:
Angular momentum in Quantum Physics: Theory and Application
1256: 66:). These functions are used in analytical solutions to 851: 74:. The spinor spherical harmonics are sometimes called 1084:, Encyclopedia of Mathematics, vol. 8, Reading: 1025: 756: 648: 129: 1109: 1043: 1008: 732: 573: 733:{\displaystyle PY_{l,sj,m}=(-1)^{l}Y_{l,s,j,m}.} 1151:Relativistic Quantum Mechanics: Wave Equations 1276: 832: 809: 8: 1072: 1070: 1068: 1283: 1269: 747:systems, they are given in matrix form by 315: 311: 1035: 1030: 1024: 983: 976: 971: 955: 936: 928: 912: 905: 900: 884: 865: 857: 846: 831: 830: 820: 808: 807: 798: 771: 761: 755: 703: 693: 656: 647: 543: 490: 480: 475: 446: 393: 383: 378: 287: 249: 238: 237: 232: 203: 150: 140: 135: 130: 128: 82:who employed them in the solution of the 1064: 1178:Elementary Theory of Angular Momentum 1112:Angular Momentum in Quantum Mechanics 7: 1237: 1235: 1181:. Dover Publications, Incorporated. 1140: 1138: 1136: 239: 233: 14: 21:Spin-weighted spherical harmonics 1239: 476: 379: 136: 98:The spinor spherical harmonics 690: 680: 536: 524: 439: 427: 346: 334: 196: 184: 1: 16:Special functions on a sphere 1255:. You can help Knowledge by 76:Pauli central field spinors 1333: 1234: 1175:Rose, M. E. (2013-12-20). 1118:Princeton University Press 52:vector spherical harmonics 32:spinor spherical harmonics 18: 1044:{\displaystyle Y_{l}^{m}} 118:angular momentum operator 60:angular momentum operator 622:azimuthal quantum number 36:spin spherical harmonics 19:Not to be confused with 1204:Quantum electrodynamics 1108:Edmonds, A. R. (1957), 1080:; Louck, J. D. (1981), 630:magnetic quantum number 1251:-related article is a 1045: 1010: 734: 575: 88:spin–orbit interaction 1317:Quantum physics stubs 1046: 1011: 735: 576: 54:. While the standard 1023: 754: 646: 127: 58:are a basis for the 1307:Rotational symmetry 1053:spherical harmonics 1040: 994: 923: 639:operation, we have 56:spherical harmonics 1041: 1026: 1006: 997: 967: 896: 730: 571: 569: 1312:Special functions 1264: 1263: 1249:quantum mechanics 1213:978-0-08-050346-2 1188:978-0-486-78879-1 1161:978-3-642-88082-7 1127:978-0-691-07912-7 1078:Biedenharn, L. C. 991: 965: 963: 944: 920: 894: 892: 873: 844: 843: 828: 779: 48:special functions 28:quantum mechanics 1324: 1285: 1278: 1271: 1243: 1236: 1226: 1225: 1199: 1193: 1192: 1172: 1166: 1165: 1142: 1131: 1130: 1115: 1105: 1099: 1098: 1074: 1050: 1048: 1047: 1042: 1039: 1034: 1015: 1013: 1012: 1007: 1002: 1001: 993: 992: 984: 975: 966: 964: 956: 945: 937: 929: 922: 921: 913: 904: 895: 893: 885: 874: 866: 858: 845: 836: 835: 829: 821: 813: 812: 803: 799: 794: 793: 780: 772: 739: 737: 736: 731: 726: 725: 698: 697: 676: 675: 615: 609: 603: 597: 580: 578: 577: 572: 570: 566: 565: 513: 512: 485: 484: 479: 469: 468: 416: 415: 388: 387: 382: 310: 309: 272: 271: 244: 243: 242: 236: 226: 225: 173: 172: 145: 144: 139: 108: 72:radial potential 40:spinor harmonics 1332: 1331: 1327: 1326: 1325: 1323: 1322: 1321: 1292: 1291: 1290: 1289: 1232: 1230: 1229: 1214: 1201: 1200: 1196: 1189: 1174: 1173: 1169: 1162: 1146:Greiner, Walter 1144: 1143: 1134: 1128: 1107: 1106: 1102: 1096: 1088:, p. 283, 1076: 1075: 1066: 1061: 1021: 1020: 996: 995: 925: 924: 847: 757: 752: 751: 699: 689: 652: 644: 643: 611: 605: 599: 585: 568: 567: 539: 514: 486: 474: 471: 470: 442: 417: 389: 377: 374: 373: 283: 273: 245: 231: 228: 227: 199: 174: 146: 134: 125: 124: 107: 99: 96: 34:(also known as 24: 17: 12: 11: 5: 1330: 1328: 1320: 1319: 1314: 1309: 1304: 1294: 1293: 1288: 1287: 1280: 1273: 1265: 1262: 1261: 1244: 1228: 1227: 1212: 1194: 1187: 1167: 1160: 1132: 1126: 1100: 1094: 1086:Addison-Wesley 1063: 1062: 1060: 1057: 1051:are the usual 1038: 1033: 1029: 1017: 1016: 1005: 1000: 990: 987: 982: 979: 974: 970: 962: 959: 954: 951: 948: 943: 940: 935: 932: 927: 926: 919: 916: 911: 908: 903: 899: 891: 888: 883: 880: 877: 872: 869: 864: 861: 856: 853: 852: 850: 842: 839: 834: 827: 824: 819: 816: 811: 806: 802: 797: 792: 789: 786: 783: 778: 775: 770: 767: 764: 760: 741: 740: 729: 724: 721: 718: 715: 712: 709: 706: 702: 696: 692: 688: 685: 682: 679: 674: 671: 668: 665: 662: 659: 655: 651: 582: 581: 564: 561: 558: 555: 552: 549: 546: 542: 538: 535: 532: 529: 526: 523: 520: 517: 515: 511: 508: 505: 502: 499: 496: 493: 489: 483: 478: 473: 472: 467: 464: 461: 458: 455: 452: 449: 445: 441: 438: 435: 432: 429: 426: 423: 420: 418: 414: 411: 408: 405: 402: 399: 396: 392: 386: 381: 376: 375: 372: 369: 366: 363: 360: 357: 354: 351: 348: 345: 342: 339: 336: 333: 330: 327: 324: 321: 318: 314: 308: 305: 302: 299: 296: 293: 290: 286: 282: 279: 276: 274: 270: 267: 264: 261: 258: 255: 252: 248: 241: 235: 230: 229: 224: 221: 218: 215: 212: 209: 206: 202: 198: 195: 192: 189: 186: 183: 180: 177: 175: 171: 168: 165: 162: 159: 156: 153: 149: 143: 138: 133: 132: 103: 95: 92: 80:Wolfgang Pauli 78:, in honor to 68:Dirac equation 15: 13: 10: 9: 6: 4: 3: 2: 1329: 1318: 1315: 1313: 1310: 1308: 1305: 1303: 1300: 1299: 1297: 1286: 1281: 1279: 1274: 1272: 1267: 1266: 1260: 1258: 1254: 1250: 1245: 1242: 1238: 1233: 1223: 1219: 1215: 1209: 1205: 1198: 1195: 1190: 1184: 1180: 1179: 1171: 1168: 1163: 1157: 1153: 1152: 1147: 1141: 1139: 1137: 1133: 1129: 1123: 1119: 1114: 1113: 1104: 1101: 1097: 1095:0-201-13507-8 1091: 1087: 1083: 1079: 1073: 1071: 1069: 1065: 1058: 1056: 1054: 1036: 1031: 1027: 1003: 998: 988: 985: 980: 977: 972: 968: 960: 957: 952: 949: 946: 941: 938: 933: 930: 917: 914: 909: 906: 901: 897: 889: 886: 881: 878: 875: 870: 867: 862: 859: 854: 848: 840: 837: 825: 822: 817: 814: 804: 800: 795: 790: 787: 784: 781: 776: 773: 768: 765: 762: 758: 750: 749: 748: 746: 727: 722: 719: 716: 713: 710: 707: 704: 700: 694: 686: 683: 677: 672: 669: 666: 663: 660: 657: 653: 649: 642: 641: 640: 638: 633: 631: 628:is the total 627: 623: 620:is the total 619: 614: 608: 602: 596: 592: 588: 562: 559: 556: 553: 550: 547: 544: 540: 533: 530: 527: 521: 518: 516: 509: 506: 503: 500: 497: 494: 491: 487: 481: 465: 462: 459: 456: 453: 450: 447: 443: 436: 433: 430: 424: 421: 419: 412: 409: 406: 403: 400: 397: 394: 390: 384: 370: 367: 364: 361: 358: 355: 352: 349: 343: 340: 337: 331: 328: 325: 322: 319: 316: 312: 306: 303: 300: 297: 294: 291: 288: 284: 280: 277: 275: 268: 265: 262: 259: 256: 253: 250: 246: 222: 219: 216: 213: 210: 207: 204: 200: 193: 190: 187: 181: 178: 176: 169: 166: 163: 160: 157: 154: 151: 147: 141: 123: 122: 121: 119: 116:of the total 115: 112: 106: 102: 93: 91: 89: 85: 84:hydrogen atom 81: 77: 73: 69: 65: 61: 57: 53: 49: 45: 44:Pauli spinors 41: 37: 33: 29: 22: 1257:expanding it 1246: 1231: 1203: 1197: 1177: 1170: 1154:. 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Index

Spin-weighted spherical harmonics
quantum mechanics
special functions
vector spherical harmonics
spherical harmonics
angular momentum operator
spin
Dirac equation
radial potential
Wolfgang Pauli
hydrogen atom
spin–orbit interaction
spinors
eigenstates
angular momentum operator
azimuthal quantum number
magnetic quantum number
parity
spin-1/2
spherical harmonics



Biedenharn, L. C.
Addison-Wesley
ISBN
0-201-13507-8
Angular Momentum in Quantum Mechanics
Princeton University Press
ISBN

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