4625:
2603:
3459:
2199:
4238:
2210:
3112:
4620:{\displaystyle {\begin{aligned}\langle n\,j\,m|x|n'\,j'\,m'\rangle &=\left\langle n\,j\,m\left|{\frac {T_{-1}^{(1)}-T_{1}^{(1)}}{\sqrt {2}}}\right|n'\,j'\,m'\right\rangle \\&={\frac {1}{\sqrt {2}}}\langle n\,j\|T^{(1)}\|n'\,j'\rangle \,{\big (}\langle j'\,m'\,1\,(-1)|j\,m\rangle -\langle j'\,m'\,1\,1|j\,m\rangle {\big )}.\end{aligned}}}
1825:
1805:
970:= +1, 0, −1). Similarly, if an electron is in one of the 4d orbitals, rotating the system will move it into a different 4d orbital. Finally, an analogous statement is true for the position operator: when the system is rotated, the three different components of the position operator are effectively interchanged or mixed.
3873:
1331:
of a spin-4 representation, two spin-3 representations, three spin-2 representations, two spin-1 representations, and a spin-0 (i.e. trivial) representation. The nonzero matrix elements can only come from the spin-0 subspace. The Wigner–Eckart theorem works because the direct product decomposition
761:
to the state. The matrix element one finds for the spherical tensor operator is proportional to a
Clebsch–Gordan coefficient, which arises when considering adding two angular momenta. When stated another way, one can say that the Wigner–Eckart theorem is a theorem that tells how vector operators
3646:
2598:{\displaystyle {\begin{aligned}{\sqrt {(j\pm m)(j\mp m+1)}}\langle j\,(m\mp 1)|T_{q}^{(k)}|j'\,m'\rangle =&{\sqrt {(j'\mp m')(j'\pm m'+1)}}\,\langle j\,m|T_{q}^{(k)}|j'\,(m'\pm 1)\rangle \\&+{\sqrt {(k\mp q)(k\pm q+1)}}\,\langle j\,m|T_{q\pm 1}^{(k)}|j'\,m'\rangle .\end{aligned}}}
442:
3454:{\displaystyle \langle j\,m|T_{q}^{(k)}|j'\,m'\rangle ={\frac {(-1)^{2k}\langle j'\,m'\,k\,q|j\,m\rangle \langle j\|T^{(k)}\|j'\rangle _{\mathrm {R} }}{\sqrt {2j+1}}}=(-1)^{j-m}{\begin{pmatrix}j&k&j'\\-m&q&m'\end{pmatrix}}\langle j\|T^{(k)}\|j'\rangle _{\mathrm {R} }.}
3096:
2194:{\displaystyle {\begin{aligned}\langle j\,m||j'\,m'\rangle ={}&\hbar {\sqrt {(j\pm m)(j\mp m+1)}}\,\langle j\,(m\mp 1)|T_{q}^{(k)}|j'\,m'\rangle \\&-\hbar {\sqrt {(j'\mp m')(j'\pm m'+1)}}\,\langle j\,m|T_{q}^{(k)}|j'\,(m'\pm 1)\rangle .\end{aligned}}}
2765:
1424:. The key qualitative aspect of the Clebsch–Gordan decomposition that makes the argument work is that in the decomposition of the tensor product of two irreducible representations, each irreducible representation occurs only once. This allows
1585:
1183:
and some or all of the 44 unknown matrix elements. Different rotations of the system lead to different algebraic relations, and it turns out that there is enough information to figure out all of the matrix elements in this way.
3687:
957:
The Wigner–Eckart theorem works because all 45 of these different calculations are related to each other by rotations. If an electron is in one of the 2p orbitals, rotating the system will generally move it into a
1574:
1227:, also called the "spin-2 representation". Similarly, the 2p quantum states form a 3-dimensional irrep (called "spin-1"), and the components of the position operator also form the 3-dimensional "spin-1" irrep.
4227:
762:
behave in a subspace. Within a given subspace, a component of a vector operator will behave in a way proportional to the same component of the angular momentum operator. This definition is given in the book
4114:
4032:
2866:
944:
of them can be used, as long as it is nonzero). Then the other 44 integrals can be inferred from that first one—without the need to write down any wavefunctions or evaluate any integrals—with the help of
3488:
73:, who developed the formalism as a link between the symmetry transformation groups of space (applied to the Schrödinger equations) and the laws of conservation of energy, momentum, and angular momentum.
1058:
1414:
1309:
866:
4243:
2653:
2215:
1830:
1590:
722:
641:
208:
281:
2956:
2648:
1177:
1106:
564:
486:
525:
253:
152:
107:
1133:
3106:
There are different conventions for the reduced matrix elements. One convention, used by Racah and Wigner, includes an additional phase and normalization factor,
2887:, where the coefficient of proportionality is independent of the indices. Hence, by comparing recursion relations, we can identify the Clebsch–Gordan coefficient
1800:{\displaystyle {\begin{aligned}&\langle j\,m||j'\,m'\rangle =\hbar {\sqrt {(k\mp q)(k\pm q+1)}}\,\langle j\,m|T_{q\pm 1}^{(k)}|j'\,m'\rangle .\end{aligned}}}
273:
228:
127:
4962:
4972:
1219:. Rotating the system transforms these states into each other, so this is an example of a "group representation", in this case, the 5-dimensional
3903:, which is a nontrivial problem. However, the Wigner–Eckart theorem simplifies the problem. (In fact, we could obtain the solution quickly using
3868:{\displaystyle \langle j\,m|T_{q}^{(k)}|j'\,m'\rangle ={\frac {\langle j'\,m'\,k\,q|j\,m\rangle \langle j\|T^{(k)}\|j'\rangle }{\sqrt {2j'+1}}}.}
4967:
4843:
Wigner, E. P. (1951). "On the
Matrices Which Reduce the Kronecker Products of Representations of S. R. Groups". In Wightman, Arthur S. (ed.).
4860:
1446:
4159:
4897:
4043:
3952:
3641:{\displaystyle \langle j\|T^{\dagger (k)}\|j'\rangle _{\mathrm {R} }=(-1)^{k+j'-j}\langle j'\|T^{(k)}\|j\rangle _{\mathrm {R} }^{*},}
4917:
3482:
factor is sometimes omitted in literature.) With this choice of normalization, the reduced matrix element satisfies the relation:
2803:
1315:
of those three representations, i.e. the spin-1 representation of the 2p orbitals, the spin-1 representation of the components of
1420:
of the corresponding abstract vector (in 45-dimensional space) onto the spin-0 subspace. The results of this calculation are the
3672:
phase factor in the definition of the reduced matrix element, it is affected by the phase convention for the
Hermitian adjoint.
1332:
contains one and only one spin-0 subspace, which implies that all the matrix elements are determined by a single scale factor.
976:
2609:
1421:
946:
644:
62:
1338:
1233:
790:
61:
can be expressed as the product of two factors, one of which is independent of angular momentum orientation, and the other a
1191:
to the states, rather than rotating the states. But this is fundamentally the same thing, because of the close mathematical
3899:. This matrix element is the expectation value of a Cartesian operator in a spherically symmetric hydrogen-atom-eigenstate
437:{\displaystyle \langle j\,m|T_{q}^{(k)}|j'\,m'\rangle =\langle j'\,m'\,k\,q|j\,m\rangle \langle j\|T^{(k)}\|j'\rangle ,}
1417:
1319:, and the spin-2 representation of the 4d orbitals. This direct product, a 45-dimensional representation of SU(2), is
671:
157:
1324:
1220:
4947:
1437:
1224:
1192:
1188:
51:
3677:
780:
590:
3091:{\displaystyle \langle j'\,m'|T_{q\pm 1}^{(k)}|j\,m\rangle \propto \langle j\,m\,k\,(q\pm 1)|j'\,m'\rangle .}
2760:{\displaystyle {\begin{aligned}\sum _{c}a_{b,c}x_{c}&=0,&\sum _{c}a_{b,c}y_{c}&=0.\end{aligned}}}
76:
Mathematically, the Wigner–Eckart theorem is generally stated in the following way. Given a tensor operator
908:
3900:
4717:, this makes the Clebsch–Gordan Coefficients zero, leading to the expectation value to be equal to zero.
1212:
1204:
963:
915:. If we do this directly, it involves calculating 45 different integrals: there are 3 possibilities for
39:
1138:
1067:
4817:
47:
1203:
To state these observations more precisely and to prove them, it helps to invoke the mathematics of
4131:
538:
453:
4749:
751:
The Wigner–Eckart theorem states indeed that operating with a spherical tensor operator of rank
4925:
4913:
4893:
4856:
4731:
3652:
890:
43:
497:
79:
4848:
4825:
3904:
1425:
55:
3922:, which is a vector. Since vectors are rank-1 spherical tensor operators, it follows that
1111:
4942:
4928:
4808:
4726:
3656:
936:
The Wigner–Eckart theorem allows one to obtain the same information after evaluating just
4821:
233:
132:
1312:
912:
784:
258:
213:
112:
4956:
66:
17:
1216:
4852:
4875:
J. J. Sakurai: "Modern quantum mechanics" (Massachusetts, 1994, Addison-Wesley).
70:
757:
on an angular momentum eigenstate is like adding a state with angular momentum
3469:
2608:
This recursion relation for the matrix elements closely resembles that of the
1328:
58:
4933:
3668:. Although this relation is not affected by the presence or absence of the
949:, which can be easily looked up in a table or computed by hand or computer.
775:
Motivating example: position operator matrix elements for 4d → 2p transition
4829:
4892:, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer,
4890:
Lie groups, Lie algebras, and representations: An elementary introduction
1810:
If we expand the commutator on the LHS by calculating the action of the
1311:. It turns out that these are transformed by rotations according to the
1207:. For example, the set of all possible 4d orbitals (i.e., the 5 states
35:
3675:
Another convention for reduced matrix elements is that of
Sakurai's
1569:{\displaystyle =\hbar {\sqrt {(k\mp q)(k\pm q+1)}}T_{q\pm 1}^{(k)},}
1335:
Apart from the overall scale factor, calculating the matrix element
973:
If we start by knowing just one of the 45 values (say, we know that
4912:
J. J. Sakurai, (1994). "Modern
Quantum Mechanics", Addison Wesley,
27:
Theorem used in quantum mechanics for angular momentum calculations
4784:
This is a special notation specific to the Wigner–Eckart theorem.
4222:{\displaystyle T_{\pm 1}^{(1)}=\mp {\frac {x\pm iy}{\sqrt {2}}}.}
4109:{\displaystyle T_{q}^{(1)}={\sqrt {\frac {4\pi }{3}}}rY_{1}^{q}}
4027:{\displaystyle x={\frac {T_{-1}^{(1)}-T_{1}^{(1)}}{\sqrt {2}}},}
1187:(In practice, when working through this math, we usually apply
2642:. We therefore have two sets of linear homogeneous equations:
3928:
must be some linear combination of a rank-1 spherical tensor
2861:{\displaystyle {\frac {x_{c}}{x_{d}}}={\frac {y_{c}}{y_{d}}}}
1064:
is also the matrix element between the rotated version of
1193:
relation between rotations and angular momentum operators
1053:{\displaystyle \langle 2p,m_{1}|r_{i}|4d,m_{2}\rangle =K}
787:
of a hydrogen atom, i.e. the matrix elements of the form
911:
that distinguish different orbitals within the 2p or 4d
1409:{\displaystyle \langle 2p,m_{1}|r_{i}|4d,m_{2}\rangle }
1304:{\displaystyle \langle 2p,m_{1}|r_{i}|4d,m_{2}\rangle }
861:{\displaystyle \langle 2p,m_{1}|r_{i}|4d,m_{2}\rangle }
4134:, which themselves are also spherical tensors of rank
3346:
4630:
The above expression gives us the matrix element for
4241:
4162:
4046:
3955:
3690:
3491:
3115:
2959:
2806:
2797:. We can only say that the ratios are equal, that is
2651:
2213:
1828:
1588:
1449:
1341:
1236:
1141:
1114:
1070:
979:
793:
674:
593:
541:
500:
456:
284:
261:
236:
216:
160:
135:
115:
82:
3907:, although a slightly longer route will be taken.)
1060:) and then we rotate the system, we can infer that
4806:Racah, G. (1942). "Theory of Complex Spectra II".
4769:provides a reminder of its rank. However, unlike
4619:
4221:
4108:
4026:
3867:
3640:
3453:
3090:
2860:
2759:
2597:
2193:
1799:
1568:
1408:
1303:
1171:
1127:
1100:
1052:
860:
716:
635:
558:
519:
480:
436:
267:
247:
222:
202:
146:
121:
101:
566:denotes an eigenstate of total angular momentum
494:-th component of the spherical tensor operator
4644:basis. To find the expectation value, we set
717:{\displaystyle \langle j\|T^{(k)}\|j'\rangle }
203:{\displaystyle \langle j\|T^{(k)}\|j'\rangle }
4605:
4500:
1179:. This gives an algebraic relation involving
783:for an electron transition from a 4d to a 2p
8:
4600:
4560:
4554:
4505:
4494:
4474:
4455:
4445:
4299:
4246:
3837:
3826:
3807:
3801:
3798:
3758:
3749:
3691:
3619:
3612:
3593:
3582:
3532:
3520:
3498:
3492:
3437:
3425:
3406:
3400:
3285:
3273:
3254:
3248:
3245:
3205:
3174:
3116:
3082:
3030:
3024:
2960:
2585:
2521:
2467:
2397:
2328:
2258:
2181:
2111:
2034:
1964:
1910:
1833:
1787:
1723:
1673:
1596:
1403:
1342:
1298:
1237:
1166:
1071:
1041:
980:
855:
794:
711:
700:
681:
675:
630:
594:
553:
428:
417:
398:
392:
389:
349:
343:
285:
197:
186:
167:
161:
2788:). It is not possible to exactly solve for
929:(−2, −1, 0, 1, 2), and 3 possibilities for
724:denotes some value that does not depend on
4845:The Collected Works of Eugene Paul Wigner
4604:
4603:
4596:
4588:
4584:
4580:
4571:
4550:
4542:
4529:
4525:
4516:
4499:
4498:
4497:
4485:
4462:
4451:
4433:
4409:
4400:
4368:
4363:
4344:
4336:
4329:
4321:
4317:
4290:
4281:
4268:
4260:
4256:
4252:
4242:
4240:
4193:
4175:
4167:
4161:
4100:
4095:
4071:
4056:
4051:
4045:
4037:where we define the spherical tensors as
4001:
3996:
3977:
3969:
3962:
3954:
3814:
3794:
3786:
3782:
3778:
3769:
3755:
3740:
3727:
3715:
3710:
3701:
3697:
3689:
3629:
3623:
3622:
3600:
3559:
3536:
3535:
3505:
3490:
3441:
3440:
3413:
3341:
3329:
3289:
3288:
3261:
3241:
3233:
3229:
3225:
3216:
3196:
3180:
3165:
3152:
3140:
3135:
3126:
3122:
3114:
3073:
3060:
3044:
3040:
3036:
3020:
3012:
3000:
2989:
2980:
2971:
2958:
2850:
2840:
2834:
2823:
2813:
2807:
2805:
2770:one for the Clebsch–Gordan coefficients (
2737:
2721:
2711:
2686:
2670:
2660:
2652:
2650:
2576:
2563:
2551:
2540:
2531:
2527:
2520:
2480:
2446:
2433:
2421:
2416:
2407:
2403:
2396:
2336:
2319:
2306:
2294:
2289:
2280:
2264:
2218:
2214:
2212:
2160:
2147:
2135:
2130:
2121:
2117:
2110:
2050:
2025:
2012:
2000:
1995:
1986:
1970:
1963:
1923:
1916:
1901:
1888:
1873:
1868:
1855:
1843:
1839:
1829:
1827:
1778:
1765:
1753:
1742:
1733:
1729:
1722:
1682:
1664:
1651:
1636:
1631:
1618:
1606:
1602:
1589:
1587:
1551:
1540:
1496:
1475:
1470:
1457:
1448:
1397:
1379:
1373:
1364:
1358:
1340:
1292:
1274:
1268:
1259:
1253:
1235:
1160:
1142:
1140:
1119:
1113:
1093:
1087:
1069:
1035:
1017:
1011:
1002:
996:
978:
962:2p orbital (usually it will wind up in a
849:
831:
825:
816:
810:
792:
688:
673:
636:{\displaystyle \langle j'm'kq|jm\rangle }
619:
592:
542:
540:
505:
499:
466:
461:
455:
405:
385:
377:
373:
369:
360:
334:
321:
309:
304:
295:
291:
283:
260:
235:
215:
174:
159:
134:
114:
87:
81:
3883:Consider the position expectation value
2204:We may combine these two results to get
4742:
2047:
1920:
1679:
1493:
275:, the following equation is satisfied:
7:
4793:
4963:Representation theory of Lie groups
2779:) and one for the matrix elements (
766:by Cohen–Tannoudji, Diu and Laloe.
65:. The name derives from physicists
3946:}. In fact, it can be shown that
3624:
3537:
3442:
3290:
1436:Starting with the definition of a
933:, so the total is 3 × 5 × 3 = 45.
109:and two states of angular momenta
25:
4973:Theorems in representation theory
4775:, it need not be an actual index.
2612:. In fact, both are of the form
1416:is equivalent to calculating the
1230:Now consider the matrix elements
1199:In terms of representation theory
1172:{\displaystyle |4d,m_{2}\rangle }
1101:{\displaystyle \langle 2p,m_{1}|}
1819:on the bra and ket, then we get
1223:("irrep") of the rotation group
1215:) form a 5-dimensional abstract
922:(−1, 0, 1), 5 possibilities for
4752:– The National Academies Press.
4707:spherical tensors. As we have
1579:which we use to then calculate
779:Let's say we want to calculate
4761:The parenthesized superscript
4589:
4543:
4539:
4530:
4469:
4463:
4375:
4369:
4351:
4345:
4269:
4261:
4182:
4176:
4063:
4057:
4008:
4002:
3984:
3978:
3821:
3815:
3787:
3728:
3722:
3716:
3702:
3607:
3601:
3556:
3546:
3515:
3509:
3420:
3414:
3326:
3316:
3268:
3262:
3234:
3193:
3183:
3153:
3147:
3141:
3127:
3061:
3057:
3045:
3013:
3007:
3001:
2981:
2564:
2558:
2552:
2532:
2515:
2497:
2494:
2482:
2464:
2447:
2434:
2428:
2422:
2408:
2391:
2363:
2360:
2338:
2307:
2301:
2295:
2281:
2277:
2265:
2253:
2235:
2232:
2220:
2178:
2161:
2148:
2142:
2136:
2122:
2105:
2077:
2074:
2052:
2013:
2007:
2001:
1987:
1983:
1971:
1958:
1940:
1937:
1925:
1889:
1885:
1880:
1874:
1848:
1844:
1766:
1760:
1754:
1734:
1717:
1699:
1696:
1684:
1652:
1648:
1643:
1637:
1611:
1607:
1558:
1552:
1531:
1513:
1510:
1498:
1487:
1482:
1476:
1450:
1380:
1365:
1275:
1260:
1143:
1094:
1018:
1003:
832:
817:
695:
689:
620:
543:
512:
506:
473:
467:
412:
406:
378:
322:
316:
310:
296:
181:
175:
94:
88:
1:
4968:Theorems in quantum mechanics
1135:, and the rotated version of
4853:10.1007/978-3-662-02781-3_42
4847:. Vol. 3. p. 614.
1211:= −2, −1, 0, 1, 2 and their
953:Qualitative summary of proof
1422:Clebsch–Gordan coefficients
966:of all three basis states,
947:Clebsch–Gordan coefficients
559:{\displaystyle |jm\rangle }
481:{\displaystyle T_{q}^{(k)}}
4989:
4674:. The selection rule for
2610:Clebsch–Gordan coefficient
1325:irreducible representation
1221:irreducible representation
1189:angular momentum operators
743:and is referred to as the
645:Clebsch–Gordan coefficient
154:, there exists a constant
63:Clebsch–Gordan coefficient
52:spherical tensor operators
3478:is often an integer, the
1438:spherical tensor operator
1108:, the rotated version of
781:transition dipole moments
3678:Modern Quantum Mechanics
2921:with the matrix element
909:magnetic quantum numbers
4929:"Wigner–Eckart theorem"
4888:Hall, Brian C. (2015),
3102:Alternative conventions
940:of those 45 integrals (
770:Background and overview
520:{\displaystyle T^{(k)}}
102:{\displaystyle T^{(k)}}
4830:10.1103/PhysRev.62.438
4621:
4223:
4110:
4028:
3869:
3642:
3472:. (Since in practice
3455:
3092:
2862:
2761:
2599:
2195:
1801:
1570:
1410:
1305:
1213:quantum superpositions
1173:
1129:
1102:
1054:
862:
745:reduced matrix element
718:
637:
560:
521:
482:
438:
269:
249:
224:
204:
148:
123:
103:
4943:Wigner–Eckart theorem
4622:
4224:
4111:
4029:
3870:
3643:
3456:
3093:
2863:
2762:
2600:
2196:
1802:
1571:
1411:
1306:
1205:representation theory
1174:
1130:
1128:{\displaystyle r_{i}}
1103:
1055:
964:quantum superposition
863:
719:
638:
561:
522:
483:
439:
270:
250:
225:
205:
149:
124:
104:
40:representation theory
32:Wigner–Eckart theorem
18:Wigner-Eckart theorem
4239:
4160:
4044:
3953:
3916:is one component of
3688:
3655:is defined with the
3489:
3113:
2957:
2950:, then we may write
2804:
2649:
2211:
1826:
1586:
1447:
1339:
1327:, instead it is the
1234:
1139:
1112:
1068:
977:
791:
672:
591:
539:
498:
454:
282:
259:
234:
214:
158:
133:
113:
80:
4822:1942PhRv...62..438R
4379:
4355:
4186:
4132:spherical harmonics
4105:
4067:
4012:
3988:
3726:
3634:
3151:
3011:
2562:
2432:
2305:
2146:
2011:
1884:
1764:
1647:
1562:
1486:
477:
320:
4926:Weisstein, Eric W.
4617:
4615:
4359:
4332:
4219:
4163:
4106:
4091:
4047:
4024:
3992:
3965:
3865:
3706:
3638:
3618:
3468:array denotes the
3451:
3394:
3131:
3088:
2985:
2858:
2757:
2755:
2716:
2665:
2595:
2593:
2536:
2412:
2285:
2191:
2189:
2126:
1991:
1864:
1797:
1795:
1738:
1627:
1566:
1536:
1466:
1406:
1301:
1169:
1125:
1098:
1050:
858:
714:
633:
556:
517:
478:
457:
434:
300:
265:
248:{\displaystyle m'}
245:
220:
210:such that for all
200:
147:{\displaystyle j'}
144:
119:
99:
4862:978-3-642-08154-5
4816:(9–10): 438–462.
4443:
4442:
4386:
4385:
4214:
4213:
4140:. Additionally,
4086:
4085:
4019:
4018:
3860:
3859:
3653:Hermitian adjoint
3311:
3310:
2856:
2829:
2707:
2656:
2518:
2394:
2256:
2108:
1961:
1720:
1534:
891:position operator
889:component of the
764:Quantum Mechanics
268:{\displaystyle q}
223:{\displaystyle m}
122:{\displaystyle j}
46:. It states that
44:quantum mechanics
16:(Redirected from
4980:
4948:Tensor Operators
4939:
4938:
4902:
4876:
4873:
4867:
4866:
4840:
4834:
4833:
4803:
4797:
4791:
4785:
4782:
4776:
4774:
4768:
4759:
4753:
4750:Eckart Biography
4747:
4716:
4706:
4697:
4686:
4680:
4673:
4663:
4653:
4643:
4635:
4626:
4624:
4623:
4618:
4616:
4609:
4608:
4592:
4579:
4570:
4546:
4524:
4515:
4504:
4503:
4493:
4484:
4473:
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4444:
4438:
4434:
4426:
4422:
4418:
4417:
4408:
4399:
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4381:
4380:
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4343:
4330:
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4272:
4264:
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4225:
4220:
4215:
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4115:
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4099:
4087:
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4025:
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4014:
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4000:
3987:
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3963:
3945:
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3927:
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3915:
3898:
3874:
3872:
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3866:
3861:
3852:
3841:
3840:
3836:
3825:
3824:
3790:
3777:
3768:
3756:
3748:
3739:
3731:
3725:
3714:
3705:
3671:
3665:
3647:
3645:
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3628:
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3611:
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3592:
3581:
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3477:
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3064:
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2999:
2984:
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2886:
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2857:
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2596:
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2257:
2219:
2200:
2198:
2197:
2192:
2190:
2171:
2159:
2151:
2145:
2134:
2125:
2109:
2098:
2087:
2073:
2062:
2051:
2040:
2033:
2024:
2016:
2010:
1999:
1990:
1962:
1924:
1917:
1909:
1900:
1892:
1883:
1872:
1860:
1859:
1847:
1818:
1806:
1804:
1803:
1798:
1796:
1786:
1777:
1769:
1763:
1752:
1737:
1721:
1683:
1672:
1663:
1655:
1646:
1635:
1623:
1622:
1610:
1592:
1575:
1573:
1572:
1567:
1561:
1550:
1535:
1497:
1485:
1474:
1462:
1461:
1415:
1413:
1412:
1407:
1402:
1401:
1383:
1378:
1377:
1368:
1363:
1362:
1310:
1308:
1307:
1302:
1297:
1296:
1278:
1273:
1272:
1263:
1258:
1257:
1178:
1176:
1175:
1170:
1165:
1164:
1146:
1134:
1132:
1131:
1126:
1124:
1123:
1107:
1105:
1104:
1099:
1097:
1092:
1091:
1059:
1057:
1056:
1051:
1040:
1039:
1021:
1016:
1015:
1006:
1001:
1000:
867:
865:
864:
859:
854:
853:
835:
830:
829:
820:
815:
814:
756:
742:
736:
729:
723:
721:
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710:
699:
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659:
653:
642:
640:
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623:
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584:
571:
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532:
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524:
523:
518:
516:
515:
493:
487:
485:
484:
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465:
443:
441:
440:
435:
427:
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415:
381:
368:
359:
342:
333:
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319:
308:
299:
274:
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254:
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251:
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209:
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196:
185:
184:
153:
151:
150:
145:
143:
128:
126:
125:
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108:
106:
105:
100:
98:
97:
56:angular momentum
54:in the basis of
21:
4988:
4987:
4983:
4982:
4981:
4979:
4978:
4977:
4953:
4952:
4924:
4923:
4909:
4900:
4887:
4884:
4879:
4874:
4870:
4863:
4842:
4841:
4837:
4809:Physical Review
4805:
4804:
4800:
4792:
4788:
4783:
4779:
4770:
4762:
4760:
4756:
4748:
4744:
4740:
4727:Tensor operator
4723:
4708:
4705:
4699:
4688:
4682:
4675:
4665:
4655:
4645:
4637:
4631:
4614:
4613:
4572:
4563:
4517:
4508:
4486:
4477:
4458:
4424:
4423:
4410:
4401:
4392:
4331:
4325:
4313:
4309:
4302:
4291:
4282:
4273:
4237:
4236:
4195:
4158:
4157:
4147:
4141:
4135:
4128:
4120:
4074:
4042:
4041:
3964:
3951:
3950:
3940:
3937:
3929:
3923:
3917:
3911:
3884:
3881:
3845:
3829:
3810:
3770:
3761:
3757:
3741:
3732:
3686:
3685:
3669:
3657:
3596:
3585:
3566:
3555:
3531:
3523:
3501:
3487:
3486:
3479:
3473:
3465:
3436:
3428:
3409:
3393:
3392:
3384:
3382:
3377:
3368:
3367:
3359:
3357:
3352:
3342:
3325:
3284:
3276:
3257:
3217:
3208:
3192:
3182:
3166:
3157:
3111:
3110:
3104:
3074:
3065:
2972:
2963:
2955:
2954:
2940:
2922:
2914:
2907:
2901:
2895:
2888:
2884:
2877:
2872:
2846:
2836:
2819:
2809:
2802:
2801:
2794:
2789:
2785:
2780:
2776:
2771:
2754:
2753:
2743:
2733:
2717:
2705:
2692:
2682:
2666:
2647:
2646:
2639:
2631:
2619:
2613:
2592:
2591:
2577:
2568:
2471:
2470:
2450:
2438:
2377:
2366:
2352:
2341:
2334:
2320:
2311:
2209:
2208:
2188:
2187:
2164:
2152:
2091:
2080:
2066:
2055:
2038:
2037:
2026:
2017:
1918:
1902:
1893:
1851:
1824:
1823:
1817:
1811:
1794:
1793:
1779:
1770:
1665:
1656:
1614:
1584:
1583:
1453:
1445:
1444:
1434:
1393:
1369:
1354:
1337:
1336:
1288:
1264:
1249:
1232:
1231:
1201:
1156:
1137:
1136:
1115:
1110:
1109:
1083:
1066:
1065:
1031:
1007:
992:
975:
974:
955:
928:
921:
906:
899:
876:
845:
821:
806:
789:
788:
777:
772:
752:
738:
731:
725:
703:
684:
670:
669:
661:
655:
648:
605:
597:
589:
588:
583:
577:
567:
537:
536:
528:
501:
496:
495:
489:
452:
451:
420:
401:
361:
352:
335:
326:
280:
279:
257:
256:
237:
232:
231:
212:
211:
189:
170:
156:
155:
136:
131:
130:
111:
110:
83:
78:
77:
28:
23:
22:
15:
12:
11:
5:
4986:
4984:
4976:
4975:
4970:
4965:
4955:
4954:
4951:
4950:
4945:
4940:
4921:
4908:
4907:External links
4905:
4904:
4903:
4899:978-3319134666
4898:
4883:
4880:
4878:
4877:
4868:
4861:
4835:
4798:
4786:
4777:
4754:
4741:
4739:
4736:
4735:
4734:
4732:Landé g-factor
4729:
4722:
4719:
4703:
4628:
4627:
4612:
4607:
4602:
4599:
4595:
4591:
4587:
4583:
4578:
4575:
4569:
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3321:
3318:
3315:
3309:
3306:
3303:
3300:
3292:
3287:
3282:
3279:
3275:
3270:
3267:
3264:
3260:
3256:
3253:
3250:
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2727:
2724:
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1659:
1654:
1650:
1645:
1642:
1639:
1634:
1630:
1626:
1621:
1617:
1613:
1609:
1605:
1601:
1598:
1595:
1593:
1591:
1577:
1576:
1565:
1560:
1557:
1554:
1549:
1546:
1543:
1539:
1533:
1530:
1527:
1524:
1521:
1518:
1515:
1512:
1509:
1506:
1503:
1500:
1495:
1492:
1489:
1484:
1481:
1478:
1473:
1469:
1465:
1460:
1456:
1452:
1433:
1430:
1405:
1400:
1396:
1392:
1389:
1386:
1382:
1376:
1372:
1367:
1361:
1357:
1353:
1350:
1347:
1344:
1313:tensor product
1300:
1295:
1291:
1287:
1284:
1281:
1277:
1271:
1267:
1262:
1256:
1252:
1248:
1245:
1242:
1239:
1225:SU(2) or SO(3)
1200:
1197:
1168:
1163:
1159:
1155:
1152:
1149:
1145:
1122:
1118:
1096:
1090:
1086:
1082:
1079:
1076:
1073:
1049:
1046:
1043:
1038:
1034:
1030:
1027:
1024:
1020:
1014:
1010:
1005:
999:
995:
991:
988:
985:
982:
954:
951:
926:
919:
904:
897:
877:is either the
872:
857:
852:
848:
844:
841:
838:
834:
828:
824:
819:
813:
809:
805:
802:
799:
796:
776:
773:
771:
768:
749:
748:
713:
709:
706:
702:
697:
694:
691:
687:
683:
680:
677:
667:
632:
629:
626:
622:
618:
615:
611:
608:
603:
600:
596:
586:
581:
555:
552:
549:
545:
534:
514:
511:
508:
504:
475:
472:
469:
464:
460:
445:
444:
433:
430:
426:
423:
419:
414:
411:
408:
404:
400:
397:
394:
391:
388:
384:
380:
376:
372:
367:
364:
358:
355:
351:
348:
345:
341:
338:
332:
329:
324:
318:
315:
312:
307:
303:
298:
294:
290:
287:
264:
243:
240:
219:
199:
195:
192:
188:
183:
180:
177:
173:
169:
166:
163:
142:
139:
118:
96:
93:
90:
86:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
4985:
4974:
4971:
4969:
4966:
4964:
4961:
4960:
4958:
4949:
4946:
4944:
4941:
4936:
4935:
4930:
4927:
4922:
4919:
4918:0-201-53929-2
4915:
4911:
4910:
4906:
4901:
4895:
4891:
4886:
4885:
4881:
4872:
4869:
4864:
4858:
4854:
4850:
4846:
4839:
4836:
4831:
4827:
4823:
4819:
4815:
4811:
4810:
4802:
4799:
4795:
4790:
4787:
4781:
4778:
4773:
4766:
4758:
4755:
4751:
4746:
4743:
4737:
4733:
4730:
4728:
4725:
4724:
4720:
4718:
4715:
4711:
4702:
4695:
4691:
4685:
4678:
4672:
4668:
4662:
4658:
4652:
4648:
4641:
4634:
4610:
4597:
4593:
4585:
4581:
4576:
4573:
4567:
4564:
4557:
4551:
4547:
4536:
4533:
4526:
4521:
4518:
4512:
4509:
4490:
4487:
4481:
4478:
4466:
4459:
4452:
4448:
4439:
4435:
4430:
4428:
4419:
4414:
4411:
4405:
4402:
4396:
4393:
4388:
4382:
4372:
4364:
4360:
4356:
4348:
4340:
4337:
4333:
4326:
4322:
4318:
4314:
4310:
4306:
4304:
4295:
4292:
4286:
4283:
4277:
4274:
4265:
4257:
4253:
4249:
4235:
4234:
4233:
4216:
4210:
4205:
4202:
4199:
4196:
4190:
4187:
4179:
4171:
4168:
4164:
4156:
4155:
4154:
4151:
4144:
4138:
4133:
4127:
4123:
4101:
4096:
4092:
4088:
4082:
4078:
4075:
4068:
4060:
4052:
4048:
4040:
4039:
4038:
4021:
4015:
4005:
3997:
3993:
3989:
3981:
3973:
3970:
3966:
3959:
3956:
3949:
3948:
3947:
3943:
3936:
3932:
3926:
3920:
3914:
3910:We know that
3908:
3906:
3902:
3896:
3892:
3888:
3878:
3862:
3856:
3853:
3849:
3846:
3842:
3833:
3830:
3818:
3811:
3804:
3795:
3791:
3783:
3779:
3774:
3771:
3765:
3762:
3752:
3745:
3742:
3736:
3733:
3719:
3711:
3707:
3698:
3694:
3684:
3683:
3682:
3680:
3679:
3673:
3667:
3664:
3660:
3654:
3635:
3630:
3615:
3604:
3597:
3589:
3586:
3577:
3574:
3570:
3567:
3563:
3560:
3552:
3549:
3543:
3527:
3524:
3512:
3506:
3502:
3495:
3485:
3484:
3483:
3476:
3471:
3448:
3432:
3429:
3417:
3410:
3403:
3395:
3388:
3385:
3379:
3374:
3371:
3363:
3360:
3354:
3349:
3343:
3336:
3333:
3330:
3322:
3319:
3313:
3307:
3304:
3301:
3298:
3280:
3277:
3265:
3258:
3251:
3242:
3238:
3230:
3226:
3221:
3218:
3212:
3209:
3200:
3197:
3189:
3186:
3177:
3170:
3167:
3161:
3158:
3144:
3136:
3132:
3123:
3119:
3109:
3108:
3107:
3101:
3085:
3078:
3075:
3069:
3066:
3054:
3051:
3048:
3041:
3037:
3033:
3027:
3021:
3017:
3004:
2996:
2993:
2990:
2986:
2976:
2973:
2967:
2964:
2953:
2952:
2951:
2947:
2944:
2938:
2934:
2930:
2926:
2918:
2911:
2904:
2898:
2892:
2885:
2878:
2851:
2847:
2841:
2837:
2831:
2824:
2820:
2814:
2810:
2800:
2799:
2798:
2795:
2786:
2777:
2750:
2747:
2745:
2738:
2734:
2728:
2725:
2722:
2718:
2712:
2708:
2702:
2699:
2696:
2694:
2687:
2683:
2677:
2674:
2671:
2667:
2661:
2657:
2645:
2644:
2643:
2638:
2634:
2630:
2626:
2622:
2618:
2611:
2588:
2581:
2578:
2572:
2569:
2555:
2547:
2544:
2541:
2537:
2528:
2524:
2512:
2509:
2506:
2503:
2500:
2491:
2488:
2485:
2477:
2475:
2461:
2458:
2454:
2451:
2442:
2439:
2425:
2417:
2413:
2404:
2400:
2388:
2385:
2381:
2378:
2374:
2370:
2367:
2356:
2353:
2349:
2345:
2342:
2331:
2324:
2321:
2315:
2312:
2298:
2290:
2286:
2274:
2271:
2268:
2261:
2250:
2247:
2244:
2241:
2238:
2229:
2226:
2223:
2207:
2206:
2205:
2184:
2175:
2172:
2168:
2165:
2156:
2153:
2139:
2131:
2127:
2118:
2114:
2102:
2099:
2095:
2092:
2088:
2084:
2081:
2070:
2067:
2063:
2059:
2056:
2044:
2042:
2030:
2027:
2021:
2018:
2004:
1996:
1992:
1980:
1977:
1974:
1967:
1955:
1952:
1949:
1946:
1943:
1934:
1931:
1928:
1913:
1906:
1903:
1897:
1894:
1877:
1869:
1865:
1861:
1856:
1852:
1840:
1836:
1822:
1821:
1820:
1814:
1790:
1783:
1780:
1774:
1771:
1757:
1749:
1746:
1743:
1739:
1730:
1726:
1714:
1711:
1708:
1705:
1702:
1693:
1690:
1687:
1676:
1669:
1666:
1660:
1657:
1640:
1632:
1628:
1624:
1619:
1615:
1603:
1599:
1594:
1582:
1581:
1580:
1563:
1555:
1547:
1544:
1541:
1537:
1528:
1525:
1522:
1519:
1516:
1507:
1504:
1501:
1490:
1479:
1471:
1467:
1463:
1458:
1454:
1443:
1442:
1441:
1439:
1431:
1429:
1427:
1426:Schur's lemma
1423:
1419:
1398:
1394:
1390:
1387:
1384:
1374:
1370:
1359:
1355:
1351:
1348:
1345:
1333:
1330:
1326:
1322:
1318:
1314:
1293:
1289:
1285:
1282:
1279:
1269:
1265:
1254:
1250:
1246:
1243:
1240:
1228:
1226:
1222:
1218:
1214:
1210:
1206:
1198:
1196:
1194:
1190:
1185:
1182:
1161:
1157:
1153:
1150:
1147:
1120:
1116:
1088:
1084:
1080:
1077:
1074:
1063:
1047:
1044:
1036:
1032:
1028:
1025:
1022:
1012:
1008:
997:
993:
989:
986:
983:
971:
969:
965:
961:
952:
950:
948:
943:
939:
934:
932:
925:
918:
914:
910:
903:
896:
892:
888:
884:
880:
875:
871:
850:
846:
842:
839:
836:
826:
822:
811:
807:
803:
800:
797:
786:
782:
774:
769:
767:
765:
760:
755:
746:
741:
734:
728:
707:
704:
692:
685:
678:
668:
664:
658:
651:
647:for coupling
646:
627:
624:
616:
613:
609:
606:
601:
598:
587:
580:
575:
570:
550:
547:
535:
531:
509:
502:
492:
470:
462:
458:
450:
449:
448:
431:
424:
421:
409:
402:
395:
386:
382:
374:
370:
365:
362:
356:
353:
346:
339:
336:
330:
327:
313:
305:
301:
292:
288:
278:
277:
276:
262:
241:
238:
217:
193:
190:
178:
171:
164:
140:
137:
116:
91:
84:
74:
72:
68:
67:Eugene Wigner
64:
60:
57:
53:
49:
45:
41:
37:
33:
19:
4932:
4889:
4871:
4844:
4838:
4813:
4807:
4801:
4789:
4780:
4771:
4764:
4757:
4745:
4713:
4709:
4700:
4693:
4689:
4683:
4676:
4670:
4666:
4660:
4656:
4650:
4646:
4639:
4632:
4629:
4231:
4149:
4142:
4136:
4125:
4121:
4118:
4036:
3941:
3934:
3930:
3924:
3918:
3912:
3909:
3894:
3890:
3886:
3882:
3676:
3674:
3662:
3658:
3650:
3474:
3463:
3105:
2945:
2942:
2936:
2932:
2928:
2924:
2916:
2909:
2902:
2896:
2890:
2880:
2873:
2870:
2790:
2781:
2772:
2769:
2636:
2632:
2628:
2624:
2620:
2616:
2607:
2203:
1812:
1809:
1578:
1435:
1428:to be used.
1334:
1320:
1316:
1229:
1217:vector space
1208:
1202:
1186:
1180:
1061:
972:
967:
959:
956:
941:
937:
935:
930:
923:
916:
901:
894:
886:
882:
878:
873:
869:
778:
763:
758:
753:
750:
744:
739:
732:
726:
662:
656:
649:
578:
573:
568:
529:
490:
446:
75:
50:elements of
31:
29:
4796:Appendix C.
4232:Therefore,
3944:∈ {−1, 0, 1
71:Carl Eckart
59:eigenstates
4957:Categories
4738:References
3666:convention
3651:where the
3470:3-j symbol
3464:where the
1440:, we have
1418:projection
1329:direct sum
576:component
4934:MathWorld
4794:Hall 2015
4601:⟩
4561:⟨
4558:−
4555:⟩
4534:−
4506:⟨
4495:⟩
4475:‖
4456:‖
4446:⟨
4357:−
4338:−
4300:⟩
4247:⟨
4200:±
4191:∓
4169:±
4079:π
3990:−
3971:−
3838:⟩
3827:‖
3808:‖
3802:⟨
3799:⟩
3759:⟨
3750:⟩
3692:⟨
3631:∗
3620:⟩
3613:‖
3594:‖
3583:⟨
3575:−
3550:−
3533:⟩
3521:‖
3507:†
3499:‖
3493:⟨
3438:⟩
3426:‖
3407:‖
3401:⟨
3372:−
3334:−
3320:−
3286:⟩
3274:‖
3255:‖
3249:⟨
3246:⟩
3206:⟨
3187:−
3175:⟩
3117:⟨
3083:⟩
3052:±
3031:⟨
3028:∝
3025:⟩
2994:±
2961:⟨
2709:∑
2658:∑
2586:⟩
2545:±
2522:⟨
2504:±
2489:∓
2468:⟩
2459:±
2398:⟨
2375:±
2350:∓
2329:⟩
2272:∓
2259:⟨
2242:∓
2227:±
2182:⟩
2173:±
2112:⟨
2089:±
2064:∓
2048:ℏ
2045:−
2035:⟩
1978:∓
1965:⟨
1947:∓
1932:±
1921:ℏ
1911:⟩
1857:±
1834:⟨
1788:⟩
1747:±
1724:⟨
1706:±
1691:∓
1680:ℏ
1674:⟩
1620:±
1597:⟨
1545:±
1520:±
1505:∓
1494:ℏ
1459:±
1404:⟩
1343:⟨
1299:⟩
1238:⟨
1167:⟩
1072:⟨
1042:⟩
981:⟨
960:different
856:⟩
795:⟨
712:⟩
701:‖
682:‖
676:⟨
631:⟩
595:⟨
554:⟩
429:⟩
418:‖
399:‖
393:⟨
390:⟩
350:⟨
344:⟩
286:⟨
198:⟩
187:‖
168:‖
162:⟨
4721:See also
4698:for the
4577:′
4568:′
4522:′
4513:′
4491:′
4482:′
4420:⟩
4415:′
4406:′
4397:′
4311:⟨
4296:′
4287:′
4278:′
3850:′
3834:′
3775:′
3766:′
3746:′
3737:′
3590:′
3571:′
3528:′
3433:′
3389:′
3364:′
3281:′
3222:′
3213:′
3171:′
3162:′
3079:′
3070:′
2977:′
2968:′
2871:or that
2582:′
2573:′
2455:′
2443:′
2382:′
2371:′
2357:′
2346:′
2325:′
2316:′
2169:′
2157:′
2096:′
2085:′
2071:′
2060:′
2031:′
2022:′
1907:′
1898:′
1784:′
1775:′
1670:′
1661:′
913:subshell
907:are the
868:, where
708:′
610:′
602:′
572:and its
527:of rank
425:′
366:′
357:′
340:′
331:′
242:′
194:′
141:′
4882:General
4818:Bibcode
4636:in the
3879:Example
785:orbital
660:to get
643:is the
488:is the
36:theorem
4916:
4896:
4859:
4692:± 1 =
4664:, and
4153:, and
3905:parity
893:, and
737:, nor
447:where
255:, and
48:matrix
4640:n j m
3939:with
3901:basis
3895:n j m
3887:n j m
3466:2 × 3
2915:± 1)|
1432:Proof
885:, or
654:with
34:is a
4914:ISBN
4894:ISBN
4857:ISBN
4712:′ =
4681:and
4669:′ =
4659:′ =
4649:′ =
4130:are
4119:and
3670:(−1)
3480:(−1)
129:and
69:and
42:and
30:The
4849:doi
4826:doi
4687:is
2939:± 1
2917:j m
2640:= 0
1323:an
1321:not
1195:.)
942:any
938:one
38:of
4959::
4931:.
4855:.
4824:.
4814:62
4812:.
4704:±1
4654:,
4148:=
3681::
3661:−
2931:′|
2927:′
2879:∝
2751:0.
2627:,
900:,
881:,
730:,
230:,
4937:.
4920:.
4865:.
4851::
4832:.
4828::
4820::
4772:q
4767:)
4765:k
4763:(
4714:m
4710:m
4701:T
4696:′
4694:m
4690:m
4684:m
4679:′
4677:m
4671:m
4667:m
4661:j
4657:j
4651:n
4647:n
4642:⟩
4638:|
4633:x
4611:.
4606:)
4598:m
4594:j
4590:|
4586:1
4582:1
4574:m
4565:j
4552:m
4548:j
4544:|
4540:)
4537:1
4531:(
4527:1
4519:m
4510:j
4501:(
4488:j
4479:n
4470:)
4467:1
4464:(
4460:T
4453:j
4449:n
4440:2
4436:1
4431:=
4412:m
4403:j
4394:n
4389:|
4383:2
4376:)
4373:1
4370:(
4365:1
4361:T
4352:)
4349:1
4346:(
4341:1
4334:T
4327:|
4323:m
4319:j
4315:n
4307:=
4293:m
4284:j
4275:n
4270:|
4266:x
4262:|
4258:m
4254:j
4250:n
4217:.
4211:2
4206:y
4203:i
4197:x
4188:=
4183:)
4180:1
4177:(
4172:1
4165:T
4150:z
4146:0
4143:T
4137:l
4126:l
4122:Y
4102:q
4097:1
4093:Y
4089:r
4083:3
4076:4
4069:=
4064:)
4061:1
4058:(
4053:q
4049:T
4022:,
4016:2
4009:)
4006:1
4003:(
3998:1
3994:T
3985:)
3982:1
3979:(
3974:1
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3960:=
3957:x
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3935:q
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3925:x
3919:r
3913:x
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3893:|
3891:x
3889:|
3885:⟨
3863:.
3857:1
3854:+
3847:j
3843:2
3831:j
3822:)
3819:k
3816:(
3812:T
3805:j
3796:m
3792:j
3788:|
3784:q
3780:k
3772:m
3763:j
3753:=
3743:m
3734:j
3729:|
3723:)
3720:k
3717:(
3712:q
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3703:|
3699:m
3695:j
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3605:k
3602:(
3598:T
3587:j
3578:j
3568:j
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3553:1
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3544:=
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3510:(
3503:T
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3430:j
3421:)
3418:k
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3411:T
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3375:m
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3337:m
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3317:(
3314:=
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3266:k
3263:(
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3252:j
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3239:j
3235:|
3231:q
3227:k
3219:m
3210:j
3201:k
3198:2
3194:)
3190:1
3184:(
3178:=
3168:m
3159:j
3154:|
3148:)
3145:k
3142:(
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3120:j
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2906:2
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2900:1
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2894:1
2891:j
2889:⟨
2883:c
2881:y
2876:c
2874:x
2852:d
2848:y
2842:c
2838:y
2832:=
2825:d
2821:x
2815:c
2811:x
2793:c
2791:x
2784:c
2782:y
2775:c
2773:x
2748:=
2739:c
2735:y
2729:c
2726:,
2723:b
2719:a
2713:c
2703:,
2700:0
2697:=
2688:c
2684:x
2678:c
2675:,
2672:b
2668:a
2662:c
2637:c
2633:x
2629:c
2625:b
2621:a
2617:c
2614:Σ
2589:.
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2516:)
2513:1
2510:+
2507:q
2501:k
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2495:)
2492:q
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2478:+
2465:)
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2440:j
2435:|
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2423:(
2418:q
2414:T
2409:|
2405:m
2401:j
2392:)
2389:1
2386:+
2379:m
2368:j
2364:(
2361:)
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2332:=
2322:m
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2308:|
2302:)
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2296:(
2291:q
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2278:)
2275:1
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2251:1
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2233:)
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2221:(
2185:.
2179:)
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2162:(
2154:j
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2115:j
2106:)
2103:1
2100:+
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2078:(
2075:)
2068:m
2057:j
2053:(
2028:m
2019:j
2014:|
2008:)
2005:k
2002:(
1997:q
1993:T
1988:|
1984:)
1981:1
1975:m
1972:(
1968:j
1959:)
1956:1
1953:+
1950:m
1944:j
1941:(
1938:)
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1926:(
1914:=
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1886:]
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1677:=
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1511:)
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1488:]
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1451:[
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1391:,
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1371:r
1366:|
1360:1
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1352:,
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1346:2
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1294:2
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1121:i
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1004:|
998:1
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990:,
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920:1
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902:m
898:1
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574:z
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513:)
510:k
507:(
503:T
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432:,
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413:)
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407:(
403:T
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379:|
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347:=
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323:|
317:)
314:k
311:(
306:q
302:T
297:|
293:m
289:j
263:q
239:m
218:m
191:j
182:)
179:k
176:(
172:T
165:j
138:j
117:j
95:)
92:k
89:(
85:T
20:)
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