1294:
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1289:{\displaystyle {\begin{aligned}\left|{\frac {1}{2}},+{\frac {1}{2}}\right\rangle \ \otimes \left|{\frac {1}{2}},+{\frac {1}{2}}\right\rangle \ &{\text{ by }}(\uparrow \uparrow ),\\\left|{\frac {1}{2}},+{\frac {1}{2}}\right\rangle \ \otimes \left|{\frac {1}{2}},-{\frac {1}{2}}\right\rangle \ &{\text{ by }}(\uparrow \downarrow ),\\\left|{\frac {1}{2}},-{\frac {1}{2}}\right\rangle \ \otimes \left|{\frac {1}{2}},+{\frac {1}{2}}\right\rangle \ &{\text{ by }}(\downarrow \uparrow ),\\\left|{\frac {1}{2}},-{\frac {1}{2}}\right\rangle \ \otimes \left|{\frac {1}{2}},-{\frac {1}{2}}\right\rangle \ &{\text{ by }}(\downarrow \downarrow )\end{aligned}}}
36:
93:
1568:
1388:
1713:
to produce a 4-dimensional representation. The 4-dimensional representation descends to the usual orthogonal group SO(3) and so its objects are tensors, corresponding to the integrality of their spin. The 4-dimensional representation decomposes into the sum of a one-dimensional trivial
1688:
1563:{\displaystyle \left.{\begin{array}{ll}|1,1\rangle &=\;\uparrow \uparrow \\|1,0\rangle &=\;{\frac {1}{\sqrt {2}}}(\uparrow \downarrow +\downarrow \uparrow )\\|1,-1\rangle &=\;\downarrow \downarrow \end{array}}\right\}\quad s=1\quad \mathrm {(triplet)} }
854:
219:
In a system with two spin-1/2 particles – for example the proton and electron in the ground state of hydrogen – measured on a given axis, each particle can be either spin up or spin down so the system has four basis states in all
419:
259:
148:, in the context of quantum mechanics, is not a mechanical rotation but a more abstract concept that characterizes a particle's intrinsic angular momentum. It is particularly important for systems at atomic length scales, such as individual
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656:
1579:
264:
using the single particle spins to label the basis states, where the first arrow and second arrow in each combination indicate the spin direction of the first particle and second particle respectively.
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573:
675:
1693:
which is antisymmetric. The result is that a combination of two spin-1/2 particles can carry a total spin of 1 or 0, depending on whether they occupy a triplet or singlet state.
274:
226:
1743:
530:
503:
476:
449:
1302:
578:
1705:, what has happened is that the two conjugate 2-dimensional spin representations of the spin group SU(2) = Spin(3) (as it sits inside the 3-dimensional
1683:{\displaystyle \left.|0,0\rangle ={\frac {1}{\sqrt {2}}}(\uparrow \downarrow -\downarrow \uparrow )\;\right\}\quad s=0\quad \mathrm {(singlet)} }
1718:, spin zero) and a three-dimensional representation (triplet, spin 1) that is nothing more than the standard representation of SO(3) on
661:
Now the total spin and its projection onto the previously defined axis can be computed using the rules for adding angular momentum in
1939:
1920:
199:
into a singlet state. This makes it kinetically nonreactive despite being thermodynamically one of the strongest oxidants.
535:
1873:
849:{\displaystyle |s,m\rangle =\sum _{m_{1}+m_{2}=m}C_{m_{1}m_{2}m}^{s_{1}s_{2}s}|s_{1}m_{1}\rangle |s_{2}m_{2}\rangle }
79:
57:
50:
183:. Most molecules encountered in daily life exist in a singlet state because all of their electrons are paired, but
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1963:
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17:
414:{\displaystyle |s_{1},m_{1}\rangle |s_{2},m_{2}\rangle =|s_{1},m_{1}\rangle \otimes |s_{2},m_{2}\rangle ,}
254:{\displaystyle \uparrow \uparrow ,\uparrow \downarrow ,\downarrow \uparrow ,\downarrow \downarrow }
44:
1958:
1745:. Thus the "three" in triplet can be identified with the three rotation axes of physical space.
61:
1968:
1702:
1375:{\textstyle \left|{\frac {1}{2}},m_{1}\right\rangle \left|{\frac {1}{2}},m_{2}\right\rangle }
651:{\textstyle \left|{\frac {1}{2}},m_{1}\right\rangle \left|{\frac {1}{2}},m_{2}\right\rangle }
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8:
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returns the possible values for total spin given along with their representation in the
175:= 1, in contrast to the more common case of two electrons aligning oppositely to give
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exists in a triplet state, which can only undergo a chemical reaction by making the
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211:, which makes it kinetically as well as thermodynamically a very strong oxidant.
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135:= 1. It has three allowed values of the spin's projection along a given axis
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1821:
Borden, Weston
Thatcher; Hoffmann, Roald; Stuyver, Thijs; Chen, Bo (2017).
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which are symmetric and a fourth state with total spin angular momentum 0:
180:
532:
are their projections onto the z axis. Since for spin-1/2 particles, the
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1839:
1822:
1795:
1823:"Dioxygen: What Makes This Triplet Diradical Kinetically Persistent?"
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basis. There are three states with total spin angular momentum 1:
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of an object such as an electron, atom, or molecule, having a
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A triplet state occurs in cases where the spins of two
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568:{\textstyle \left|{\frac {1}{2}},m\right\rangle }
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575:basis states span a 2-dimensional space, the
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142:= −1, 0, or +1, giving the name "triplet".
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658:basis states span a 4-dimensional space.
214:
80:Learn how and when to remove this message
1861:
1827:Journal of the American Chemical Society
478:are the spins of the two particles, and
91:
43:This article includes a list of general
1930:Shankar, R. (1994). "chapter 14-Spin".
14:
1951:
1868:. New York: McGraw-Hill. p. 149.
1865:A modern approach to quantum mechanics
859:substituting in the four basis states
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49:it lacks sufficient corresponding
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1909:Introduction to Quantum Mechanics
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1932:Principles of Quantum Mechanics
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1907:Griffiths, David J. (2004).
7:
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1714:representation (singlet, a
667:Clebsch–Gordan coefficients
10:
1985:
1934:(2nd ed.). Springer.
1862:Townsend, John S. (1992).
27:Quantum state of a system
1697:A mathematical viewpoint
64:more precise citations.
1897:Spin and Spin–Addition
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1738:{\displaystyle R^{3}}
1703:representation theory
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525:{\displaystyle m_{2}}
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498:{\displaystyle m_{1}}
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171:= 1/2, align to give
96:Examples of atoms in
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1840:10.1021/jacs.7b04232
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1786:Spin quantum number
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167:, each having spin
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1629:↓ ↑
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1510:↓ ↓
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165:unpaired electrons
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1941:978-0-306-44790-7
1922:978-0-13-111892-8
1833:(26): 9010–9018.
1781:Spin multiplicity
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1911:(2nd ed.).
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189:room temperature
185:molecular oxygen
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891:
888:
883:
880:
874:
862:
861:
860:
838:
834:
828:
824:
810:
806:
800:
796:
785:
780:
776:
770:
766:
760:
755:
751:
745:
741:
736:
730:
727:
722:
718:
714:
709:
705:
700:
696:
690:
687:
684:
672:
671:
670:
669:. In general
668:
664:
659:
644:
638:
634:
630:
625:
622:
616:
611:
605:
601:
597:
592:
589:
583:
561:
557:
554:
549:
546:
540:
517:
513:
490:
486:
463:
459:
436:
432:
408:
400:
396:
392:
387:
383:
374:
366:
362:
358:
353:
349:
340:
332:
328:
324:
319:
315:
301:
297:
293:
288:
284:
271:
270:
269:
266:
245:
239:
233:
223:
222:
221:
212:
210:
209:singlet state
206:
202:
201:Photochemical
198:
190:
186:
182:
178:
174:
170:
166:
161:
159:
155:
151:
147:
143:
138:
134:
131:
127:
126:quantum state
123:
119:
118:triplet state
115:
107:
103:
99:
94:
84:
81:
73:
70:December 2010
63:
59:
53:
52:
46:
41:
32:
31:
19:
1969:Spectroscopy
1931:
1908:
1892:
1864:
1857:
1830:
1826:
1816:
1701:In terms of
1700:
1692:
1572:
1298:
858:
660:
423:
267:
263:
218:
181:spin singlet
176:
172:
168:
162:
144:
136:
132:
130:quantum spin
122:spin triplet
121:
117:
111:
105:
76:
67:
48:
18:Spin triplet
1796:Spin tensor
203:or thermal
62:introducing
1953:Categories
1808:References
665:using the
205:activation
45:references
1766:Diradical
1626:−
1602:⟩
1501:⟩
1495:−
1444:⟩
1414:⟩
1246:−
1225:⊗
1204:−
1121:⊗
1100:−
1038:−
1017:⊗
913:⊗
844:⟩
816:⟩
701:∑
694:⟩
406:⟩
375:⊗
372:⟩
338:⟩
307:⟩
158:electrons
124:, is the
1884:23650343
1849:28613073
1791:Spin-1/2
1749:See also
1711:tensored
1369:⟩
1336:⟩
1260:⟩
1218:⟩
1156:⟩
1114:⟩
1052:⟩
1010:⟩
948:⟩
906:⟩
645:⟩
612:⟩
562:⟩
1709:) have
179:= 0, a
154:protons
108:states.
106:triplet
102:doublet
98:singlet
58:improve
1938:
1919:
1882:
1872:
1847:
1801:Spinor
1716:scalar
1264:
1222:
1160:
1118:
1056:
1014:
952:
910:
424:where
104:, and
47:, but
156:, or
150:atoms
120:, or
1936:ISBN
1917:ISBN
1880:OCLC
1870:ISBN
1845:PMID
505:and
451:and
146:Spin
116:, a
1835:doi
1831:139
191:, O
112:In
1955::
1915:.
1878:.
1843:.
1829:.
1825:.
160:.
152:,
100:,
1944:.
1925:.
1886:.
1851:.
1837::
1731:3
1727:R
1677:)
1674:t
1671:e
1668:l
1665:g
1662:n
1659:i
1656:s
1653:(
1648:0
1645:=
1642:s
1637:}
1632:)
1620:(
1614:2
1610:1
1605:=
1599:0
1596:,
1593:0
1589:|
1557:)
1554:t
1551:e
1548:l
1545:p
1542:i
1539:r
1536:t
1533:(
1528:1
1525:=
1522:s
1517:}
1506:=
1498:1
1492:,
1489:1
1485:|
1477:)
1471:+
1465:(
1459:2
1455:1
1449:=
1441:0
1438:,
1435:1
1431:|
1419:=
1411:1
1408:,
1405:1
1401:|
1363:2
1359:m
1355:,
1350:2
1347:1
1341:|
1330:1
1326:m
1322:,
1317:2
1314:1
1308:|
1280:)
1274:(
1254:2
1251:1
1243:,
1238:2
1235:1
1229:|
1212:2
1209:1
1201:,
1196:2
1193:1
1187:|
1179:,
1176:)
1170:(
1150:2
1147:1
1142:+
1139:,
1134:2
1131:1
1125:|
1108:2
1105:1
1097:,
1092:2
1089:1
1083:|
1075:,
1072:)
1066:(
1046:2
1043:1
1035:,
1030:2
1027:1
1021:|
1004:2
1001:1
996:+
993:,
988:2
985:1
979:|
971:,
968:)
962:(
942:2
939:1
934:+
931:,
926:2
923:1
917:|
900:2
897:1
892:+
889:,
884:2
881:1
875:|
839:2
835:m
829:2
825:s
820:|
811:1
807:m
801:1
797:s
792:|
786:s
781:2
777:s
771:1
767:s
761:m
756:2
752:m
746:1
742:m
737:C
731:m
728:=
723:2
719:m
715:+
710:1
706:m
697:=
691:m
688:,
685:s
681:|
639:2
635:m
631:,
626:2
623:1
617:|
606:1
602:m
598:,
593:2
590:1
584:|
558:m
555:,
550:2
547:1
541:|
518:2
514:m
491:1
487:m
464:2
460:s
437:1
433:s
409:,
401:2
397:m
393:,
388:2
384:s
379:|
367:1
363:m
359:,
354:1
350:s
345:|
341:=
333:2
329:m
325:,
320:2
316:s
311:|
302:1
298:m
294:,
289:1
285:s
280:|
246:,
240:,
234:,
193:2
177:S
173:S
169:s
140:S
137:m
133:S
83:)
77:(
72:)
68:(
54:.
20:)
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