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Spinor spherical harmonics
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<h2 id="properties">Properties</h2> <p>The spinor spherical harmonics <i>Y</i><i>l, s, j, m</i> are the <a href="/facts/Spinor/sKFNnEwP">spinors</a> <a href="/facts/Eigenstate/7jbxd7Dx">eigenstates</a> of the total <a href="/facts/Angular_momentum_operator/d55sXn5u">angular momentum operator</a> squared: </p> j 2 Y l , s , j , m = j ( j + 1 ) Y l , s , j , m j z Y l , s , j , m = m Y l , s , j , m ; m = − j , − ( j − 1 ) , ⋯ , j − 1 , j l 2 Y l , s , j , m = l ( l + 1 ) Y l , s , j , m s 2 Y l , s , j , m = s ( s + 1 ) Y l , s , j , m {\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}}} <p>where j = l + s, where j, l, and s are the (dimensionless) total, orbital and spin angular momentum operators, <i>j</i> is the total <a href="/facts/Azimuthal_quantum_number/vHSIzbFC">azimuthal quantum number</a> and <i>m</i> is the total <a href="/facts/Magnetic_quantum_number/LvFY0dVV">magnetic quantum number</a>. </p><p>Under a <a href="/facts/Parity_(physics)/emSbJ4hD">parity</a> operation, we have </p> P Y l , s j , m = ( − 1 ) l Y l , s , j , m . {\displaystyle PY_{l,sj,m}=(-1)^{l}Y_{l,s,j,m}.} <p>For <a href="/facts/Spin-1%2f2/3I929M4X">spin-1/2</a> systems, they are given in matrix form by<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a><a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a><a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a> </p> Y l , ± 1 2 , j , m = 1 2 ( j ∓ 1 2 ) + 1 ( ± j ∓ 1 2 ± m + 1 2 Y l m − 1 2 j ∓ 1 2 ∓ m + 1 2 Y l m + 1 2 ) . {\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}}.} <p>where Y l m {\displaystyle Y_{l}^{m}} are the usual <a href="/facts/Spherical_harmonics/bn98Gv2L">spherical harmonics</a>. </p> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Biedenharn, L. C.; Louck, J. D. (1981), Angular momentum in Quantum Physics: Theory and Application, Encyclopedia of Mathematics, vol. 8, Reading: Addison-Wesley, p. 283, ISBN 0-201-13507-8 <a href="0-201-13507-8" target="_blank">0-201-13507-8</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Edmonds, A. R. (1957), Angular Momentum in Quantum Mechanics, Princeton University Press, ISBN 978-0-691-07912-7 {{citation}}: ISBN / Date incompatibility (help) <a href="978-0-691-07912-7" target="_blank">978-0-691-07912-7</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Greiner, Walter (6 December 2012). "9.3 Separation of the Variables for the Dirac Equation with Central Potential (minimally coupled)". Relativistic Quantum Mechanics: Wave Equations. Springer. ISBN 978-3-642-88082-7. <a href="978-3-642-88082-7" target="_blank">978-3-642-88082-7</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Rose, M. E. (2013-12-20). Elementary Theory of Angular Momentum. Dover Publications, Incorporated. ISBN 978-0-486-78879-1. <a href="978-0-486-78879-1" target="_blank">978-0-486-78879-1</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Greiner, Walter (6 December 2012). "9.3 Separation of the Variables for the Dirac Equation with Central Potential (minimally coupled)". Relativistic Quantum Mechanics: Wave Equations. Springer. ISBN 978-3-642-88082-7. <a href="978-3-642-88082-7" target="_blank">978-3-642-88082-7</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>Biedenharn, L. C.; Louck, J. D. (1981), Angular momentum in Quantum Physics: Theory and Application, Encyclopedia of Mathematics, vol. 8, Reading: Addison-Wesley, p. 283, ISBN 0-201-13507-8 <a href="0-201-13507-8" target="_blank">0-201-13507-8</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>Biedenharn, L. C.; Louck, J. D. (1981), Angular momentum in Quantum Physics: Theory and Application, Encyclopedia of Mathematics, vol. 8, Reading: Addison-Wesley, p. 283, ISBN 0-201-13507-8 <a href="0-201-13507-8" target="_blank">0-201-13507-8</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> <li id="fn:8"><p>Greiner, Walter (6 December 2012). "9.3 Separation of the Variables for the Dirac Equation with Central Potential (minimally coupled)". Relativistic Quantum Mechanics: Wave Equations. Springer. ISBN 978-3-642-88082-7. <a href="978-3-642-88082-7" target="_blank">978-3-642-88082-7</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li> <li id="fn:9"><p>Berestetskii, V. B.; E. M. Lifshitz; L. P. Pitaevskii (2008). Quantum electrodynamics. Translated by J. B. Sykes; J. S. Bell (2nd ed.). Oxford: Butterworth-Heinemann. ISBN 978-0-08-050346-2. OCLC 785780331. <a href="978-0-08-050346-2" target="_blank">978-0-08-050346-2</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li> </ol>