Spinor spherical harmonics: Difference between revisions

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In quantum mechanics, spinor spherical harmonics $Y l, s, j, m$ are spinors eigenstates of the total angular momentum operator squared:

{\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}\end{aligned}}

where $j = l + s$ . They are the natural spinorial analog of vector spherical harmonics.

For spin-½ systems, they are given in matrix form by^[1]^[2]

Y_{j\pm {\frac {1}{2}},{\frac {1}{2}},j,m}={\frac {1}{\sqrt {2{\bigl (}j\pm {\frac {1}{2}}{\bigr )}+1}}}{\begin{pmatrix}\mp {\sqrt {j\pm {\frac {1}{2}}\mp m+{\frac {1}{2}}}}Y_{j\pm {\frac {1}{2}}}^{m-{\frac {1}{2}}}\\{\sqrt {j\pm {\frac {1}{2}}\pm m+{\frac {1}{2}}}}Y_{j\pm {\frac {1}{2}}}^{m+{\frac {1}{2}}}\end{pmatrix}}.

Spinor spherical harmonics are used in analytical solutions to Dirac equation in a radial potential.^[2]

The spin spherical harmonics are sometimes called Pauli central field spinors, in honor to Wolfgang Pauli who employed them in the solution of the hydrogen atom with spin–orbit interaction.^[1]

References

^ ^a ^b 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 ^b Greiner, Walter. "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.

@@ Line 9: / Line 9: @@
 where {{math|'''j''' {{=}} '''l''' + '''s'''}}.  They are the natural spinorial analog of [[vector spherical harmonics]].
-For [[spin-½]] systems, they are given in matrix form by<ref name=":0">{{Citation | last1=Biedenharn | first1=L. C. | authorlink=Lawrence Biedenharn | last2=Louck | first2=J. D. | title=Angular momentum in Quantum Physics: Theory and Application | publisher=[[Addison-Wesley]] | place=Reading | volume=8 | series=Encyclopedia of Mathematics | isbn=0-201-13507-8 | year=1981 | page=283}}</ref>
+For [[spin-½]] systems, they are given in matrix form by<ref name=":0">{{Citation | last1=Biedenharn | first1=L. C. | authorlink=Lawrence Biedenharn | last2=Louck | first2=J. D. | title=Angular momentum in Quantum Physics: Theory and Application | publisher=[[Addison-Wesley]] | place=Reading | volume=8 | series=Encyclopedia of Mathematics | isbn=0-201-13507-8 | year=1981 | page=283}}</ref><ref name=":1">{{Cite book|last=Greiner|first=Walter|url=https://books.google.fr/books/about/Relativistic_Quantum_Mechanics.html?id=a6_rCAAAQBAJ&source=kp_book_description&redir_esc=y|title=Relativistic Quantum Mechanics: Wave Equations|publisher=Springer|year=|isbn=978-3-642-88082-7|location=|pages=|language=en|chapter=9.3 Separation of the Variables for the Dirac Equation with Central Potential (minimally coupled)|author-link=Walter Greiner}}</ref>
 :<math>
   Y_{j \pm \frac{1}{2}, \frac{1}{2}, j, m}
@@ Line 19: / Line 19: @@
 </math>
-Spinor spherical harmonics are used in analytical solutions to [[Dirac equation]] in a [[Particle in a spherically symmetric potential|radial potential]].
+Spinor spherical harmonics are used in analytical solutions to [[Dirac equation]] in a [[Particle in a spherically symmetric potential|radial potential]].<ref name=":1" />
 The spin spherical harmonics are sometimes called '''Pauli central field spinors''', in honor to [[Wolfgang Pauli]] who employed them in the solution of the [[hydrogen atom]] with [[spin–orbit interaction]].<ref name=":0" />

Revision as of 10:21, 13 October 2020

References

Further reading