Spinor spherical harmonics
{{Short description|Special functions on a sphere}}
{{Distinguish|Spin-weighted spherical harmonics}}
In quantum mechanics, the spinor spherical harmonics (also known as spin spherical harmonics,{{Citation | last1=Edmonds | first1=A. R. | title=Angular Momentum in Quantum Mechanics | publisher=Princeton University Press | isbn=978-0-691-07912-7 | year=1957 | url-access=registration | url=https://archive.org/details/angularmomentumi0000edmo }}
spinor harmonics and Pauli spinors{{Cite book|last=Rose|first=M. E.|url=https://books.google.com/books?id=1c7lngEACAAJ&q=angular+momentum+rose|title=Elementary Theory of Angular Momentum|date=2013-12-20|publisher=Dover Publications, Incorporated|isbn=978-0-486-78879-1|language=en}}) are special functions defined over the sphere. The spinor spherical harmonics are the natural spinor analog of the vector spherical harmonics. While the standard spherical harmonics are a basis for the angular momentum operator, the spinor spherical harmonics are a basis for the total angular momentum operator (angular momentum plus spin). These functions are used in analytical solutions to Dirac equation in a radial potential. The spinor spherical harmonics are sometimes called Pauli central field spinors, in honor of Wolfgang Pauli who employed them in the solution of the hydrogen atom with spin–orbit interaction.
Properties
The spinor spherical harmonics {{math|Yl, s, j, m}} are the spinors eigenstates of the total angular momentum operator squared:
:
\begin{align}
\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} &= m Y_{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{align}
where {{math|j {{=}} l + s}}, where {{math|j}}, {{math|l}}, and {{math|s}} are the (dimensionless) total, orbital and spin angular momentum operators, j is the total azimuthal quantum number and m is the total magnetic quantum number.
Under a parity operation, we have
:
P Y_{l, s j, m}
= (-1)^{l}Y_{l,s, j, m}.
For spin-1/2 systems, they are given in matrix form by{{Citation | last1=Biedenharn | first1=L. C. | author-link=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}}{{Cite book|last=Greiner|first=Walter|url=https://books.google.com/books?id=a6_rCAAAQBAJ|title=Relativistic Quantum Mechanics: Wave Equations|publisher=Springer|isbn=978-3-642-88082-7|language=en|chapter=9.3 Separation of the Variables for the Dirac Equation with Central Potential (minimally coupled)|date=6 December 2012|author-link=Walter Greiner}}{{Cite book |last=Berestetskii |first=V. B. |title=Quantum electrodynamics |date=2008 |publisher=Butterworth-Heinemann |author2=E. M. Lifshitz |author3=L. P. Pitaevskii |translator=J. B. Sykes |translator2=J. S. Bell |isbn=978-0-08-050346-2 |edition=2nd |location=Oxford |oclc=785780331}}
:
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}.
where are the usual spherical harmonics.
References
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