In quantum mechanics, the spinor spherical harmonics (also known as spin spherical harmonics, spinor harmonics and Pauli spinors) 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 Yl, s, j, m are the spinors eigenstates of the total angular momentum operator squared:
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}}}where j = l + s, where j, l, and 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 . {\displaystyle PY_{l,sj,m}=(-1)^{l}Y_{l,s,j,m}.}For spin-1/2 systems, they are given in matrix form by789
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}}.}where Y l m {\displaystyle Y_{l}^{m}} are the usual spherical harmonics.
References
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 0-201-13507-8 ↩
Edmonds, A. R. (1957), Angular Momentum in Quantum Mechanics, Princeton University Press, ISBN 978-0-691-07912-7 {{citation}}: ISBN / Date incompatibility (help) 978-0-691-07912-7 ↩
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. 978-3-642-88082-7 ↩
Rose, M. E. (2013-12-20). Elementary Theory of Angular Momentum. Dover Publications, Incorporated. ISBN 978-0-486-78879-1. 978-0-486-78879-1 ↩
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. 978-3-642-88082-7 ↩
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 0-201-13507-8 ↩
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 0-201-13507-8 ↩
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. 978-3-642-88082-7 ↩
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. 978-0-08-050346-2 ↩