Spinor spherical harmonics Special functions on a sphere
In quantum mechanics , the spinor spherical harmonics [ 1] spin spherical harmonics ,[ 2] spinor harmonics[ 3] and Pauli spinors [ 4] 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 .[ 3] Pauli central field spinors , in honor of Wolfgang Pauli who employed them in the solution of the hydrogen atom with spin–orbit interaction .[ 1]
Properties
The spinor spherical harmonics Y l, s, j, m 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 j l s 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 by[ 1] [ 3] [ 5]
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}}
spherical harmonics .
References
^ a b c 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 ^ Edmonds, A. R. (1957), Angular Momentum in Quantum Mechanics Princeton University Press , ISBN 978-0-691-07912-7 ^ a b c 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 ISBN 978-3-642-88082-7 ^ Rose, M. E. (2013-12-20). Elementary Theory of Angular Momentum ISBN 978-0-486-78879-1 ^ 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 .
