Spinor spherical harmonics
In quantum mechanics, the spinor spherical harmonics[1] (also known as spin spherical harmonics,[2] spinor harmonics[3] and Pauli spinors[4]) 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.[3] The spinor 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]
Properties
The spinor spherical harmonics Yl, s, j, m are the spinors eigenstates of the total angular momentum operator squared:
- [math]\displaystyle{ \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} }[/math]
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
- [math]\displaystyle{ P Y_{l, s j, m} = (-1)^{l}Y_{l,s, j, m}. }[/math]
For spin-½ systems, they are given in matrix form by[1][3][5]
- [math]\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}. }[/math]
where [math]\displaystyle{ Y_{l}^{m} }[/math] are the usual spherical harmonics.
References
- ↑ 1.0 1.1 1.2 Biedenharn, L. C.; Louck, J. D. (1981), Angular momentum in Quantum Physics: Theory and Application, Encyclopedia of Mathematics, 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, https://archive.org/details/angularmomentumi0000edmo
- ↑ 3.0 3.1 3.2 Greiner, Walter (6 December 2012). "9.3 Separation of the Variables for the Dirac Equation with Central Potential (minimally coupled)" (in en). Relativistic Quantum Mechanics: Wave Equations. Springer. ISBN 978-3-642-88082-7. https://books.google.com/books?id=a6_rCAAAQBAJ.
- ↑ Rose, M. E. (2013-12-20) (in en). Elementary Theory of Angular Momentum. Dover Publications, Incorporated. ISBN 978-0-486-78879-1. https://books.google.com/books?id=1c7lngEACAAJ&q=angular+momentum+rose.
- ↑ Berestetskii, V. B.; E. M. Lifshitz; L. P. Pitaevskii (2008). Quantum electrodynamics (2nd ed.). Oxford: Butterworth-Heinemann. ISBN 978-0-08-050346-2. OCLC 785780331.
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