Spinor spherical harmonics

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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:

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

For spin-1/2 systems, they are given in matrix form by [1] [3] [5]

where are the usual spherical harmonics.

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References

  1. 1 2 3 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
  2. Edmonds, A. R. (1957), Angular Momentum in Quantum Mechanics , Princeton University Press, ISBN   978-0-691-07912-7
  3. 1 2 3 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.
  4. Rose, M. E. (2013-12-20). Elementary Theory of Angular Momentum. Dover Publications, Incorporated. ISBN   978-0-486-78879-1.
  5. 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.