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Australian Journal of Physics

Commutation Rules for Generalized Pauli Spin Matrices

EA Jeffery

Published Oct 1, 1978• 0 citations
Publisher

Abstract

The algebra is developed for matrices involved in 2(2j+ I)-component arbitrary spin equations. These matrices can act as generators for the unitary group, and are shown to deserve the name 'generalized Pauli spin matrices'. Their commutation and anticommutation rules are derived from those for the ordinary Pauli spin matrices by a method termed mixed induced multiplication.

DOI
10.1071/ph780367

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The algebra is developed for matrices involved in 2(2j+ I)-component arbitrary spin equations. These matrices can act as generators for the unitary group, and are shown to deserve the name 'generalized Pauli spin matrices'. Their commutation and anticommutation rules are derived from those for the ordinary Pauli spin matrices by a method termed mixed induced multiplication.
This chapter demonstrates how the two kinds of spin can be represented in terms of spin up and spin down operators. The Pauli matrices have many interesting properties. The operators and representing the components of the electron spin can be written in terms of the spin raising and spin lowering operators. Considering the Pauli matrices representing the spin operators, this chapter helps to prove that the operators are Hermitian. This result is what the readers could expect as the operators represent a physical (measurable) quantity, the electron spin. It shows that the operator each has eigenvalues +1, -1. The chapter also shows that the operators obey the commutation relation. These commutation relations show that the three components of the spin cannot be measured simultaneously with the same precision. The result is a confirmation of the conservation of the total spin of the system that the magnitude of the total spin vector is constant.