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College Physics

Universal ladder operators for angular momentum

Ni Zhi

Published Jan 1, 2000• 0 citations
OpenAlex

Abstract

Using the latest developed theory on the nonlinear algebra,an algebraic equation which the general ladder operators for angular momentum must satisfy is obtained.These ladder operators are constructed which can shift the angular quantum numbers and the magnetic quantum numbers for all kinds of angular momentum eigenstates.

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The method of self-adjoint ladder operators, developed in Parts I and II, is applied to the solution of the generalized angular-momentum problem. This reveals many interesting aspects of this approach to eigenvalue problems and, in particular, its relationship to addition of angular momentum. The complete set of irreducible unitary representations of the underlying algebra is obtained and also the corresponding Clebsch-Gordan (Wigner) coefficients for the addition of spin and angular momentum in a space of arbitrary dimension.
The theory of self-adjoint ladder operators is outlined, a comparison being made with the more usual type of ladder operator. It is shown how the method applies to the nnn-dimensional orbital angular momentum problem, for which it is found that the ladder operator may be expressed as a linear combination of the components of angular momentum, the expansion coefficients being the elements of a generalized spin algebra. Its relationship to the Infeld-Hull factorization method is discussed, the factorization appearing as part of a more general scheme. The physical significance of the ladder operators is emphasized through their application to the relativistic quantum theory of Dirac.
Using the method of nonlinear Lie algebra,the tensors of ladder operators are obtained.These tensors can shift the angular quantum numbers in ±2.Also the properties of the tensors' components and the relationships between angular momentum ladder operator vectors and the components are discussed in this paper.Thus, a conclusion is reached that the components of the 2 rank spherical noncancellation tensor operators are special linear combinations of the vector ladder operators' tensor products. They can shift the angular quantum numbers in 2 or-2,while the magnetic quantum numbers are changed in 0,±1,±2.