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Reviews of Modern Physics

Self-Adjoint Ladder Operators. III

A. JOSEPH

Published Oct 1, 1968• 7 citations
Publisher

Abstract

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.

DOI
10.1103/revmodphys.40.845

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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.
The approach to quantum mechanics which we have used to derive the matrix treatment of spin from first principles is now employed to treat systems of compounded angular momentum. A general treatment is first given, which is then applied to the concrete cases of a spin-0 and a spin-1 system obtained by adding the spins of two spin-1/2 systems. Thus the probability amplitudes for measurements on the systems are derived, as well as the matrix vectors and operators corresponding to the systems. The matrix operators and states obtained are different from the standard forms and are much more generalized. The new results are applied to the case of joint measurements on the subsystems of such a system; this is a problem that has been made very topical by the high level of interest in the foundations of quantum mechanics. As a consequence of the insights arising from this treatment, we show that the Clebsch-Gordan coefficients are amenable to generalization, and we give the generalized forms for these cases.
We develop a systematic approach to deriving addition theorems for, and some other bilocal sums of, spin spherical harmonics. In this first part we establish some necessary technical results. We discuss the factorization of orbital and spin degrees of freedom in certain products of Clebsch-Gordan coefficients, and obtain general explicit results for the matrix elements in configuration space of tensor products of arbitrary rank of the position and angular-momentum operators. These results are the basis of the addition theorems for spin spherical harmonics obtained in part II.
Abstract We review the irreducible representation of an angular momentum vector operator constructed in terms of spinor algebra. We generalize the idea of spinor approach to study the coupling of the eigenstates of two independent angular momentum vector operators. Utilizing the spinor algebra, we are able to develop a simple way for calculating the SU(2) Clebsch-Gordan (CG) coefficients. The explicit expression for the SU(2) CG coefficients is worked out, and some simple physical examples are presented to illustrate the spinor approach.
The addition theorem of angular momentum states that the angular momentum operator of a coupled state is the vector sum of the angular momentum operators of the uncoupled states. Given that, the eigenfunction of the angular momentum of a system in the coupled state can be represented as a linear combination of eigenfunctions of the system in its uncoupled state and, in turn, the eigenfunctions of the uncoupled state are a direct product of eigenfunctions of the individual particles. Clebsch-Gordan coefficients emerge as an overlap integral in the series expansion of the coupled state in the basis of uncoupled states. This chapter provides a detailed derivation of the addition theorem of angular momentum, including obtaining the lower and upper boundaries of the quantum numbers, deriving the orthonormality conditions for Clebsch-Gordan coefficients, and using shift operators to generate the members of a multiplet. It also derives a general expression for the evaluation of Clebsch-Gordan coefficients, provides a detailed analysis of the addition of angular momenta and electron spin to build the resulting Clebsch-Gordan matrix, and shows that a rotation transformation of the direct product of eigenfunctions of the angular momentum and spin leads to the recurrence relations for the Wigner matrix.
We develop a systematic approach to deriving addition theorems for, and some other bilocal sums of, spin spherical harmonics. In this first part we establish some necessary technical results. We discuss the factorization of orbital and spin degrees of freedom in certain products of Clebsch–Gordan coefficients, and obtain general explicit results for the matrix elements in configuration space of tensor products of arbitrary rank of the position and angular-momentum operators. These results are the basis of the addition theorems for spin spherical harmonics obtained in part II (2011 J. Phys. A: Math. Theor. 44 165302 ).