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Reviews of Modern Physics

Self-Adjoint Ladder Operators (I)

A. JOSEPH

Published Oct 1, 1967• 53 citations
Publisher

Abstract

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 $n$-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.

DOI
10.1103/revmodphys.39.829

Related papers

Self-Adjoint Ladder Operators. III

A. JOSEPH

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.

Reviews of Modern Physics•Oct 1, 1968•7 citations

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American Journal of Physics•Apr 1, 1986•4 citations

The perturbed ladder operator method: closed form expressions of perturbed wavefunctions and matrix elements

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Universal ladder operators for angular momentum

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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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Orbital Angular Momentum

Reinhold A. Bertlmann, Nicolai Friis

Abstract In this chapter we introduce the concept of orbital angular momentum into the mathematical language of quantum mechanics. We discuss the commutation relations of the corresponding operators and their roles as generators of rotations in the context of the theory of Lie groups and Lie algebras, focusing on the three-dimensional rotation group SO(3). We further examine how scalar and vector-valued operators transform under spatial rotations. We then turn to the derivation of the eigenfunctions and eigenvalues of the angular-momentum operators, for which we follow a dual approach in analogy to that employed for the harmonic oscillator in Chapter 5. Using both the algebraic ladder-operator method and an analytical method using spherical polar coordinates we discover the relevant azimuthal and magnetic quantum numbers and their relation to the spherical harmonics and the Legendre polynomials

•Sep 7, 2023•1 citation

Quantum formulation of angle and orbital angular momentum

Jörg B. Götte, Stephen Mark Barnett

The quantum mechanical description of the azimuthal rotation angle and the orbital angular momentum around the polar axis differs greatly from the description of position and momentum. This has led to a controversy over the existence of a self-adjoint angle operator in the literature. It is possible to circumvent the problems of defining a self-adjoint operator for a periodic variable by using trigonometric functions of operators as a basis for the quantum mechanical description. This approach, however, does not allow us to study the properties of the angle operator itself. By using a state space of an arbitrarily large yet finite number of dimensions, it is possible to introduce angle and orbital angular momentum as a conjugate pair of variables both represented by Hermitian matrices. After physical and measurable quantities have been calculated in this state space the number of dimensions is allowed to tend to infinity in a limiting procedure. The first part of this chapter reviews the difficulties associated with the quantum mechanical formulation of angle and orbital angular momentum and presents a way to overcome these problems using a finite-dimensional state space. In the second part this formulation is generalised to include the phenomenon of non-integer orbital angular momentum. This part is based on a chapter of the PhD thesis of Götte [1], where further details may be found. Introduction The correct description of a periodic variable such as a rotation angle or the optical phase has been a long standing problem in quantum mechanics. At the root of the problem is the question of whether the variable itself is restricted in a 2 π radian range or whether it evolves continuously without bound.

Cambridge University Press eBooks•Nov 8, 2012•2 citations
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 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.
By mapping the perturbation scheme onto the ladder operator formalism, the field of application of the Schrodinger-Infeld-Hull factorisation method is enlarged. It is shown how, at each order of the perturbation, perturbed ladder operators can be constructed. Thus, without having to calculate explicitly either the excited unperturbed functions or any matrix element, one obtains analytical expressions of the perturbed eigenvalues in terms of the quantum numbers of the factorisable unperturbed problem. A three-terms recurrence relation, valid at any rank of the perturbation, is derived and leads to closed form expressions of the perturbed eigenfunctions. Consequently, a closed form expression of any matrix element on the basis of the perturbed eigenfunctions is easily obtained from the calculation of one unique particular integral.
Abstract In this chapter we introduce the concept of orbital angular momentum into the mathematical language of quantum mechanics. We discuss the commutation relations of the corresponding operators and their roles as generators of rotations in the context of the theory of Lie groups and Lie algebras, focusing on the three-dimensional rotation group SO(3). We further examine how scalar and vector-valued operators transform under spatial rotations. We then turn to the derivation of the eigenfunctions and eigenvalues of the angular-momentum operators, for which we follow a dual approach in analogy to that employed for the harmonic oscillator in Chapter 5. Using both the algebraic ladder-operator method and an analytical method using spherical polar coordinates we discover the relevant azimuthal and magnetic quantum numbers and their relation to the spherical harmonics and the Legendre polynomials
The quantum mechanical description of the azimuthal rotation angle and the orbital angular momentum around the polar axis differs greatly from the description of position and momentum. This has led to a controversy over the existence of a self-adjoint angle operator in the literature. It is possible to circumvent the problems of defining a self-adjoint operator for a periodic variable by using trigonometric functions of operators as a basis for the quantum mechanical description. This approach, however, does not allow us to study the properties of the angle operator itself. By using a state space of an arbitrarily large yet finite number of dimensions, it is possible to introduce angle and orbital angular momentum as a conjugate pair of variables both represented by Hermitian matrices. After physical and measurable quantities have been calculated in this state space the number of dimensions is allowed to tend to infinity in a limiting procedure. The first part of this chapter reviews the difficulties associated with the quantum mechanical formulation of angle and orbital angular momentum and presents a way to overcome these problems using a finite-dimensional state space. In the second part this formulation is generalised to include the phenomenon of non-integer orbital angular momentum. This part is based on a chapter of the PhD thesis of Götte [1], where further details may be found. Introduction The correct description of a periodic variable such as a rotation angle or the optical phase has been a long standing problem in quantum mechanics. At the root of the problem is the question of whether the variable itself is restricted in a 2 π radian range or whether it evolves continuously without bound.