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arXiv:2007.02885 · Journal of Mathematical Physics

Converting translation operators into plane polar and spherical coordinates and their use in determining quantum-mechanical wavefunctions in a representation-independent fashion

Michael Rushka, M. A. Esrick, W. N. Mathews, J. K. Freericks

Published Jul 1, 2021• 9 citations
arXivHTMLPDF

Abstract

Quantum mechanics is often developed in the position representation, but this is not necessary, and one can perform calculations in a representation-independent fashion, even for wavefunctions. In this work, we illustrate how one can determine wavefunctions, aside from normalization, using only operators and how those operators act on state vectors. To do this in plane polar and spherical coordinates requires one to convert the translation operator into those coordinates. As examples of this approach, we illustrate the solution of the Coulomb problem in two and three dimensions without needing to express any operators in position space.

DOI
10.1063/5.0021013

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Quantum mechanics is often developed in the position representation, but this is not necessary, and one can perform calculations in a representation-independent fashion, even for wavefunctions. In this work, we illustrate how one can determine wavefunctions, aside from normalization, using only operators and how those operators act on state vectors. To do this in plane polar and spherical coordinates requires one to convert the translation operator into those coordinates. As examples of this approach, we illustrate the solution of the Coulomb problem in two and three dimensions without needing to express any operators in position space.
The factorization method of Schrödinger shows us how to determine the energy eigenstates without needing to determine the wavefunctions in position or momentum space. A strategy to convert the energy eigenstates to wavefunctions is well known for the one-dimensional simple harmonic oscillator by employing the Rodrigues formula for the Hermite polynomials in position or momentum space. In this work, we illustrate how to generalize this approach in a representation-independent fashion to find the wavefunctions of other problems in quantum mechanics that can be solved by the factorization method. We examine three problems in detail: (i) the one-dimensional simple harmonic oscillator; (ii) the three-dimensional isotropic harmonic oscillator; and (iii) the three-dimensional Coulomb problem. This approach can be used in either undergraduate or graduate classes in quantum mechanics.
A general expression for spherical harmonics as an explicit function of the angle coordinates is derived using standard methods of group-representation theory. This expression is more useful to physicists than a corresponding expression available in the mathematical literature for two reasons: (1) Its normalization is the one most commonly used by physicists, rather than one of several other normalization conventions used mainly by mathematicians. (2) Its form is especially convenient for use in the operator equations which occur in quantum mechanics. Furthermore, the derivation serves the pedagogical purpose of providing a simple illustration of the methods of group-representation theory. A set of relations useful in solving the Dirac equation are listed, both because their verification provides an example of the utility of the expression derived, and because the form in which these relations are stated in the literature is apt to result in confusion stemming from the various different normalization conventions in use.
The usual explicit expressions for spherical harmonics are derived by a method which does not involve the solution of differential equations or the use of angular momentum operators in spherical polar form. Instead a raising operator constructed from an arbitrary vector is used together with the usual angular momentum raising and lowering operators. The method is first presented as simply as possible in Schrödinger representation using position as the vector. Then it is generalized to an arbitrary vector and arbitrary integral angular momentum without using a particular representation. Finally it is shown how a conserved vector (other than angular momentum) of a central system can be used to reveal properties of the energy eigenstates. As an example, eigenstates of the hydrogen atom are found using the Runge-Lenz vector.
Abstract In Chapter 5 we considered the factorization of a radial Hamiltonian for a particle in a spherically symmetric potential which is independent of angular momentum. This factorization provides shift operators for energy and angular momentum in the eigenkets of the radial Hamiltonian for the oscillator and Coulomb potentials. These operators are linear functions of the momentum operator *p and they are also nonlinear functions of the position operator r: in the coordinate representation of wave mechanics they yield first-order differential operators. In this chapter we show that for the three-dimensional isotropic harmonic oscillator and the Coulomb potential, one can also construct shift operators for energy and angular momentum that are linear functions* of r and nonlinear functions of p. In the momentum representation of wave mechanics the shift operators yield first-order differential operators. Applications to the calculation of energy eigenvalues, matrix elements, and radial momentum-space wavefunctions are considered.
Quantum mechanics is traditionally taught in a specific representation—the position-space representation—because the time-independent Schrödinger equation is simplest to formulate there. But this leads to a number of conceptual issues both associated with language and with the formalism, especially if one starts more formally and then emphasizes representations for calculations. In this work, we describe a number of these issues and argue why restricting to a representation-independent formalism (that is, working with abstract operators and state vectors independent of any specific representation) fixes many of these issues. Then representations can be introduced later when needed for specific applications (but we find the need to do this can be sharply reduced from current paradigms).
While groups of point transformations such as rotations, translations and permutations have been extensively used in quantum mechanics, we cannot say the same of the more general groups of canonical transformations. To begin with, for the latter groups we have to determine their unitary representation in appropriate spaces. But once this is achieved canonical transformations can play as important a role in quantum mechanics as point transformations. We illustrate this in relation with the harmonic oscillator in one and many dimensions and for one and many particles. We show how radial matrix elements with respect to harmonic oscillator states can be determined group theoretically both in the Schroedinger and Heisenberg pictures. We discuss the application of canonical transformations to the three body nuclear problem. We briefly outline in the concluding section applications to Coulomb problems, many body systems, accidental degeneracy, and generalized integral transforms.