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arXiv:hep-th/9711118

Integrability, Coordinate Systems, and Separation of Variables in Path Integrals

Ch. Grosche

Published Nov 17, 1997• 0 citations
arXivHTMLPDF

Abstract

In this contribution I summarize the achievements of separation of variables in integrable quantum systems from the point of view of path integrals. This includes the free motion on homogeneous spaces, and motion subject to a potential force, and I would like to propose systematic investigations of parametric coordinate systems on homogeneous spaces.

DOI
10.48550/arxiv.hep-th/9711118

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In this contribution I summarize the achievements of separation of variables in integrable quantum systems from the point of view of path integrals. This includes the free motion on homogeneous spaces, and motion subject to a potential force, and I would like to propose systematic investigations of parametric coordinate systems on homogeneous spaces.
In quantum mechanics, as well as in classical mechanics, the number of variables involved in the equations of motion may be reduced by utilizing the constants of motion of the system. For a set of N particles moving in a field-free space there are six integrals if N is greater than two. These correspond to the laws of conservation of linear and of angular momentum. It is the purpose of this paper to eliminate the coordinates of rotation and of translation of the total system from the Schrodinger equation and to obtain a new set of differential equations involving only 3N-6 variables.1
The momentum representation is seldom used in quantum mechanics courses. Some students are thence surprised by the change in viewpoint when, in doing advanced work, they have to use the momentum rather than the coordinate representation. In this work, we give an introduction to quantum mechanics in momentum space, where the Schrödinger equation becomes an integral equation. To this end we discuss standard problems, namely, the free particle, the quantum motion under a constant potential, a particle interacting with a potential step, and the motion of a particle under a harmonic potential. What is not so standard is that they are all conceived from momentum space and hence they, with the exception of the free particle, are not equivalent to the coordinate space ones with the same names. All the problems are solved within the momentum representation making no reference to the systems they correspond to in the coordinate representation.
Chapter 2 constructed a path integral representation of the matrix elements of the statistical operator e-βH. This chapter extends the construction to hamiltonians which are general functions of phase space variables. This results in integrals over trajectories or paths in phase space. When the hamiltonian is quadratic in the momentum variables, the integral over momenta is gaussian and can be performed. In the separable example, the path integral of Chapter 2 is recovered. In the case of the charged particle in a magnetic field a more general form is found, which is somewhat ambiguous, reflecting the problem of order between quantum operators. Hamiltonians which are general quadratic functions of momentum variables provide other important examples, and these are analyzed thoroughly. Such hamiltonians arise in the quantization of the motion on Riemannian manifolds. The analysis is illustrated by the quantization of the free motion on the sphere SN-1.