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Physical Review A

Generalized canonical transformations and path integrals

L. Chétouani, Larbi Guechi, Théophile F. Hammann

Published Aug 1, 1989• 32 citations
Publisher

Abstract

Some time-dependent physical systems do not admit, in the general case, either an invariant or auxiliary equation. The study of these systems is then in general made easier by space-time transformation of coordinates. This is true for the case for a rectangular well with a moving wall, which generalized canonical transformations reduce, in the path-integral formalism, to the case of a variable-frequency oscillator with fixed walls. The variable-frequency harmonic oscillator is an example where the invariant and the auxiliary equations can be calculated. All calculations are done in phase space.

DOI
10.1103/physreva.40.1157

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Some time-dependent physical systems do not admit, in the general case, either an invariant or auxiliary equation. The study of these systems is then in general made easier by space-time transformation of coordinates. This is true for the case for a rectangular well with a moving wall, which generalized canonical transformations reduce, in the path-integral formalism, to the case of a variable-frequency oscillator with fixed walls. The variable-frequency harmonic oscillator is an example where the invariant and the auxiliary equations can be calculated. All calculations are done in phase space.
The path integral formulation of quantum mechanics constructs the propagator by evaluating the action S for all classical paths in coordinate space. A corresponding momentum path integral may also be defined through Fourier transforms in the endpoints. Although these momentum path integrals are especially simple for several special cases, no one has, to my knowledge, ever formally constructed them from all classical paths in momentum space. I show that this is possible because there exists another classical mechanics based on an alternate classical action R. Hamilton's Canonical equations result from a variational principle in both S and R. S uses fixed beginning and ending spatial points while R uses fixed beginning and ending momentum points. This alternative action's classical mechanics also includes a Hamilton-Jacobi equation. I also present some important points concerning the beginning and ending conditions on the action necessary to apply a Canonical transformation. These properties explain the failure of the Canonical transformation in the phase space path integral. It follows that a path integral may be constructed from classical position paths using S in the coordinate representation or from classical momentum paths using R in the momentum representation. Several example calculations are presented that illustrate the simplifications and practical advantages made possible by this broader view of the path integral. In particular, the normalized amplitude for a free particle is found without using the Schrodinger equation, the internal spin degree of freedom is simply and naturally derived, and the simple harmonic oscillator is calculated.
The Feynman path integral for the generalized harmonic oscillator is reviewed, and it is shown that the path integral can be used to nd a complete set of wave functions for the oscillator. Harmonic oscillators with dierent (time-dependent) parameters can be related through unitary transformations. The existence of generalized coherent states for a simple harmonic oscillator can then be interpreted as the result of a (formal) invariance under a unitary transformation which relates the same harmonic oscillator. In the path integral formalism, the invariance is reflected in that the kernels do not depend on the choice of classical solutions.
The Feynman path integral for the generalized harmonic oscillator is reviewed, and it is shown that the path integral can be used to find a complete set of wave functions for the oscillator. Harmonic oscillators with different (time-dependent) parameters can be related through unitary transformations. The existence of generalized coherent states for a simple harmonic oscillator can then be interpreted as the result of a (formal) {\em invariance} under a unitary transformation which relates the same harmonic oscillator. In the path integral formalism, the invariance is reflected in that the kernels do not depend on the choice of classical solutions.