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Application of Path Integration to Three-Dimensional Harmonic Oscillator Using General Canonical Transformations

Sara Fadhel, Nacira Bedida, Mohammed Tayeb Meftah, Mosbah Difallah

Published Feb 12, 2020• 0 citations
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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.
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.