Canonical Transformations and Quantum Mechanics

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.