Invariants and wavefunctions for some time-dependent harmonic oscillator-type Hamiltonians

Abstract

Recently the author has shown that the Hamiltonian, H= (1/2) ωT A (t) ω+B (t)Tω+C (t), in which A (t) is a positive definite symmetric matrix and ωμ=qi, μ=1,n, i=1,n, ωμ=pi, μ=n+1,2n, i=1,n, may be transformed to the time-independent Hamiltonian, H̄= (1/2) ω̄Tω̄, by a time-dependent linear canonical transformation, ω̄=Sω+r. H̄ is an exact invariant of the motion described by H. A matrix invariant may also be constructed which provides a basis for the generators of the dynamical symmetry group SU(n) which may always be associated with H, usually as a noninvariance group. In this paper we examine, by way of example, an oscillator with source undergoing translation, the two-dimensional anisotropic oscillator, general one- and two-dimensional oscillators with Hamiltonians of homogeneous quadratic form and obtain explicit invariants and Schrödinger wavefunctions with the aid of the linear canonical transformations.