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

Abstract

Recently the author has shown that the Hamiltonian, H= (1/2) ω^T A (t) ω+B (t)^Tω+C (t), in which A (t) is a positive definite symmetric matrix and ω^μ=q_i, μ=1,n, i=1,n, ω^μ=p_i, μ=n+1,2n, i=1,n, may be transformed to the time‐independent Hamiltonian, H̄= (1/2) ω̄^Tω̄, 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.