Displacement-Operator Squeezed States. I. Time-Dependent Systems Having Isomorphic Symmetry Algebras

Abstract

In this paper we use the Lie algebra of space-time symmetries to construct states which are solutions to the time-dependent Schrödinger equation for systems with potentials $V(x,τ)=g^{(2)}(τ)x^2+g^{(1)}(τ)x +g^{(0)}(τ)$. We describe a set of number-operator eigenstates states, $\{Ψ_n(x,τ)\}$, that form a complete set of states but which, however, are usually not energy eigenstates. From the extremal state, $Ψ_0$, and a displacement squeeze operator derived using the Lie symmetries, we construct squeezed states and compute expectation values for position and momentum as a function of time, $τ$. We prove a general expression for the uncertainty relation for position and momentum in terms of the squeezing parameters. Specific examples, all corresponding to choices of $V(x,τ)$ and having isomorphic Lie algebras, will be dealt with in the following paper (II).