Group Theoretical Approach to the Coherent and the Squeeze States of a Time-Dependent Harmonic Oscillator with a Singular Term

TL;DR

This paper employs group theory, specifically Lie algebras $SU(2)$ and $SU(1,1)$, to analyze the coherent and squeeze states of a time-dependent harmonic oscillator with a singular term, providing a unified algebraic framework.

Contribution

It introduces a group-theoretical method to construct eigenstates and coherent states for a harmonic oscillator with a singular term, extending the analysis to both time-dependent and time-independent cases.

Findings

Constructed the generalized invariant using $SU(2)$ algebra.

Derived the evolution operator within the Lie algebra framework.

Showed the squeeze operator's unitary transformation of the basis.

Abstract

For a time-dependent harmonic oscillator with an inverse squared singular term, we find the generalized invariant using the Lie algebra of $SU(2)$ and construct the number-type eigenstates and the coherent states using the spectrum-generating Lie algebra of $SU(1,1)$. We obtain the evolution operator in both of the Lie algebras. The number-type eigenstates and the coherent states are constructed group-theoretically for both the time-independent and the time-dependent harmonic oscillators with the singular term. It is shown that the squeeze operator transforms unitarily the time-dependent basis of the spectrum-generating Lie algebra of $SU(1,1)$ for the generalized invariant, and thereby evolves the initial vacuum into a final coherent vacuum.