Making squeezed-coherent states concrete by determining their wavefunction

TL;DR

This paper presents three methods to explicitly construct the wavefunction of squeezed-coherent states, aiming to enhance understanding and teaching of these quantum states in educational settings.

Contribution

It introduces three different approaches to derive the wavefunction of squeezed-coherent states, making the concept more accessible for instructional purposes.

Findings

Wavefunctions of squeezed-coherent states are Gaussian in position and momentum.

Three methods: differential equations, harmonic oscillator expansion, and operator approach.

Facilitates teaching and understanding of squeezed states in quantum mechanics.

Abstract

With the successes of the Laser Interferometer Gravitational-wave Observatory, we anticipate increased interest in working with squeezed states in the undergraduate and graduate quantum-mechanics classroom. Because squeezed-coherent states are minimum uncertainty states, their wavefunctions in position and momentum space must be Gaussians. But this result is rarely discussed in treatments of squeezed states in quantum textbooks or quantum optics textbooks. In this work, we show three different ways to construct the wavefunction for squeezed-coherent states: (i) a differential equation-based approach; (ii) an approach that uses an expansion in terms of the simple-harmonic oscillator wavefunctions; and (iii) a fully operator-based approach. We do this to illustrate that the concept of the wavefunction can be introduced no matter what methodology an instructor wishes to use. We hope that working with the wavefunction will help demystify the concept of a squeezed-coherent state.