Notes on squeezed states in x-space representation

TL;DR

This paper provides an elementary, operator-free approach to understanding squeezed states in quantum mechanics using the Schrödinger equation, suitable for undergraduate education and including detailed derivations.

Contribution

It introduces a straightforward method to analyze squeezed states without operators, emphasizing explicit derivations in the position representation for educational purposes.

Findings

Explicit derivations of squeezed states in x-space

Analysis of pure and mixed squeezed states

Educational approach suitable for undergraduates

Abstract

In these notes, we discuss squeezed states using the elementary quantum language based on one-dimensional Schr\"odinger equation. No operators are used. The language of quantum optics is mentioned only for a hint to solve a differential equation. Sections II and III (squeezed vacuum and pure squeezed states) can be used in a standard undergraduate course on quantum mechanics after discussion of a harmonic oscillator. Section IV discusses density matrix of mixed squeezed states in $x$-representation. These notes present explicit (therefore somewhat lengthy) step-by-step derivations. These notes are not intended for publication as a journal paper.