Free expansion of a Gaussian wavepacket using operator manipulations

Abstract

The free expansion of a Gaussian wavepacket is a problem commonly discussed in undergraduate quantum classes by directly solving the time-dependent Schrödinger equation as a differential equation. In this work, we provide an alternative way to calculate the free expansion by recognizing that the Gaussian wavepacket can be thought of as the ground state of a harmonic oscillator with its frequency adjusted to give the initial width of the Gaussian, and the time evolution, given by the free-particle Hamiltonian, being the same as the application of a time-dependent squeezing operator to the harmonic oscillator ground state. Operator manipulations alone (including the Hadamard lemma and the exponential disentangling identity) then allow us to directly solve the problem. As quantum instruction evolves to include more quantum information science applications, reworking this well-known problem using a squeezing formalism will help students develop intuition for how squeezed states are used in quantum sensing.