Free Fall of a Quantum Many-Body System

TL;DR

This paper demonstrates a simplified method using gauge transformation to analyze the quantum free fall problem for single particles and many-body systems, making the problem analytically tractable and pedagogically valuable.

Contribution

It introduces a gauge transformation approach that simplifies the quantum free fall problem for many-body systems, enabling analytical solutions and educational insights.

Findings

The gauge transformation removes the gravitational potential from the Schrödinger equation.

The method applies to both single particles and many-body systems with interparticle interactions.

Provides a pedagogical example of analytically solvable quantum many-body dynamics.

Abstract

The quantum version of the free fall problem is a topic often skipped in undergraduate quantum mechanics courses because its discussion usually requires wavepackets built on the Airy functions -- a difficult computation. Here, on the contrary, we show that the problem can be nicely simplified both for a single particle and for general many-body systems by making use of a gauge transformation that corresponds to a change of reference frame from the laboratory frame to the one comoving with the falling system. Using this approach, the quantum mechanics problem of a particle in an external gravitational potential reduces to a much simpler one where there is no longer any gravitational potential in the Schr\"{o}dinger equation. It is instructive to see that the same procedure can be used for many-body systems subjected to an external gravitational potential and a two-body interparticle potential that is a function of the distance between the particles. This topic provides a helpful and pedagogical example of a quantum many-body system whose dynamics can be analytically described in simple terms.