Ideal Gas Law for a Quantum Particle

TL;DR

This paper investigates how the ideal gas law applies to a single quantum particle in a 2D cavity, revealing that classical thermodynamic behavior emerges more accurately in chaotic and isotropic systems, especially with a particular pressure definition.

Contribution

It introduces two definitions of quantum pressure and analyzes their validity for different cavity geometries, connecting quantum eigenstates with classical thermodynamic laws.

Findings

IGL holds exactly in isotropic systems with the first pressure definition

Deviations from IGL are reduced in chaotic systems and coherent states

Second pressure definition aligns well with IGL across geometries

Abstract

The question of how classical thermodynamic laws emerge from the underlying quantum substrate lies at the foundations of physics. Here, we examine the validity of the ideal gas law (IGL) for a single quantum particle confined within a two-dimensional cavity. By interpreting the quantum wave function as a probability density analogous to that of an ideal gas, we employ the energy equipartition principle to define the temperature of the quantum state. For the mean pressure we take two definitions, one straightforwardly based on the radiation pressure concept and the other taking advantage of a quasi-orthogonality relation valid for billiard eigenstates. We analyze systems with regular dynamics-the circular and rectangular billiards-and compare them with the classically chaotic Bunimovich stadium. We find that the IGL for the first definition of pressure holds exactly in isotropic systems (as the circular case), while for anisotropic geometries, quantum eigenfunctions generally conform to the IGL only on average, exhibiting meaningful deviations. These deviations are diminished in the presence of chaotic dynamics and for coherent states. This observation is consistent with the Eigenstate Thermalization Hypothesis (ETH). Notably, the second definition of pressure allows for a good matching with the IGL.