Thermodynamics of free bosons and fermions in the hyperball

TL;DR

This paper analyzes the thermodynamics of free bosons and fermions confined in a D-dimensional spherical potential well, providing insights into spectral properties and shell structures, with implications for quantum many-particle systems.

Contribution

It extends the theory of ideal quantum gases to spherical confinement, solving the Schrödinger equation with zero boundary conditions and exploring the shell structure due to SO(D) symmetry.

Findings

Spectral properties of particles in spherical confinement derived.

Shell structure emerges from angular momentum quantization.

Results approach known thermodynamics in the thermodynamic limit.

Abstract

Many-particle systems pose commonly known computational challenges in quantum theory. The obstacles arise from the difficulty in finding sets of eigenvalues and eigenvectors of the underlying Hamiltonian while enforcing fermion or boson statistics, not to mention the prohibitive increase in the computational cost with the system's size. The first obvious step in this direction is to elaborate the theory for Fermi or Bose gases without inter-particle interactions. The traditional approach to the work is with the ideal gases confined in a cubic container with impenetrable walls (in arbitrary dimensions). This approach allows one to find the particle's spectra and compute all thermodynamic quantities of the confined gas. In the present work, we consider the gas confined in a spherical container (in other words, an infinitely deep spherical potential well in D dimensions), solving the corresponding Schroedinger equation using zero boundary conditions. We address the case of a finite number of particles N, either bosons or fermions, in the spherical potential box, as well as the thermodynamic limit. Owing to Weyl's relations, in the latter limit, the results do not depend on the shape of the box and thus approach the commonly known ones valid in the infinite space. Owing to the underlying SO(D) symmetry, we are dealing with particles carrying a well-defined angular momentum that, together with sorted energy eigenvalues, imparts a shell structure to the system.