# Derivation of Bose’s Entropy Spectral Density from the Multiplicity of Energy Eigenvalues

## TL;DR

This paper re-examines the derivation of Bose’s entropy spectral density by focusing on the multiplicity of energy eigenvalues in a photon cavity.

## Contribution

A new derivation of energy and entropy spectral densities based on the multiplicity of energy eigenvalues is presented.

## Key findings

- The multiplicity of energy eigenvalues plays a central role in deriving entropy spectral density.
- A discrete-to-continuous transition is achieved using eigenfrequency distributions of energy and entropy.

## Abstract

The modern textbook analysis of the thermal state of photons inside a three-dimensional reflective cavity is based on the three quantum numbers that characterize photon’s energy eigenvalues coming out when the boundary conditions are imposed. The crucial passage from the quantum numbers to the continuous frequency is operated by introducing a three-dimensional continuous version of the three discrete quantum numbers, which leads to the energy spectral density and to the entropy spectral density. This standard analysis obscures the role of the multiplicity of energy eigenvalues associated to the same eigenfrequency. In this paper we review the past derivations of Bose’s entropy spectral density and present a new analysis of energy spectral density and entropy spectral density based on the multiplicity of energy eigenvalues. Our analysis explicitly defines the eigenfrequency distribution of energy and entropy and uses it as a starting point for the passage from the discrete eigenfrequencies to the continuous frequency.