Discrete Density of States

TL;DR

This paper introduces a discrete density of states concept for quantum particles in a box, providing exact counts and analytical expressions that improve understanding of quantum state distributions across dimensions.

Contribution

It presents a novel discrete DOS framework, deriving analytical expressions that connect to classical continuum results and analyzing error behaviors under various conditions.

Findings

Discrete DOS accurately counts quantum states in a box.

Analytical expressions converge to classical results at high NOS.

Weyl's conjecture-based formulas maintain low errors across conditions.

Abstract

By considering the quantum-mechanically minimum allowable energy interval, we exactly count number of states (NOS) and introduce discrete density of states (DOS) concept for a particle in a box for various dimensions. Expressions for bounded and unbounded continua are analytically recovered from discrete ones. Even though substantial fluctuations prevail in discrete DOS, they're almost completely flatten out after summation or integration operation. It's seen that relative errors of analytical expressions of bounded/unbounded continua rapidly decrease for high NOS values (weak confinement or high energy conditions), while the proposed analytical expressions based on Weyl's conjecture always preserve their lower error characteristic.