Random Matrix theory approach to Quantum mechanics

TL;DR

This paper introduces a novel approach connecting quantum mechanics with random matrix theory via the quantum Hamilton-Jacobi formalism, revealing analogies in bound state problems and extending to exceptional polynomials.

Contribution

It establishes a new framework linking quantum bound states to Gaussian ensembles and extends random matrix theory to include exceptional polynomials.

Findings

Bound state problems are analogous to Gaussian unitary ensemble solutions.

Potential functions are identified as super potentials in the joint probability distribution.

Extension of random matrix theory to exceptional polynomials.

Abstract

In this paper, we give random matrix theory approach to the quantum mechanics using the quantum Hamilton-Jacobi formalism. We show that the bound state problems in quantum mechanics are analogous to solving Gaussian unitary ensemble of random matrix theory. This study helps in identify the potential appear in the joint probability distribution function in the random matrix theory as a super potential. This approach allows to extend the random matrix theory to the newly discovered exceptional polynomials.