A Random Matrix Approach to Quantum Mechanics

TL;DR

This paper establishes an equivalence between the quantum Hamilton-Jacobi method for bound state problems and the Gaussian orthogonal ensemble in random matrix theory, revealing shared Berry connections and super potentials.

Contribution

It introduces a novel link between quantum mechanics and random matrix theory using the Hamilton-Jacobi approach and super potentials, expanding the theoretical framework.

Findings

Quantum Hamilton-Jacobi approach is equivalent to Gaussian orthogonal ensemble.

Berry connection is identical in both problems.

Super potential enables application to exceptional polynomials.

Abstract

We show that the quantum Hamilton Jacobi approach to a class of quantum mechanical bound state problems and the Gaussian orthogonal ensemble of random matrix theory are equivalent. The Berry connection for both problems is identical to their quantum momentum function.The potential that appears in the joint probability distribution function in the random matrix theory is a super potential allowing us to apply it to exceptional polynomials.