[Paper Review] Random Matrix theory approach to Quantum mechanics

Shiv Chaitanya|arXiv (Cornell University)|Jan 27, 2015

Random Matrices and Applications1 references3 citations

TL;DR

This paper introduces a random matrix theory (RMT) approach to quantum mechanics using the quantum Hamilton-Jacobi formalism, demonstrating that bound state problems correspond to the Gaussian unitary ensemble in RMT. It identifies the potential in the joint probability distribution as a superpotential, enabling extension of RMT 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.

Motivation & Objective

• To establish a correspondence between bound state problems in quantum mechanics and the Gaussian unitary ensemble in random matrix theory.
• To identify the potential in the joint probability distribution of RMT as a superpotential in quantum systems.
• To extend the framework of random matrix theory to include newly discovered exceptional polynomials.
• To provide a novel mathematical bridge between quantum mechanics and random matrix theory using the quantum Hamilton-Jacobi formalism.

Proposed method

• Utilizes the quantum Hamilton-Jacobi formalism to map quantum bound state problems onto random matrix ensembles.
• Analyzes the joint probability distribution function in RMT to extract a superpotential analogous to quantum potentials.
• Applies the structure of the Gaussian unitary ensemble to model quantum systems with known bound states.
• Establishes a correspondence between the eigenvalue distribution in RMT and the energy spectrum of quantum systems.
• Uses the superpotential derived from RMT to generate quantum potentials compatible with exceptional orthogonal polynomials.
• Extends RMT formalism to accommodate exceptional polynomials by leveraging the superpotential structure.

Experimental results

Research questions

• RQ1 How can the quantum Hamilton-Jacobi formalism be used to relate quantum bound states to random matrix ensembles?
• RQ2 What is the physical interpretation of the potential in the joint probability distribution of RMT in quantum mechanical terms?
• RQ3 Can the superpotential derived from RMT be used to construct quantum potentials compatible with exceptional polynomials?
• RQ4 How does the Gaussian unitary ensemble in RMT correspond to the energy levels of quantum systems?
• RQ5 What is the role of exceptional polynomials in extending the scope of random matrix theory in quantum mechanics?

Key findings

• Bound state problems in quantum mechanics are mathematically equivalent to the Gaussian unitary ensemble in random matrix theory.
• The potential in the joint probability distribution of RMT is identified as a superpotential in quantum mechanics.
• The superpotential derived from RMT enables the construction of quantum potentials that support exceptional polynomials.
• The framework successfully extends random matrix theory to include systems described by exceptional orthogonal polynomials.
• The correspondence between RMT and quantum mechanics provides a new analytical tool for studying quantum systems with complex potentials.
• The method establishes a consistent mapping between eigenvalue statistics in RMT and energy spectra in quantum systems.