Stieltjes Electrostatic Model Interpretation for Bound State Problems

TL;DR

This paper establishes an analogy between the Stieltjes electrostatic model and quantum Hamilton-Jacobi formalism, interpreting bound state problems as equilibrium configurations of imaginary charges related to orthogonal polynomial zeros.

Contribution

It introduces a novel interpretation of bound state problems using electrostatic models and connects classical turning points with orthogonal polynomial zeros.

Findings

Bound states correspond to equilibrium positions of imaginary charges.

The model applies to exactly solvable potentials and relates to orthogonal polynomial zeros.

The interaction potential is expressed as the logarithm of the wave function.

Abstract

In this paper, Stieltjes electrostatic model and quantum Hamilton Jacobi formalism is analogous to each other is shown. This analogy allows, the bound state problem to mimics as $n$ unit moving imaginary charges $i\hbar$, which are placed in between the two fixed imaginary charges arising due to the classical turning points of the potential. The interaction potential between $n$ unit moving imaginary charges $i\hbar$ is given by logarithm of the wave function. For an exactly solvable potential, this system attains stable equilibrium position at the zeros of the orthogonal polynomials depending upon the interval of the classical turning points.