Supersymmetric Quantum Potentials Analogs of Classical Electrostatic Fields

TL;DR

This paper establishes a mathematical analogy between classical electrostatic fields and supersymmetric quantum potentials, enabling the construction of quantum analogs of electrostatic configurations through a supersymmetric framework.

Contribution

It introduces a method to derive supersymmetric quantum potentials from electrostatic fields using a confluent algorithm and an ansatz for the electrostatic potential.

Findings

Electrostatic fields satisfy Bernoulli's equation in this framework.

Given a charge density, the corresponding electrostatic and quantum potentials can be explicitly obtained.

The approach allows for the determination of charge density and field profiles for solvable quantum potentials.

Abstract

A relation between classical electrostatic fields and Schr\"odinger-like Hamiltonians is evidenced. Hence, supersymmetric quantum potentials analogous to classical electrostatic fields can be constructed. Proposing an ansatz for the electrostatic potential as the natural logarithm of a nodeless function, it is demonstrated that the electrostatic fields fulfil the Bernoulli equation associated to a second-order confluent supersymmetric transformation. By using the so-called confluent algorithm, it is possible, given a charge density, to find the corresponding electrostatic field as well as the supersymmetric potentials. Furthermore, the associated charge density and the electrostatic field profile of Schr\"odinger-like solvable potentials can be determined.