Duality between Poisson and Schr\"odinger equations in three dimensions

TL;DR

This paper generalizes a duality between electrostatic and quantum mechanical problems from one dimension to three dimensions, enabling the determination of ground state solutions via electrostatic results for s-wave Schr"odinger equations with central potentials.

Contribution

It introduces a new transformation technique applicable to three-dimensional s-wave Schr"odinger equations, linking electrostatic potentials to quantum wave functions.

Findings

Establishes a relationship between electrostatic potential and quantum wave function in 3D.

Demonstrates the method's applicability to central potentials in three dimensions.

Provides a way to compute quantum energies from electrostatic energy densities.

Abstract

A duality between an electrostatic problem in a three dimensional world and a quantum mechanical problem in a one dimensional world which allows one to obtain the ground state solution of the Schr\"odinger equation by using electrostatic results is generalized to three dimensions. Here, it is demonstrated that the same transformation technique is also applicable to the s-wave Schr\"odinger equation in three dimensions for central potentials. This approach leads to a general relationship between the electrostatic potential and the s-wave function and the electric energy density to the quantum mechanical energy.