Effective polar potential in the central force Schrodinger equation

TL;DR

This paper reformulates the angular part of the Schrödinger equation for a central potential into a one-dimensional form, revealing a family of polar potentials that influence the spatial distribution of quantum states.

Contribution

It introduces a novel polar potential perspective linked to the magnetic quantum number, providing new insights into angular solutions of the Schrödinger equation.

Findings

Polar potential acts as a confining potential increasing with m

Variation in angular distribution linked to 'squeezing' by the polar potential

Connection established between polar potential and associated Legendre functions

Abstract

The angular part of the Schrodinger equation for a central potential is brought to the one-dimensional 'Schrodinger form' where one has a kinetic energy plus potential energy terms. The resulting polar potential is seen to be a family of potentials characterized by the square of the magnetic quantum number m. It is demonstrated that this potential can be viewed as a confining potential that attempts to confine the particle to the xy-plane, with a strength that increases with increasing m. Linking the solutions of the equation to the conventional solutions of the angular equation, i.e. the associated Legendre functions, we show that the variation in the spatial distribution of the latter for different values of the orbital angular quantum number l can be viewed as being a result of 'squeezing' with different strengths by the introduced 'polar potential'.