Eigenfunctions of spinless particles in a one-dimensional linear potential well

TL;DR

This paper derives the eigenfunctions of spinless particles in a one-dimensional linear potential well by solving the Klein-Gordon equation, revealing solutions in terms of parabolic cylinder and confluent hypergeometric functions.

Contribution

It provides an analytical solution for the eigenfunctions of spinless particles in a linear potential well using the Klein-Gordon framework, connecting to special functions.

Findings

Eigenfunctions are expressed as parabolic cylinder functions.

Solutions reduce to confluent hypergeometric functions under boundary conditions.

The approach clarifies the quantum behavior in linear potential wells.

Abstract

In the present paper, we work out the eigenfunctions of spinless particles bound in a one-dimensional linear finite range, attractive potential well, treating it as a time-like component of a four-vector. We show that the one-dimensional stationary Klein-Gordon equation is reduced to a standard differential equation, whose solutions, consistent with the boundary conditions, are the parabolic cylinder functions, which further reduce to the well-known confluent hypergeometric functions.