A physicist's guide to the solution of Kummer's equation and confluent hypergeometric functions

TL;DR

This paper provides a comprehensive guide to solving Kummer's equation, clarifying special cases and solutions, with applications to quantum physics problems like hydrogenic atoms, aiding physics education and research.

Contribution

It systematically details all cases and solutions of the confluent hypergeometric equation, including special parameter conditions often encountered in physics.

Findings

Explicit formulas for solutions in special cases

Application to hydrogenic atom bound states

Clarification of solution independence in special parameter regimes

Abstract

The confluent hypergeometric equation, also known as Kummer's equation, is one of the most important differential equations in physics, chemistry, and engineering. Its two power series solutions are the Kummer function, M(a,b,z), often referred to as the confluent hypergeometric function of the first kind, and z^{1-b}M(1+a-b,2-b,z), where a and b are parameters that appear in the differential equation. A third function, the Tricomi function, U(a,b,z), sometimes referred to as the confluent hypergeometric function of the second kind, is also a solution of the confluent hypergeometric equation that is routinely used. All three of these functions must be considered in a search for two linearly independent solutions of the confluent hypergeometric equation. There are situations, when a, b, and a - b are integers, where one of these functions is not defined, or two of the functions are not linearly independent, or one of the linearly independent solutions of the differential equation is different from these three functions. Many of these special cases correspond precisely to cases needed to solve physics problems. This leads to significant confusion about how to work with confluent hypergeometric equations, in spite of authoritative references such as the NIST Digital Library of Mathematical Functions. Here, we carefully describe all of the different cases one has to consider and what the explicit formulas are for the two linearly independent solutions of the confluent hypergeometric equation. Our results are summarized in Table I in Section 3. As an example, we use these solutions to study the bound states of the hydrogenic atom, going beyond the standard treatment in textbooks. We also briefly consider the cutoff Coulomb potential. We hope that this guide will aid physics instruction that involves the confluent hypergeometric differential equation.