Solution of the quantum harmonic oscillator plus a delta-function potential at the origin: The oddness of its even-parity solutions

TL;DR

This paper analytically solves the quantum harmonic oscillator with an added delta-function potential at the origin, highlighting the impact on even-parity solutions and correcting common textbook misconceptions.

Contribution

It provides a detailed derivation of energy levels for even-parity states with a delta potential, clarifying boundary conditions and correcting textbook inaccuracies.

Findings

Derived energy levels for even-parity solutions

Highlighted boundary condition modifications at the origin

Corrected common textbook misconceptions

Abstract

We derive the energy levels associated with the even-parity wave functions of the harmonic oscillator with an additional delta-function potential at the origin. Our results bring to the attention of students a non-trivial and analytical example of a modification of the usual harmonic oscillator potential, with emphasis on the modification of the boundary conditions at the origin. This problem calls the attention of the students to an inaccurate statement in quantum mechanics textbooks often found in the context of solution of the harmonic oscillator problem.