The variational method applied to the harmonic oscillator in presence of a delta function potential

TL;DR

This paper applies the variational method to approximate the ground state energy of a harmonic oscillator with a delta function potential, achieving results closer to the exact solutions than previous approximations.

Contribution

It demonstrates the effectiveness of simple trial wave functions in the variational method for a harmonic oscillator with a delta potential, improving upon earlier approximate results.

Findings

Ground state energy estimates are closer to exact values.

The method works for both repulsive and attractive delta potentials.

Provides a practical approach to visualize complex eigenfunctions.

Abstract

The problem of the harmonic oscillator with a centrally located delta function potential can be exactly solved in one dimension where the eigenfunctions are expressed as superpositions of the Hermite polynomials or as confluent hypergeometric functions in general. The eigenfunctions obtained exactly are difficult to visualise and hence to gain more insight, one can attempt using model wave functions which are explicitly and simply expressed. Here we apply the variational method to verify how close one can approach the exact ground state eigenvalues using such trial wave functions. We obtain the estimates of the ground state energies which are closer to the exact values in comparison to earlier approximate results for both the repulsive and attractive delta potentials.