  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.
