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

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