<p> We study a many particle system contained within a harmonic oscillator potential, with the single particle spacings equispaced and nondegenerate. We consider either fermions or bosons within this harmonic oscillator potential and derive the density of states for this system, with low excitation energy corrections. We extend our method for more than one species of fermions or bosons within this harmonic oscillator potential and compare our results with number-theoretic techniques. We move on to the degenerate harmonic oscillator potential and show that, for a fermion system, shell effects are contained within low excitation energy terms. We attempt to arrive at the density of states for an infinite number of bosons within the same potential. </p> / Thesis / Master of Science (MSc)