Using the commutation relations of a and a^dag, we show that the eigenvalues of the harmonic oscillator hamiltonian are hbar omega (N + 1/2), where N is an integer greater than or equal to zero. The eigenvectors for each N are nondegenerate. The operator a acting on the Nth eigenvector produces an eigenvector with eigenvalue N-1. Similarly, the operator a^dag acting on the Nth eigenvector produces an eigenvector with eigenvalue N+1. Thus, the operator a is called a lowering operator, and a^dag is called a raising operator.