All the eigenvectors of the harmonic oscillator hamiltonian can be created by repeatedly acting on the ground state with the raising operator, a^dag. Moreover, because a^dag acting on the nth state is proportional to the (n+1)th state, the only nonzero elements of a^dag in a matrix representation are below the diagonal. Similarly, the only nonzero elements of the lowering operator, a, are above the diagonal. Since the raising and lowering operators are linearly related to the position and momentum operators, the position and momentum expectation values can be computed easily using a and a^dag. One finds that the position and momentum expectation values of a general wave function oscillate sinusoidally as for the classical harmonic oscillator. For the nth eigenvector, the expectation value for the position is zero, and the root mean square position, Delta x, is the classical maximum amplitude divided by the square root of 2.