Exact Quantum Theory of the Harmonic Oscillator with the Classical Solution in the Form of Mathieu Functions

Using the dynamical invariant operator method, we obtain the exact wave function, uncertainty relation, and energy eigenvalues for the harmonic oscillator with the classical equation of motion in the form of Mathieu functions. The probability density varies as a function of position, but is almost constant in time. The uncertainty relations satisfy the minimum uncertainty, and the energy eigenvalues oscillate slowly or rapidly depending on the frequency. The quantum and classical energies oscillate in a similar fashion with respect to frequency and time. KCI Citation Count: 10