A New Description of Quantum Behaviors for a Simple Harmonic Oscillator

We describe quantum behaviors of a simple harmonic oscillator, starting from the classical mechanics. By imposing two conditions on the phase points generated from a symplectic algorithm, we obtain discrete energy levels, satisfying $E_n \tau_n = h$ where $E_n$ is the total energy of the oscillator and $\tau_n$ is the time step for the closed orbit of $n$-polygon in phase space. We can thus successfully integrate classical and quantum mechanics into a single frame, if we assume that time is discrete.