18.104.22.168 Solution schrodsol-b
For the one-dimensional harmonic oscillator, the energy eigenvalues are
Write out the coefficients for those energies.
Now classically, the harmonic oscillator has a natural frequency . That means that whenever is a whole multiple of , the harmonic oscillator is again in the same state as it started out with. Show that the coefficients of the energy eigenfunctions have a natural frequency of ; must be a whole multiple of for the coefficients to return to their original values.
The coefficients are
Now if is , the argument of the exponential equals times an odd multiple of . That makes the exponential equal to minus one. It takes until until the exponential returns to its original value one.