First consider the simpler bosonic case of the photons of the quantum harmonic oscillator. Start with the Schrödinger equation for the one-dimensional time independent quantum harmonic oscillator, ( − − ℏ ℏ 2 2 m d 2 d x 2 + 1 2 m ω ω 2 x 2 ) ψ ψ ( x ) = E ψ ψ ( x ) .