How does a quantum harmonic oscillator oscillate?

[closet mathemetician]

A classical harmonic oscillator follows a smooth, sinusoidal path of oscillation. Since on a quantum level energy levels are discrete, does a quantum harmonic oscillator actually oscillate in the everyday sense?

 

[tom.stoer]

does a quantum harmonic oscillator actually oscillate in the everyday sense?

No, it doesn't.

In classical physics x=x(t) is moving around. In qm what is moving around with the classical frequency is the qm operator x(t) in the Heisenberg picture.