Note on the Forced and Damped Oscillator in Quantum Mechanics

The wave equations for the forced, damped, and forced and damped oscillators are solved in closed form for an arbitrary forcing function; the solutions produced being in one-to-one correspondence with the stationary states of the unforced, undamped Hamiltonian H 0 . The quantal motion is closely connected with the classical: for the forced oscillator the probability density is that of H 0 but moves as a whole with the classical motion; for the forced and damped oscillator this motion is accompanied by a contraction progressing eventually into a delta function at the classical position. Transition probabilities between states of H 0 are computed in the case of forced motion and depend solely on the classically acquired energy of the oscillator at any time. The transition probability vanishes strictly only when this energy has a value falling at the roots of a Laguerre polynomial associated with the transition. The classical dipole radiation emitted by a disturbed oscillator is, when the damping force is identified with the force of radiation damping, that of the classical oscillator: a shifted and broadened line.