S-Matrix Solution for the Forced Harmonic Oscillator

The exact solution for a forced, undamped quantum harmonic oscillator is obtained by S-matrix techniques. The force $F(t)$ is assumed to vanish at $t = ±∞$⁠. The only restriction is that the Fourier transform of $F(t)$ must exist. The transition probabilities are obtained in closed form in terms of Laguerre polynomials. The mean energy transfer to the oscillator is found to be independent of the initial state and is in agreement with the classical result for an oscillator originally at rest. This problem provides a good example of field theoretical procedures in an elementary context. Therefore, all field theoretical concepts are carefully defined and explained as they are introduced in order that the discussion may be self-contained.