### Abstract:

A classical system of two masses connected by a spring and subjected to a time dependent
pulse is used to model a kicked quantum harmonic oscillator. This model
is of interest because it is related to the M¨ossbauer effect where the kick is due to
the emission of a photon [1]-[3]. Furthermore, the model can be solved analytically
and results in coherent states. The solution to this model is found by writing the
Schr¨odinger equation in terms of center of mass and relative coordinates [4]. The
wave function is found after performing two extended Galilean transformations [5]
and applying separation of variables [6]. The force applied to the system is specified
as a Gaussian pulse, and the expectation value of the energy is calculated to agree
with Ehrenfest’s theorem [4]. The probability that the system is in the unperturbed
harmonic oscillator state is found to have a form that indicates coherent states
[7]. We find that any force acting on the system in a stationary state will produce
coherent states once the force has gone to zero, except in the case where the force is
turned off at the moment when the oscillator is in its equilibrium position with zero
velocity. In this case, the wave function returns to the original stationary state. We
further show that the wave packet of the system keeps its shape with time, which is
another indicator of coherent states for the harmonic oscillator. Finally, we examine
the case of a delta-function pulse acting on the system for the harmonic oscillator
potential as well as a general potential. We find a form for the wave function of the
harmonic oscillator case using three techniques and determine that the wave packet
does not keep its shape for the general potential case.