Unitary transformation for the system of a particle in a linear potential

A unitary operator, which relates the system of a particle in a linear potential with time-dependent parameters to that of a free particle, has been given. This operator, closely related to the one which is responsible for the existence of coherent states for a harmonic oscillator, is used to find a general wave packet described by an Airy function. The kernel ( propagator) and a complete set of Hermite-Gaussian-type wave functions are also given.