Lewis-Riesenfeld approach to the solutions of the Schrödinger equation in the presence of a time-dependent linear potential

We reexamine the general solution of a Schrödinger equation in the presence of a time-dependent linear potential in configuration space based on the Lewis-Riesenfeld framework. For comparison, we also solve the problem in momentum space and then Fourier transform the solution to get the general wave function. Appropriately choosing the weight function in the latter method, we can obtain the same wave function as the former method. It is found that a non-Hermitian time-dependent linear invariant can be used to obtain Gaussian-type wave-packet solutions of the time-dependent system. This operator is a specific linear combination of the initial momentum and initial position operators. This fact indicates that the constants of integration such as the initial position and initial momentum that determine the classical motion play important roles in the time-dependent quantum system. The eigenfunction of the linear invariant is interpreted as a wave packet with a "center of mass" moving along the classical trajectory, while the ratio between the coefficients of the initial position and initial momentum determines the width of the wave packet.