Time dependent quantum mechanics of systems from one to three dimensions

Donald Gary Swanson

In the previous chapter, the postulates helped us to use the wave function to find the expected results of experimental measurements by letting a dynamical quantity F (x, p, t) → F (xˆ, pˆ, t) and integrating over space. In the examples, however, the wave functions were typically given and not derived from first principles. We now endeavor to find the appropriate wave functions for a variety of physical systems in one dimension by solving the Schro¨dinger equation where for time-independent cases, the total energy is an eigenvalue. In subsequent chapters, we will extend the method to three dimensions and develop the appropriate operators, eigenvalues, and eigenfunctions for angular momentum. We will also develop methods for examining time-dependent systems. The latter chapters address special systems of interest that may be referred to as applications of quantum mechanics.

James P. Lavine

Gary Bowman

The time evolution of an arbitrary one-dimensional initial quantum state, subject to a time-independent, spatially uniform classical force, is shown to be that of a free-particle state, plus an overall motion arising from the classical force. Bohmian mechanics is then used to extend this result to an arbitrarily time-dependent, uniform force in three dimensions. The resulting solution is completed and confirmed using conventional quantum mechanics.

Gary E. Bowman

The time evolution of an arbitrary one-dimensional initial quantum state, subject to a time-independent, spatially uniform classical force, is shown to be that of a free-particle state, plus an overall motion arising from the classical force. Bohmian mechanics is then used to extend this result to an arbitrarily time-dependent, uniform force in three dimensions. The resulting solution is completed and confirmed using conventional quantum mechanics.

R. S. Kaushal