Simple quantum systems in the momentum representation

Abstract

The momentum representation is little used in quantum mechanics courses. Some students are hence surprised by the change in viewpoint when, in doing advanced work, they have to use the momentum rather than the coordinate representation. In this work, we give an introduction to quantum mechanics in momentum spate. To this end we discuss standard "elementary" problems, namely, the free particle, the quantum motion under a constant potential function, a particle interacting with a potential step function, and the motion of a particle under a oscillatory potential function. What is not standard is that they are all conceived from momentum space and hence they, with the exception of the free particle, are not equivalent to the coordinate space ones with the same names. We solve for the bound eigenstates of the problems working within the momentum representation trying not to make explicit reference to the systems they correspond to in the coordinate representation.