Motion in One Dimension

In this chapter, we will study the quantum mechaincs of one-dimensional systems. We study such problems for two reasons. Not only is it interesting to study the simplest cases to demonstrate some of the general properties of quantum mechanical systems, but it also turns out that there are two- and three-dimensional systems that can be reduced to effective one-dimensional problems, such as a particle in a central potential (also see the problem below).

We will discuss both bound and scattering states of one-dimensional potentials, thus illustrating the basic features of each, such as the discrete energy spectrum of the bound states, as opposed to the continuous scattering state spectrum. We also give an introduction to scattering from one-dimensional potentials, and show how to calculate the probabilities of transmission and reflection from such a potential. Finally, we treat two special cases, namely the Dirac delta function potential and periodic potentials, the latter of which will introduce the concept of energy bands, a topic of great importance in, for example, the study of the electronic properties of crystalline solids.

Problem

(Based on Problem 3.19 in Schaum's Theory and Problems of Quantum Mechanics)

Consider a particle of mass $m\!$ in a three dimensional potential of the form, $V(x,y,z) = X(x)+Y(y)+Z(z).\!$ Show that we can treat the problem as three independent one-dimensional problems. Relate the energy of the three-dimensional state to the effective energies of the one-dimensional problems.