10 - One-Dimensional Piecewise Constant Potentials

Summary

This chapter solves the one-dimensional Schrödinger equation in a potential which (i) is constant everywhere or (ii) jumps discontinuously from one constant value to another at a finite number of points. The importance of studying such simple potentials lies in the fact that the motion of a particle in them exhibits certain exclusive quantum features, such as the possibility of finding the particle in the classically forbidden region or tunnelling through it. They also idealize several realistic potentials and provide useful insight in to the properties of the motion of a particle in realistic situations.

Constant Potential

Consider first the potential which has the same value on the entire real axis. A particle moving in such a potential would not experience any force and hence propagate freely. Without loss of generality, we take the potential to be zero. If the mass of the particle is m, then its wave function of definite energy E is the solutions of the Schroödinger equation (9.1) corresponding to U(x) = 0:

Recall from Section 8.6 that the energy of the particle cannot be less than the global minimum of the potential in which it moves. Since the potential in the present case is zero everywhere, we must have. The two linearly independent solutions of the equation above then read