Boundary Conditions

Abstract

In this chapter we describe the elastic reflection of particles in the presence of impenetrable obstacles. Instead of describing walls and obstacles by electrostatic forces (which would have to be infinitely strong and concentrated on the surface of the obstacle), it is more appropriate to interpret an impenetrable barrier as a boundary condition. Starting with the simplest example—a solid wall in one dimension—we discuss Dirichlet boundary conditions, which exert a strongly repulsive influence, and Neumann boundary conditions, which are more neutral toward the particle.

A very interesting problem is the description of particles in a box. The surrounding walls confine the particle for all times to a finite region. Thus, the behavior of a particle in a box is quite different from the free motion. Instead of propagating wave packets we find an orthonormal basis of stationary states, which can be described as eigenvectors of the Hamiltonian operator. As a consequence, the quantum-mechanical energy of a particle in a box cannot have arbitrary values. The only possible energies are given by a discrete set of eigenvalues of the Hamiltonian operator—a fact that cannot be understood by classical mechanics. In particular, the lowest possible energy (the energy of the ground state) is greater than zero, that is, a confined particle is never really at rest. By forming superpositions of eigenstates, we can describe the motion of arbitrary initial states. The motion is always periodic in time and can be very complicated, as illustrated by the mathematically interesting example showing the unit function in a Dirichlet box.

The accompanying CD-ROM contains many movies of wave packets hitting walls and obstacles in various geometric configurations. Of particular interest is the double slit—a wall with two holes through which the particle can reach the other side. Behind the wall, the wave function shows a nice interference pattern which vanishes as soon as one of the slits is closed. More generally, one can say that the interference vanishes as soon as one attempts to determine through which of the holes the particle actually goes. We use this behavior to illustrate once more how quantum mechanics contradicts the classical picture of localized particles.