Wave Mechanics

Until now we have represented quantum mechanical plane waves by sine and cosine functions, just as with other types of waves. However, plane matter waves cannot be truly represented by sines and cosines. We need instead mathematical functions in which the wave displacement is expressed in terms of complex rather than real numbers. This requires the introduction of a bit of new mathematics, which we tackle first. Then we show how this new math is used to represent matter waves, first for free particles, and then for particles subject to a conservative force represented by a potential energy. We then examine an important special case in which particles are confined to a particular region.

Particle confinement can happen in a number of ways. We first look at the so-called ``particle in a box'' in one spatial dimension. We find that confined particles can take on only discrete energy values. When confinement isn't perfect we see how a quantum mechanical particle can leak through a potential energy barrier which is classically impenetrable. Movement of a particle on a circular ring leads us to another form of confinement and the introduction of angular momentum. This brings us finally to a discussion of the intrinsic or spin angular momentum of elementary particles.