6.1 The Harmonic Oscillator Potential

We know from our experience with classical mechanics that a particle of mass m subject to a linear restoring force Fx = ? ? x, where ? is the force constant, results in one-dimensional simple harmonic motion with an oscillation frequency. . The potential the particle moves in is quadratic V( x) = ?x ²/2, and so in this case the potential has a minimum at position x = 0. The idea that a quadratic potential may be used to describe a local minimum in an otherwise more complex potential turns out to be a very useful concept in both classical and quantum mechanics. An underlying reason why it is of practical importance is that a local potential minimum often describes a point of stability in a system. For example, the positions of atoms that form a crystal are stabilized by the presence of a potential that has a local minimum at the location of each atom. If we wish to understand how the vibrational motion of atoms in a crystal determines properties such as the speed of sound and heat transfer, then we need to develop a model that describes the oscillatory motion of an atom about a local potential minimum. The same is true if we wish to understand the vibrational behavior of atoms in molecules.

As a starting point for our investigation of the vibrational properties of atomic systems, let us assume a static...