Page 115 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

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Harmonic oscillator

In this chapter we treat in detail the quantum behavior of the harmonic
oscillator. This physical system serves as an excellent example for illustrating
the basic principles of quantum mechanics that are presented in Chapter 3. The
È
Schrodinger equation for the harmonic oscillator can be solved rigorously and
exactly for the energy eigenvalues and eigenstates. The mathematical process
for the solution is neither trivial, as is the case for the particle in a box, nor
excessively complicated. Moreover, we have the opportunity to introduce the
ladder operator technique for solving the eigenvalue problem.
The harmonic oscillator is an important system in the study of physical
phenomena in both classical and quantum mechanics. Classically, the harmonic
oscillator describes the mechanical behavior of a spring and, by analogy, other
phenomena such as the oscillations of charge ¯ow in an electric circuit, the
vibrations of sound-wave and light-wave generators, and oscillatory chemical
reactions. The quantum-mechanical treatment of the harmonic oscillator may
be applied to the vibrations of molecular bonds and has many other applica-
tions in quantum physics and ®eld theory.

4.1 Classical treatment

The harmonic oscillator is an idealized one-dimensional physical system in
which a single particle of mass m is attracted to the origin by a force F
proportional to the displacement of the particle from the origin
F ˆÿkx (4:1)
The proportionality constant k is known as the force constant. The minus sign
in equation (4.1) indicates that the force is in the opposite direction to the
direction of the displacement. The typical experimental representation of the
oscillator consists of a spring with one end stationary and with a mass m

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