BioPHYSICAL chemistry

In this chapter the properties of vibrational motion are described. First,
the classical treatment of the simple harmonic oscillator is reviewed. The
classical theory is followed by the equivalent analysis of vibrational motion
using Schrödinger’s equation. This chapter has a detailed derivation of the
equation and you are encouraged to work through the analysis to get a
better understanding of how to solve problems in quantum mechanics. After
the derivation, the various properties of the simple harmonic oscillator are
discussed, such as the calculation of expectation values. The usefulness of
these concepts is demonstrated by examining how the structure and func-
tion of proteins can be probed by measuring their vibrational properties
using infrared spectroscopy.

Simple harmonic oscillator: classical theory

Classically, when a mass, m, is attached to an immovable object by a
spring, the mass undergoes harmonic motion; that is, the mass will
vibrate back and forth as the spring alternatively pulls and pushes on
the mass (Figure 11.1). This motion is expressed using the classical laws
by describing the properties of the spring with the spring constant, k. The
stiffer the spring the larger the spring constant will be and the faster the
mass will vibrate. The motion is described using Newton’s laws. As the mass
stretches the spring, the spring exerts a force,
F, in the opposite direction to the motion.
Since the force opposes the motion the mass
slows down until it reaches the maximal
extension, or amplitude, A. The mass does not
stay at that position, owing to the force that
pulls the mass back to the center. When the

11 Vibrational motion and infrared spectroscopy

11 Vibrational motion and infrared spectroscopy

AA

k

m

Figure 11.1Classical

model for the simple

harmonic oscillator

showing the

amplitude of motion,

A, the mass, m, and

the spring constant, k.