BioPHYSICAL chemistry

Note that the second derivate of x yields x multiplied by a constant.
Substituting this into eqn 11.3 yields:

(11.7)

So the mass oscillates with a frequency ωthat is equal to the square root
of the spring constant divided by the mass. The frequency will be slower
for stiffer springs or for larger masses.

Potential energy for the simple harmonic oscillator

To solve Schrödinger’s equation for this problem, we must first determine
the classical expression for the potential energy, V. Classically, for a given
force, the potential energy is given by the integral of the force over the
displacement:

(11.8)

Thus, the potential energy of the simple har-
monic oscillator is parabolic. When there is
no displacement xhas a value of zero and the
potential energy is zero (Figure 11.2). As the
spring is either expanded or compressed, the
potential energy increases and the mass begins
to slow down. When the mass reaches the point
where the kinetic energy is zero and all of the
energy is potential energy, the mass stops and
turns back toward the center of the motion.

Simple harmonic oscillator: quantum theory

Using the potential energy for the harmonic
oscillator (eqn 11.8), Schrödinger’s equation
can written as:

(11.9)

The solutions of this equation are a series of func-
tions called Hermite polynomials, with energies

−+=

Z^22

2

2

22 mx

x

k

xx Ex

d

d

ψψψ() () ()

VFx kxxkxx
k
=−∫∫ddd=− −∫()==x
2

2

mxkx
k
m

()−=−ωω^2 or =

CHAPTER 11 VIBRATIONAL MOTION 223

0

Displacement, x

Potential energy,

v

Figure 11.2The potential energy of a simple

harmonic oscillator has a parabolic dependence

on the displacement of the mass.