BioPHYSICAL chemistry

An understanding of the properties of biological molecules requires know-

ledge of quantum mechanics. This chapter presents two examples of how

Schrödinger’s equation can be used to solve problems. First, we consider

the one-dimensional particle in a box, in which a particle is trapped inside

a box with infinite walls. The second problem is the simple harmonic

oscillator. For this problem, we consider how quantum mechanics describes

vibrational motion. Using these problems as typical examples, we show

how to use Schrödinger’s equation and discuss how various properties

of objects can be described. In both cases we see that quantum theory

makes surprising predictions, such as tunneling. The usefulness of these

problems is shown by their application to understanding the properties

of conjugated molecules commonly found in biological systems, such as

carotenoids, as well as the vibrational properties of molecules.

One-dimensional particle in a box

A particle of mass mis confined between two walls at x=0 and x=L.

The potential energy is zero inside the box but rises immediately to infinity

at the walls (Figure 10.1). Because the potential is infinite

outside the box, the particle cannot be outside the box

and the wavefunction is zero there. To determine the

properties of the particle inside the box, Schrödinger’s

equation is solved. In this case, the potential energy V(x)

is zero inside of the box, so the equation can be simply

written as:

(10.1)

The solutions to this equation can be written in terms

of exponential, sine, or cosine functions. The choice of

−=

Z^22

2 mx^2 xEx

d

d

ψψ() ()

10 Particle in a box and tunneling

10 Particle in a box and tunneling

V  0

x  0 x  L

V   V  

Figure 10.1
A particle is confined
to be within a box
with a potential of
zero inside the box
and an infinite
potential outside
the box.