Concise Physical Chemistry

c16 JWBS043-Rogers September 13, 2010 11:28 Printer Name: Yet to Come

THE PARTICLE IN A ONE-DIMENSIONAL BOX 253

16.5 THE PARTICLE IN A ONE-DIMENSIONAL BOX

Facility in working quantum mechanical energy problems can be gained by going

from simple problems to more complicated ones. The problem usually chosen as the

starting point is theparticle in a one-dimensional boxof lengthl:

x=0| • |x= 1

Since there is only one dimension, the wave function is(x). The Schrodinger ̈

equation in this case is

−

h ̄^2

2 m

∇^2 (x)+V(x)=E(x)

We stipulate that the potential energy is zero inside the boxV=0 and that it is

infinite outside the box. These conditions mean that the particle cannot escape from

the box. We note that the one-dimensional operator is∇^2 =

d^2 (x)

dx^2

,so

−

h ̄^2

2 m

d^2 (x)

dx^2

=E(x)

inside the box, or

d^2 (x)

dx^2

=−

2 mE

̄h^2

(x)

which is a typical wave equation and is an eigenvalue problem.

From our knowledge of wave equations (Section 16.2), we know that a function

like(x)=Asin

2 πx

λ

will satisfy this equation. The second derivative of(x)is

d^2 (x)

dx^2

=−A

4 π^2

λ^2

sin

2 πx

λ

=−

4 π^2

λ^2

(x)

butitisalsotruethat

d^2 (x)

dx^2

=−

2 mE

h ̄^2

(x), so

4 π^2

λ^2

=

2 mE

h ̄^2

We also know that bound waves must have wavelengths that go up as half-integers

of the limits placed on the oscillatory excursion—that is, the length of a vibrating

guitar string (Fig. 16.1) or the lengthlof a box. Wavelengths allowed by the boundary

conditions areλ=^1 nλfundamental(Section 16.1), whereλfundamental= 2 l. It follows that