Concise Physical Chemistry

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

THE PARTICLE IN A CUBIC BOX 255

n12 3

Pnx( ) sin n x()()^2 n^2

2

4

6

8

10

P(1 x)

P(2 x)

P(3 x)

x

FIGURE 16.3 AMathcad©Csketch of the born probability densities at the first three levels

of the particle in a box. The lowest wave function has no internal nodes, the second wave

function has one, and the third has two.

nodes of a square plate or membrane. It can be solved in 3-space to produce solutions

to the vibratory motions in a solid cube. Other geometries are possible such as a

circle, rectangle, parallelepiped, cylinder, and so on. Each problem lends insight and

is recommended to the interested reader.

16.6 THE PARTICLE IN A CUBIC BOX

In the case of a particle confined to the interior of a cube, we have a three-dimensional

wave equation of the form

−

h ̄^2

2 m

∇^2 (x,y,z)+V(x,y,z)=E(x,y,z)