Concise Physical Chemistry

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

THE PARTICLE IN A CUBIC BOX 257

x

y

z

FIGURE 16.4 The ground state orbital of a particle confined to a cubic box.

16.6.1 Orbitals

The ground state probability density has a maximum in the center when viewed from

each of the three dimensions (Fig. 16.4). Thus there is a high probability density at the

center of the cube diminishing symmetrically in all directions (spherical symmetry).

This is the geometry of ansatomicorbital.

16.6.2 Degeneracy

If the particle is excited to then=2 state in thexdirection while its motion in both

theyandzdirections remain at the lowest energy, the resulting orbital has an internal

node in thexdirection but no nodes in theyandzdirections (Fig. 16.5). This gives

the geometry of the firstporbital which we denotepx. What was said for thex

direction can be said for theyandzdirections, so we have three orbitals with energies

that are identical. Different orbitals with the same energy are said to bedegenerate

(Fig. 16.6). Perhaps we remember from elementary chemistry that theporbitals of

hydrogen are three-fold degenerate.

16.6.3 Normalization

One can also find the constantAin(x)=Asin

2 πx

λ

bynormalizingthe wave

function. Normalization requires setting the integral of^2 over all space equal to 1.0

x

y

z

FIGURE 16.5 The first excited state of a particle confined to a cubic box.