Concise Physical Chemistry

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

254 WAVE MECHANICS OF SIMPLE SYSTEMS

2

4

6

8

10

sin(x) 1

sin(2x) 4

sin(3x) 9

x

FIGURE 16.2 Wave forms for the first three wave functions of the particle in a box. The

waves are drawn at the lowest three energy levels:n= 1 , 2 ,3; (n^2 = 1 , 4 ,9).

λ=^2 nl. A little algebra goes from

−

4 π^2

( 2

nl

) 2 =−

2 mE

̄h^2

to the energyspectrum(Fig. 16.2):

E=

n^2 ̄h^2 π^2

2 ml^2

=

n^2 h^2

8 ml^2

Except for the lowest one, each wave function has internal values ofxornodes

at which(x)=0. The Born probability density|(x)|^2 of finding the particle

precisely at one of these nodes is zero. Not counting the nodes at the extremities of

the wave function, the number of internal nodes goes up as 0, 1, 2,.... Each internal

node in the wave function(x) yields an internal zero point shown as a minimum in

the probability function in Fig. 16.3. There aren−1 internal nodes.

By elaboration of the methods used for the particle in a one-dimensional box,

the problem can be solved in two dimensions to produce solutions for the vibratory