bei48482_FM

The Solid State 373

waves. For clarity we consider electrons moving in the xdirection; extending the

argument to any other direction is straightforward. When ka, as we have

seen, the waves are Bragg-reflected back and forth, and so the only solutions of

Schrödinger’s equation consist of standing waves whose wavelength is equal to the

periodicity of the lattice. There are two possibilities for these standing waves for

n1, namely,

 1 A sin (10.23)

 2 A cos (10.24)

The probability densities  1 ^2 and  2 ^2 are plotted in Fig. 10.45. Evidently  1 ^2

has its minima at the lattice points occupied by the positive ions, while  2 ^2 has its

x

a

x

a

• 4 π
  a
  
  
  • 3 π
    a
    
    
    • 2 π
      a
      
      
      • π
        a
      
      
      
      
    
    
    
    
  
  
  
  

0 π

a

2 π

a

3 π

a

4 π

a

k

E

Allowed

energies

Forbidden

energies

E =^

(^2) k 2
2 m
h
Figure 10.44Electron energy Eversus wave number kin the kxdirection. The dashed line shows how
Evaries with kfor a free electron, as given by Eq. (10.22).
(b)
x
|
2 |^2
x
(a)
|
1 |^2
Figure 10.45Distributions of the probability densities  1 ^2 and  2 ^2.
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