Page 63 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

54 Schro Èdinger wave mechanics
2
d ø(x) 2m
ˆ (V 0 ÿ E)ø(x) (2:49)
dx 2 " 2
for which the general solution is

ø II ˆ Ce âx ‡ De ÿâx (2:50)
where C and D are integration constants and â is the abbreviation
p 
2m(V 0 ÿ E)
⠈ (2:51)
"
The term exp[iáx] in equations (2.47) indicates travel in the positive x-
direction, while exp[ÿiáx] refers to travel in the opposite direction. The
coef®cient A is, then, the amplitude of the incident wave, B is the amplitude of
the re¯ected wave, and F is the amplitude of the transmitted wave. In region
III, the particle moves in the positive x-direction, so that G is zero. The relative
probability of tunneling is given by the transmission coef®cient T
2
jFj
T ˆ 2 (2:52)
jAj
and the relative probability of re¯ection is given by the re¯ection coef®cient R
2
jBj
R ˆ (2:53)
jAj 2
The wave function for the particle is obtained by joining the three parts ø I ,
ø II , and ø III such that the resulting wave function ø(x) and its ®rst derivative
ø9(x) are continuous. Thus, the following boundary conditions apply
ø I (0) ˆ ø II (0), ø9 I (0) ˆ ø9 II (0) (2:54)

ø II (a) ˆ ø III (a), ø9 II (a) ˆ ø9 III (a) (2:55)
These four relations are suf®cient to determine any four of the constants A, B,
C, D, F in terms of the ®fth. If the particle were con®ned to a ®nite region of
space, then its wave function could be normalized, thereby determining the ®fth
and ®nal constant. However, in this example, the position of the particle may
range from ÿ1 to 1. Accordingly, the wave function cannot be normalized,
the remaining constant cannot be evaluated, and only relative probabilities such
as the transmission and re¯ection coef®cients can be determined.
We ®rst evaluate the transmission coef®cient T in equation (2.52). Applying
equations (2.55) to (2.47 b) and (2.50), we obtain
Ce âa ‡ De ÿâa ˆ Fe iáa

â(Ce âa ÿ De ÿâa ) ˆ iáFe iáa
from which it follows that