Page 65 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

56 Schro Èdinger wave mechanics

where again equation (A.46) has been used. Combining this result with
equations (2.52) and (2.53), we ®nd that
2 2 2 2
jBj (á ‡ â ) 2
R ˆ T ˆ T sinh âa
2 2
jFj 2 4á â
Substitution of equations (2.48), (2.51), and (2.58) yields
V 2 p 
2
0 sinh ( 2m(V 0 ÿ E) a=")
4E(V 0 ÿ E)
R ˆ 2 (2:59)
V p 
2
1 ‡ 0 sinh ( 2m(V 0 ÿ E) a=")
4E(V 0 ÿ E)
The transmission coef®cient T in equation (2.58) is the relative probability
that a particle impinging on the potential barrier tunnels through the barrier.
The re¯ection coef®cient R in equation (2.59) is the relative probability that
the particle bounces off the barrier and moves in the negative x-direction. Since
the particle must do one or the other of these two possibilities, the sum of T
and R should equal unity
T ‡ R ˆ 1
which we observe from equations (2.58) and (2.59) to be the case.
We also note that the (relative) probability for the particle being in the region
0 < x < a is not zero. In this region, the potential energy is greater than the
total particle energy, making the kinetic energy of the particle negative. This
concept is contrary to classical theory and does not have a physical signi®-
cance. For this reason we cannot observe the particle experimentally within the
potential barrier. Further, we note that because the particle is not con®ned to a
®nite region, the boundary conditions on the wave function have not imposed
any restrictions on the energy E. Thus, the energy in this example is not
quantized.
In this analysis we considered the relative probabilities for tunneling and
re¯ection for a single particle. The conclusions apply equally well to a beam of
particles, each of mass m and total energy E, traveling initially in the positive
x-direction. In that case, the transmission coef®cient T in equation (2.58) gives
the fraction of incoming particles that tunnel through the barrier, and the
re¯ection coef®cient R in equation (2.59) gives the fraction that are re¯ected
by the barrier.
If the potential barrier is thick (a is large), the potential barrier is high
compared with the particle energy E (V 0 E), the mass m of the particle is
large, or any combination of these characteristics, then we have

1 e âa
sinh âa ˆ (e âa ÿ e ÿâa )
2 2