Let us assume that the potential barrier Uis high relative to the energy Eof the
incident particles. If this is the case, then k 2 k 1 k 1 k 2 and (5.96)Let us also assume that the barrier is wide enough for (^) IIto be severely weakened
between x0 and xL. This means that k 2 L1 and
ek^2 L ek^2 L
Hence Eq. (5.95) can be approximated by
e
(ik 1 k 2 )L (5.97)
The complex conjugate of (AF), which we need to compute the transmission prob-
ability T, is found by replacing iby iwherever it occurs in (AF):
- e(ik^1 k^2 )L (5.98)
Now we multiply (AF) and (AF) to give
e^2 k^2 L
Here IIIIso III 1 1 in Eq. (5.83), which means that the transmission
probability is
T
1
e^2 k^2 L (5.99)
From the definitions of k 1 , Eq. (5.77), and of k 2 , Eq. (5.86), we see that
2
1 (5.100)
This formula means that the quantity in brackets in Eq. (5.99) varies much less with
Eand Uthan does the exponential. The bracketed quantity, furthermore, always is of
the order of magnitude of 1 in value. A reasonable approximation of the transmission
probability is therefore
Te^2 k^2 L (5.101)
as stated in Sec. 5.10.
Approximate
transmission
probability
U
E
2 m(UE) 2
2 mE 2
k 2
k 1
16
4 (k 2 k 1 )^2
AA
FF
FFIII
AAI
Transmission
probability
k^22
16 k^21
1
4
AA
FF*
ik 2
4 k 1
1
2
A
F
ik 2
4 k 1
1
2
A
F
k 2
k 1
k 1
k 2
k 2
k 1
196 Appendix to Chapter 5
bei48482_ch05.qxd 1/17/02 12:17 AM Page 196