Tunneling

4
Substituting, we can calculate the transmission coefficient T,
T = S tr
S in
= v trψ * tr ψ tr
v in ψ * in ψ in
=
T = F A
k 1
m F * e−ik 1x Fe
ik 1 x
k 1
m A * e−ik 1 x Ae ik 1 x
2
.
Using the relations obtained from matching the wavefunctions and first derivatives at the boundaries, the ratio of
F/A may be determined. For this system, the transmission coefficient is
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T = F A
2
,
or T =
1 +
1
sinh 2 .
( k 2 L)
4E ⎛⎛
1 − E ⎞⎞
⎜⎜ ⎟⎟
V 0 ⎝⎝ V 0 ⎠⎠
In this equation, the function sinh is the hyperbolic sine function. It is defined
€
sinhbx = ebx − e −bx
.
2
Though not appearing in this expression, the hyperbolic cosine function, cosh, is defined in a similar fashion,
€
coshbx = ebx + e −bx
.
2
For values of the energy less than V 0 , the transmission coefficient only gets large for energies close to the top of the
barrier, as illustrated in Figure 5.
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€
0.020
0.018
T
0.016
0.014
0.012
0.010
0.008
0.006
0.004
0.002
0.000
0.0 0.2 0.4 0.6 0.8 1.0 1.2
E/V 0
Figure 5. Transmission coefficient for a particle tunneling through a finite barrier in one dimension.