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In the previous example, the quantum particle could penetrate for some
distance through walls which were
infinitely thick. What would happen if a quantum particle were to propagate towards walls
which were not infinitely thick ? The particle in fact has a finite
probability of passing through such a
wall. This is known as the tunneling effect, and it is a purely quantum phenomenon.
Figure 16 depicts a particle incident from the left approaching a barrier.
The height of the
barrier is larger than the energy of the particle, . Some of the amplitude of the
quantum particle is reflected, while some passes through the barrier (nonclassical behaviour),
and emerges to continue travelling towards the right on the other side.
Figure 16:
A particle with an energy below the barrier height approaches the barrier. Some
amplitude is reflected, and some amplitude tunnels through to the other side.

The potential is :

(87) 
It is clear, as in the previous problem, that in the potential free areas we will
have oscillatory solutions, while in the area where the particle energy is less than the barrier
height , we will have exponential solutions.
Using the identities
we can write the oscillatory solution as
where

(92) 
This is a more suitable form to describe wave functions moving in a
particular direction. For example,

(93) 
is a wave function in region I incident from the left. At , the wave
strikes the barrier and is partially reflected, as the
wave function

(94) 
A classical particle would rebound off the barrier completely, not partially. However,
we already know that some of the amplitude of the wave function will
penetrate into the barrier in the classically nonallowed region II.
Inside the barrier (region II from ), we know that , and
therefore the Schrödinger wave equation will have not have
oscillatory solutions, but exponential solutions.

(95) 
where

(96) 
If the barrier is not infinitely thick, then some amplitude may still
remain on the far side of the barrier.
On the far side of the barrier, in region III (), the transmitted wave would again be
moving towards the right.

(97) 
Applying the boundary conditions, we demand that the wave functions in
all three regions match smoothly to each other, in both value and slope.

(98) 
at the left hand side of the barrier and

(99) 
Specifying these conditions, we find we must solve the equations
Considering figure 16, we can see that the interesting
quantity
to calculate is the amount of transmitted wave amplitude relative to
the incident amplitude.
This is know as the transmission coefficient
.
(The steps to realise the transmission coefficient from here on
are presented for your interest, and are not examinable.)
If the barrier is much higher than the particle energy, then
so that

(102) 
Suppose also the barrier is wide enough so that the transmitted
amplitude is weak

(103) 
We can now simplify the expression for the transmission probability

(104) 
finally

(105) 
As the bracketed quantity is more slowly varying than the exponential,
we can simplify the transmission probability further to

(106) 
The transmission of a quantum particle through a barrier is shown in
figure 17 below.
Figure 17:
A quantum particle tunneling through a barrier.

Next: Applications of ``tunneling''
Up: Simple Quantum Systems
Previous: Applications of the ``particle
Simon Connell
20060221