Wave Propagation in Linear Media | re-examined

2.2 Quantum mechanical tunnelling
e ect. In quantum mechanics, a particle is described in terms of its corresponding complex
wave function (x; t), which satis es a particular wave equation, the Schrodinger equation:
~ 2
2m
+ j~ @
@t
, V =0: (2.10)
In this equation, ~ = h=(2 ) is Planck's quantum of action, m is the mass of the particle, and
V is a usually time-invariant potential. In the absence of transient processes the probability
density j (x; t)j 2 is independent of time, and one can obtain a simpler version by separating
the variables and setting
(x; t) = (x)e ,j!t ; (2.11)
which gives the time-independent Schrodinger equation or energy eigenequation
~ 2
2m
+ E ,V =0; (2.12)
where E = !~ is the energy of the particle. Obviously, this equation has one-dimensional
solutions of the form
(x) =Ae
j 1
~ xp 2m(E,V) : (2.13)
The amplitude A is a normalisation constant and chosen such that the overall probability density
of the particle equals one, which means that the particle is guaranteed to be somewhere.
Hence,
Z 1
,1
(x) (x) dx =1: (2.14)
pThe case we are interested in occurs whenever E < V , so that the wave number k =
2m(E , V )=~ becomes imaginary. Then the oscillation of the stationary wave turns into an
exponential decay and we have an e ect similar to evanescence in the electromagnetic case.
The simplest and classical arrangement leading to the tunnel e ect is a rectangular potential
barrier with
V =
8
><
>:
0 if ,1