Physically, (27) corresponds to a source current of
one particle impinging on the
step per unit time, which then results in a current of
particles reflected per unit time and
particles transmitted per unit
time. The probabilities
of reflection and transmission are thus respectively 
and 
.
We now have two cases to consider. In the case where 
,
both k and 
are real numbers. We then have,
It is always good practice to check that the transmission and reflection probabilities sum to unity. In this case, we have
In the case where 
, the solution in Region t is a decaying
exponential. Such a function is a real function and the expression
for the current 
gives
zero when 
is a real function. Thus 
. For 
we must
take care to note that the wave vector in Region t is now imaginary,
, where
We verify our choice of by noting that with this choice,
, a proper exponentially decaying solution.
With determined in this way, we then have
Thus, also when , we find 
.
Our full formula for the reflection probability may be written out as
where T is the kinetic energy of the incoming particle. This quantity sets the basic energy scale for the problem. Plotting the scattering probabilities in terms of V/T, which is the strength of the potential in terms of the kinetic energy of the incoming particles, gives the results in Figure 11.
Figure 11: Scattering probabilities from a potential step of height
Vo as a function of Vo/T, where T is the kinetic
energy of the incoming particles.
First we notice, that as expected, when there is no step ( 
),
there is perfect transmission, 
and 
. For negative
values of 
, we have a downward step, and surprisingly (in
classical terms) there is a non-zero probability of reflection. In
fact, as 
, 
, and nearly
all particles are reflected. From the wave-propagation point of view,
this is not surprising. The k's describe the nature of propagation
of the wave in the different regions. A large step 
results is a
large difference in how the waves propagate in each region, which
generates a large reflection.
For 
, we are studying reflections from an upward
step in potential. As increases 
increases from zero
until we reach the special
point where 
. After this point, the kinetic energy of the
incoming particle is less than the height of the potential barrier, so that
as we have seen, the reflection probability becomes 
.
