The potential barrier according to quantum mechanics

We require for this problem an initial state which corresponds as
closely as possible to the classical situation, i.e. a particle on the
right of the barrier and moving towards it with a velocity v. To this
end we take a wavefunction with a magnitude that is peaked in the
neighbourhood of the initial position and with an angular variation
such that the average velocity is equal to v. There will of course
be an uncertainty in both the position and the velocity, according
to equation (2.1). A possible form for the square of the magnitude,
which we recall is proportional to the probability, is shown in figure
ll(a). Since we are dealing with one particle the area under this
peak will be equal to one.
The Schrodinger equation now determines the subsequent
behaviour of this wavefunction. We shall not discuss the method
of solving the equation but merely state the results. The peak in the
wavefunction moves towards the barrier with a velocity approximately
v-this is very similar to the classical motion of a particle
where there are no forces. There is, in addition, a small increase in
the width of the peak, so the situation at a later time is shown in
figure 1 l(b). When the peak reaches the barrier, where the effect of
the force begins to be felt, it spreads out more rapidly and then
splits into two peaks, as seen in figure 1 l(c). These two peaks then
move away from the barrier in opposite directions, so a little later we have the situation shown in figure 1 l(d). Our wavefunction has
separated into two peaks, one reflected and one transmitted by the
barrier.
It is a consequence of the Schrodinger equation that, throughout
the motion, the total area under the graph of the square of the length of the wavefunction remains equal to one. In fact we know
that this has to be true for consistency with the probability
interpretation-the particle always has to be somewhere. The probability
that it is on the right of the barrier, i.e. that it has been
reflected, is given by the area under the right-hand peak, whereas
the probability for transmission is given by the area under the peak
on the left. Thus the calculation allows us to predict these
probabilities and to compare with the results of experiments as
discussed in $1.3. In all cases where calculations using the
Schrodinger equation have been compared with experiment the
agreement is perfect. In particular, it is worth mentioning that we
obtain agreement with the classical result for a very high or very
low potential barrier, namely almost 100% reflection or transmission
respectively.
We must now look more closely at what our calculation for the
potential barrier experiment really tells us. After collision with the
barrier the wavefunction, and hence the probability, is the sum of
two pieces. Here we are ignoring the fact that the two parts are in
practice joined because the wavefunction is never quite zero, just
very small, between them. What, then, happens when we make an
observation which tells us whether the particle has been reflected?
Clearly, in some sense, we ‘select’ one of the two peaks in the wavefunction.
In other words, we might say that the wavefunction has
jumped from having two peaks to having only one. This process is
referred to as ‘reduction of the wave packet’. What it means, whether
it happens and, if so, how, are topics to which we shall return.
To close this section we emphasise that the wavefunction is
determined from the initial conditions in a completely deterministic
way. Knowing the initial wavefunction exactly (e.g. figure 1 l ( ~ ) ) ,
we can calculate, without any uncertainty, the wavefunction at all
later times and hence the probability of transmission or reflection.
The non-deterministic, probabilistic, aspects of the potential
barrier experiment arise because we do not observe wavefunctions
but rather particles; in particular, we can observe the position of
an individual particle after it has interacted with the barrier.