Consider the potential barrier shown below. In classical mechanics,
an incoming particle with energy
Let us revisit the finite step potential,
“Scattering” :
“Tunnelling” :
The scattering regime
The Hamiltonian eigenfunctions take the form
The wave functions are not square-normalizable and to extract
physical information we instead compute ratios of probability currents.
The probability current is
Now consider
Notice that the above wave function can be obtained from the
scattering problem by replacing
In summary, we conclude that
This is consistent with the limit
It coincides with the classical expectation that the particle is
always reflected when
Despite the fact that
This has important consequences if we were to add a step down to
Let us now consider the finite
barrier potential
In this case, we will see that there is a non-zero probability to
find the particle to the right of the barrier even when
The scattering regime corresponds again to
Since the potential remains finite at
The limit
The limit
The function has trigonometric dependence on
The tunnelling regime is
As before, the coefficients
The limit
The limit
Note the exponential rather than trigonometric dependence on
Finally, we can combine the results in the scattering and tunnelling
regimes to sketch the reflection / transmission probabilities across the
entire range
Consider the same problem but with
How do the hamiltonian eigenfunctions change in the region
Explain why the probability current
Show that
Sketch the probability density in the region
(This problem is similar to problem 9 of the May 2019 exam and
provided here to illustrate the method required to solve it. This
problem was not part of the 2020-2021 module). Consider the finite
potential well
Consider the following ansatz for “bound state" wavefunctions,
Find constants
Explain why there are no terms in the ansatz proportional to
What boundary conditions do the wavefunction obey at
Impose the boundary conditions and eliminate
Illustrate solutions of the quantisation condition graphically and show that
There is at least one solution independent of
Show that you reproduce the spectrum of the infinite potential
well in the limit
Solution ▶
The Hamiltonian operator for
These solutions diverge as
Since the potential remains finite both the wavefunction
Although not asked for in the question, we can show this as follows.
First, continuity of
We now impose the boundary conditions at
We first rearrange to find a quadratic equation for the ratio
There is a discrete number of solutions, which increases with
the dimensionless parameter
The limit