Required math: calculus
Required physics: Schrödinger equation
Reference: Griffiths, David J. (2005), Introduction to Quantum Mechanics, 2nd Edition; Pearson Education – Problem 2.34.
A variant of the finite square well is the finite step, which has the potential
where is a positive constant energy.
There are two distinct cases here:
- Energy below the barrier:
- Energy greater than the barrier:
We’ll consider first the case where .
In this case, the Schrödinger equation for is
This has solution
To keep the solution finite as we must have so the solution is an exponentially decaying wave function:
To the left of the barrier, the Schrödinger equation is
Assuming particles coming in from the left, we have
Since the potential is finite everywhere, both and are continuous everywhere, which gives us two boundary conditions at .
This has solution
The reflection coefficient is
That is, the probability of an incoming particle being reflected is 1. This is because the wave function for is exponentially decaying, so the probability of a particle reaching infinity is zero, thus no particles can be transmitted.
For the Schrödinger equation for is
We now get travelling wave solutions instead of exponentially decaying ones:
Assuming incoming particles arrive only from the left, we can set . Applying the boundary conditions, we get
In this case, the reflection coefficient is
Substituting the expressions for and we get
From this we can get the transmission coefficient
Note that this is not equal to . Have we done something wrong?
The answer lies in the fact that the wave for is not the same as the wave for , since the net energy on the right is while on the left it is just . One way of looking at it is in terms of the probability current for the free particle. The probability current must be conserved; this is just a way of saying that particles cannot vanish into, nor arise from, thin air. Since the probability current for a free particle with stationary state
the conservation law implies, for the case of the step potential
That is, the influx of particles from the left minus the reflected beam must equal the transmitted beam. Dividing through by we get
The first term is the reflection coefficient we calculated in 26. The second term is the transmission coefficient, which works out to
which is what we got earlier.