This post describes a way to calculate tunneling probabilities for one dimensional quantum barriers. This method is easy to code up, and is very fast.
Consider the following barrier. If your energy is less than 3 eV, you’ll just reflect off. But above 3, weird things happen. How do you calculate the reflection and transmission coefficients?
Quantum tunneling is a favorite conceptual topic for students. It is a notion of something so very different from what is expected classically that can be described so easily by invoking memories of throwing balls at walls. Students are encouraged to find connections with frustrated total internal reflection in order to further cement their understanding of matter as waves. Both in optics and quantum mechanics instructors can note how the continuity of the wave function and its derivative across boundaries implies that the wave cannot abruptly go to zero. This enables students to see why a (typically small) portion of the wave can tunnel through a barrier.
On the other hand, the quantitative aspects of tunneling are a different story for students. As usual very simple situations can be calculated by hand like the square barrier but more typical barriers found in the lab require the use of a computer and an algorithm that can apply the conceptual physics they have learned (the boundary conditions) in an iterative fashion.
In this post I’ll talk about a different algorithm to calculate the tunneling probability of a particle with known energy through an arbitrary one-dimensional potential barrier. It is both fast and accurate but it also uses tools that most students are familiar with in their studies of the Schroedinger equation. Specifically it involves the direct integration of the Schroedinger equation in a manner very similar to the shooting method employed to find the eigenstates of an arbitrary potential well. However, instead of needing to adjust parameters to find a particular eigenstate, students can directly inspect the results for any given particle energy and determine both the tunneling probability and the shape and nature of the wavefunction inside the barrier.
The most common approach to calculating tunneling probabilities is to consider the barrier to be a collection of square barriers. In the WKB approach, only the exponentially decaying portion of the wavefunction is kept and integrated through all the slices (Simmons 2007). In the matrix transfer method, the boundary conditions among all the slices are carefully calculated (Alexpoulos 2007, Mendez 1994, Morelhao 2007 (pdf), Probst 2002, Zhang 2000). Specifically, at every boundary between the square slices the wavefunction and its slope are continuous. In each slice the wavefunction is composed of two components: either a right and left traveling wave with a wavelength determined from the kinetic energy (the difference between the total energy and the barrier height); or a growing and decaying exponential whose growth rate is determined from the (negative) kinetic energy. Often these boundary condition equations are described in a matrix formalism as they are simple linear equations relating the incoming and outgoing wavefunctions along with the barrier heights of the slices. The effect on the incoming wave by the barrier is then modeled by a single matrix that can used to solve for the tunneling probability.
There are also some approaches in the literature that have more directly integrated the Schroedinger equation, but all do a forward propagation as opposed to the backward one described below (Ban 2000, Yunpeng 1996). These approaches use both numeric and analytical methods to determine the phase of the incoming wave that enables solely a right-traveling wave in the transmission region. The method below does not require such adjustments and simply gives both the wavefunction in the tunneling region and the tunneling probability after a single direct integration.
Consider a tunneling situation as laid out in the figure at the top of this post. The first and third regions have a constant potential while the middle region can have any form, including discontinuities and regions where the particle is classically allowed. Region I can have right- and left-traveling waves
while Region III only has a right traveling wave:
Using a fourth-order Runge-Kutta technique I numerically integrate the real and imaginary parts of the Schrödinger equation from the right edge of Region II (x=L) to the left edge (x=0). Note that since the Schrödinger equation does not have any single derivatives in it a Numeroff approach can also be used. Note also that in Mathematica you can integrate complex numbers with just one call to the Runge-Kutta solver (NDSolve). Since both the wavefunction and its slope will be the same on both sides of the boundary between Regions II and III, the initial conditions are determined by arbitrarily setting F=1 and using the form from Region III:
To determine the transmission probability, T, we need to find the value of A:
This is done by investigating the value of the wavefunction and its slope at where, according to Eq. (1):
Once again we have used the equality of the value and slope of the wavefunction across a boundary.
Once the Schrödinger equation has been numerically integrated, the transmission probability is easily calculated.
This method employs many techniques used when teaching the numerical solution of eigenstates for arbitrary barriers. In those situations students are taught to employ the shooting method to find energies that produce physically allowable wavefunctions. (like I’ve posted about before). The major difference in this new application is that both the real and imaginary parts need to be integrated, as is illustrated in the figure below. If you only do the real part (as is often done in the shooting method application) you are unable to calculate the transmission coefficient as seen in Eq. (8) above.
Examples and Comparisons
The transmission coefficient () as a function of particle energy for the potential shown in Figure (1) is given in Figure (3) below. The top curve is the result of the current method while the lower lines use the transfer matrix method with varying number of slices of Region~II. Ultimately both approaches converge to the same result at every energy.
It is interesting to compare the transfer matrix method with the new method where the number of slices is compared with the number of steps that the Runge-Kutta method employs. The transmission probability versus energy for the arbitrary barrier shown in Figure (1) is given for the total step number ranging from 5 to 12 in the inset of Figure (3) above. The curve with 300 steps is also shown. It is clear that the number of steps needed for the Runge-Kutta method is far less than the number of slices needed in the transfer matrix method to achieve the same accuracy. Note, however, that one should really compare the number of calculations involved when doing these comparisons. A fairer comparison would need to multiply the number of Runge-Kutta steps by four, though this still shows that the current approach compares favorably to the transfer matrix method.
As an example to show the pedagogical uses of the current method, I consider resonant tunneling. Specifically I compare the wavefunction in the barrier region to the eigenstates expected for a simply-shaped barrier.
Consider the potential shown below.
This parabolic potential barrier is parabola centered at x=1 but chopped off at x=0 and 2. The analogous potential well that is not truncated has resonant energies at
The transmission probability as a function of energy is shown here:
The resonance peaks shown correspond very nearly with the eigenenergies of a parabolic well. At lower energies where the resonance peaks are very sharp the energies are the same as the eigen energies. As the peaks become broader, the resonant energies become larger than the eigenenergies by as much as 20% for n=11.
The reason the resonant energies grow larger than the eigenenergies as the energy increases is due to the boundary conditions that the wavefunction has to match at x=0 and x=1. This can be seen by comparing the resulting wavefunctions (both the tunneling wavefunction and the eigenfunction for the parabola) as seen here:
Close inspection of the wavefunctions near the boundaries indicates the differences between the tunneling wavefunction and the eigenstate. While the eigenstate is decaying to zero in all cases, the tunneling wavefunction is forced to match the boundary condition at the right edge. At low energies there is little difference as the exponential rise is very steep but at higher energies there is a higher bend needed that explains the rise in the energy compared to the analogous eigenenergy.
I have discussed a new method for calculating both transmission probabilities and wavefunctions for a particle tunneling through an arbitrary one-dimensional barrier. The approach is applicable at the undergraduate level as it uses common tools related to the shooting method for finding potential well eigenstates. It is fast and accurate and enables the study of complex phenomena like resonant tunneling.
Here are some starters for you:
- This is really useful. I plan to use it in . . .
- This is dumb, I’d never use it and here’s why . . .
- This reads like an article you wrote for the American Journal of Physics that got denied with a reviewer saying since it was so easy to code up it wasn’t worth publishing.
- Wait, so you integrate from right to left, I didn’t think that was allowed!
- So you just assume that something makes it through and work back to see what could have caused it? Weird.
- You lost me at Mathematica
- Can you help me implement this in . . .