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The algorithm described above was programmed using the
Mathematica software package.
As the number of barrier segments increases the
calculation of the transmission coefficient should converge to
some limiting value close to the result using the WKB
approximation.
Figure 3 shows the result for the calculation of the
transmission coefficient through the barrier shown in Figure 2 for
different numbers of adjacent rectangular barriers.
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after dividing the
potential barrier into 250 adjacent rectangular barriers.
This
calculation took about one second on a 400 MHz PC running Linux.
The solid line in the
figure is the result of a calculation of the transmission coefficient
using the WKB approximation.
It is within a factor of three of the value the
transfer matrix calculation approaches.
This level of
agreement is about the same as one finds in comparing the
calculation of the transmission coefficient through a
rectangular barrier using the two methods.7
In the classroom students were asked to generate the algorithm for solving the rectangular potential barrier problem using Mathematica. The method was developed in lecture and then a laboratory was used to introduce the necessary commands. In a later session the extension to barriers of arbitrary shape was made and another laboratory used to introduce any new commands. The results of their transmission coefficient calculations were used to generate the half-lives for a series of radioactive isotopes in the manner of Gurney and Condon and then compared with the measured values found in the literature.11,12
The pedagogical impact of the method was significant. The simplicity of the algorithm makes it accessible to undergraduates. They were able to program the algorithm with little outside help and as a result spent their time investigating the physics of the problem rather than debugging code. The transition from the rectangular barrier problem to potential barriers of arbitrary shape was transparent. It is analogous to the approach used to introduce the notion of an integral to many of our students so it can be developed with a minimum of class time. The validity of the approach was confirmed by having each student test the convergence of their algorithm as shown in Figure 3.
The incorporation of computational laboratories (of which this method is part)
has been successful.
We recently began assessing our graduating seniors' competence
using the Educational Testing Service's Field of Study test.13During the period 1994-1997 when we started this project and also began
testing, the average performance of our seniors was in the
percentile in the quantum mechanics portion of the test.
A numerical integration of the Schrödinger equation was explored to solve the tunneling problem. Methods like this one are generally less familiar to undergraduates and hence less accessible to them. One must invest considerable class time to develop the algorithms or else run the risk of treating the solution of differential equations as a mere `push button' function on a sophisticated calculator like Mathematica.
To conclude, the transfer matrix method provides a means for solving some of the typical quantum mechanics problems in a coherent, powerful fashion that can be readily extended to problems inside and outside the usual realm of introductory quantum mechanics courses. 8,9,14,15The technique is readily accessible to undergraduates and its connection to the simple rectangular barrier problem is transparent. It also acquaints them with powerful matrix methods in a new context and is a vehicle for developing the students' skills with the computational tools that are used outside the classroom.
The author wishes to acknowledge the support of the National Science Foundation's Instrumentation and Laboratory Improvement Program and the University of Richmond.