Introduction

Tunneling through a one-dimensional potential energy barrier is a topic routinely investigated in introductory quantum mechanics and provides one of the most striking departures of quantum physics from classical physics. A particle that is bound by some attractive nuclear force (for example, a $\rm ^4He$ nucleus moving inside a larger atomic nucleus) is able to escape from the parent system even though it lacks the energy to overcome the attractive force. Classical physics predicts that such behavior is impossible. The $\rm ^4He$ nucleus or $\alpha$ particle would have to acquire enough additional energy from some source to reach `escape velocity' before it could leave the parent nucleus. However, the `fuzziness' of Nature at the sub-atomic scale implies that precise knowledge of the $\alpha$ particle's trajectory within the nucleus is unobtainable. This uncertainty means the particle has a small, but non-zero probability of being outside the nucleus where the Coulomb repulsion will push it away from the residual nucleus. We say it has `tunneled' through a potential energy barrier created by the attractive nuclear force. The treatment of this phenomenon in many introductory texts has become rather standard.<sup>1-6</sup> In this paper a method is presented for investigating quantum mechanical tunneling that is readily accessible to undergraduates, permits the solution of a broad range of problems, and takes advantage of increasingly common computational tools. It is part of a program at the University of richmond to incorporate these tools using a `hands-on' laboratory environment.

The phenomenon is usually approached by first considering the problem for a highly idealized potential energy curve, the rectangular barrier shown in Figure 1.

 

Figure 1: 50pt 50pt Potential energy curve for the rectangular potential barrier.

The problem is dealt with by solving a set of simultaneous equations generated by applying the appropriate boundary conditions to the solution of the one-dimensional Schrödinger equation shown here.^1-6

$${-\hbar^2 \over 2m} {\partial^2 \psi(x) \over \partial x^2} + V(x)\psi(x) = E\psi(x)$$ (1)

The constants E and m are the energy and mass of a particle moving in a potential V(x)and $\hbar$ is Planck's constant divided by $2 \pi$. Potential barriers of arbitrary shape are then treated using the Wentzel, Kramers, Brillouin (WKB) approximation.^1-6This pattern of development has several drawbacks. The rectangular barrier problem has limited applicability (few potential barriers are described adequately by it) while the WKB approximation requires either precious class time to rigorously justify the method or resorting to a less substantial, `cookbook' approach. In addition, the two methods of solution are inconsistent with one another. Applying the WKB approximation to the rectangular barrier does not recover the original result obtained by solving the set of simultaneous equations.⁷Finally, from an undergraduate's perspective, the solutions to the two problems use different techniques (solving a set of simultaneous equations versus performing an integral) whose underlying connections are often unseen or misunderstood.

The use of transfer matrices to solve the rectangular barrier problem is a well established technique used to solve the rectangular barrier problem.^8,9The method can be extended quickly and naturally to potential barriers of arbitrary shape while retaining a transparent connection to the original rectangular barrier problem. It also has the pedagogically useful features of introducing powerful matrix methods in a new context and takes advantage of current teaching technologies.

In Section 2 an overview of the solution of the rectangular barrier problem with the transfer matrix formalism will be developed and extended to potential barriers of arbitrary shape. More detailed discussions of the technique have been done by others.^8,9,10 In Section 3 some results from the application of the technique to a specific example will be displayed, the response of students to this approach discussed, and a comparison made with other computational methods.