A New Teaching Approach

To Quantum Mechanical Tunneling

G.P.Gilfoyle
Physics Department
University of Richmond, Virginia
ggilfoyl@richmond.edu

Outline

1.
Background and motivation.

2.
The transfer matrix approach to tunneling.

3.

4.
Classroom results.

5.
Conclusions.

Background and Motivation

1.
Why bother?

• Use computational tools.
• Use a hands-on' laboratory approach.

2.
The Program

• Present the observations.
• Build a model.
• Use the tools.
• Use this program regularly.

3.
Is it physically rigorous?

• Advantages over WKB and finite difference methods.

4.
Did it work?

• Easy, consistent, engaging method of investigating quantum mechanics.

Solving the Schrödinger Equation

The one-dimensional, nonrelativistic Schrödinger equation is

The general solution is

Express this solution as a matrix.

Get the Particular Solution

Match the wave function and its derivative at the boundaries.

Write the result in matrix form.

Use the definition of the matrix inverse and get

which can be expressed in matrix form as

The discontinuity matrix!

Make Things Simple

Shift coordinates.

Match the wave function and its derivative at the right-hand-side boundary in the new coordinates.

Relate the two coordinate systems.

And obtain the following result.

The propagation matrix!

Pull Things Together

Build the transfer matrix

The final result.

Get the transmission coefficient.

Pull Things Together

Gather the pieces.

Build the transfer matrix.

The final result.

Get the transmission coefficient.

Multiple Barriers

Consider a series of rectangular barriers.

Build up the transfer matrix.

And get the transmission coefficient as before.

Programming the Algorithm
in Mathematica

Define a function to calculate the transmission coefficient.

RadDecay[Znuc_,Anuc_,Eout_,Ndiv_] := (

(* constants *)
Mout  = Aout*AtoMeV;
R0    = r0*(Anuc - Aout)^(1/3);
Rmax  = Zout*(Znuc - Zout)*e2/Eout;
step  = (Rmax - R0)/Ndiv;

(* table of barrier heights *)
Ve    = Table[{r, Zout*Znuc*e2/r},{r, R0, Rmax-step,step}];

(* put zeros at each end of the array. *)
PrependTo[Ve, {R0-step, 0}];
AppendTo[Ve, {Rmax, 0}];

(* initialize the transfer matrix. *)
trans = { {1,0},
{0,1} };

(* loop over the barriers *)
Do[
V1  = Ve[[i,2]];
V2  = Ve[[i+1,2]];
k1  = Sqrt[2*Mout*(Eout - V1)/(hbarc^2) ];
k2  = Sqrt[2*Mout(Eout - V2)/(hbarc^2) ];
d   = 0.5*{ {1 + (k2/k1), 1 - (k2/k1)},
{1 - (k2/k1), 1 + (k2/k1)} };
p   = { {E^(I*k2*step),             0},
{0            ,E^(-I*k2*step)} };
trans = trans.d.p;,
{i,1,Ndiv+1}
];

(* get the transmission coefficient. *)
tr = Abs[1/(Conjugate[ trans[[1,1]] ]*trans[[1,1]])];
Print["Transmission Coefficient: ",tr]
)


Comparison With Other Methods

1.
The WKB approximation.

• It does not reduce to the rectangular barrier correctly while the transfer matrix method does.

• The connection between the rectangular barrier and potentials of arbitrary shape is less transparent for the WKB than the transfer matrix method.

2.
Numerical integration of the Schrödinger equation.

• It can be more difficult than the transfer matrix method.

• It's less accessible to undergraduates. Students are unfamiliar with numerical methods, but are well acquainted with matrix operations.

Classroom Results

1.
The method is accessible.

• Matrix operations are familiar.

2.
The student response is good.

3.

• Used to study the rectangular barrier, -decay, and one-dimensional solids.

4.
Computational laboratories work.
• Students score in the percentile in the quantum mechanics portion of the Educational Testing Service's Field of Study physics test (1994-1997).

Conclusions

1.
Physically rigorous.

• More consistent than the WKB approximation.

2.
It's pedagogically appealing.

• It's accessible and engaging to undergraduates.
• Uses modern computational tools.

3.
Part of a successful program of incorporating a hands-on' computational laboratory experience.

Support provided by the United States National Science Foundation and the University of Richmond.

1998-09-14