**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.
- Application to radioactive decay.
- 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.

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

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.

Build the transfer matrix

The final result.

Get the transmission coefficient.

Gather the pieces.

Build the transfer matrix.

The final result.

Get the transmission coefficient.

Consider a series of rectangular barriers.

Build up the transfer matrix.

And get the transmission coefficient as before.

in

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] )

- 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.

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

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

- 1.
- The method is accessible.
- Undergraduates can do it.
- Matrix operations are familiar.

- Undergraduates can do it.
- 2.
- The student response is good.
- 3.
- It's broadly applicable.
- 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).

- 1.
- Physically rigorous.
- More consistent than the WKB approximation.
- Broadly applicable.

- 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.