Exercise: Transmission Resonances for a Square Barrier

The applet simulates electron scattering from a square barrier; the barrier height U = 10.0 eV and width 2a = 1.00 Å appear in the Equation View of the applet, along with the electron mass = 511 keV/c². On the Graphics: [x] tab the square barrier potential energy V(x) is plotted over the interval [–5 Å, +5 Å]. The listing to the right of the graph includes placeholders for a single scattering (stationary) state of the electron and its reflected component; the energy of this state is adjustable. In this exercise, we will determine the two lowest electron energies that result in perfect transmission across the barrier, and explore the corresponding wavefunctions to elicit the connection between the barrier width and electron wavelength in the barrier region.

1. Show the scattering waveform ψ₀ by right-clicking its placeholder in the list, clicking the visibility icon in the Display Options field, then choosing the "OK" button. This is a stationary state of the electron in this environment, constructed as a purely transmitted wave to the right of the barrier to model the case where electrons are incident on the barrier from the left. The electron energy defaults to 10.0 eV, the barrier height. For this complex-valued waveform, the plotting style defaults to a color-for-phase scheme (see Technical Notes below). Note the uniform amplitude of the transmitted wave. In contrast, the wave to the left of the barrier is a mixture of incident and reflected components.
2. Show the reflected component of ψ₀ by right-clicking the remaining placeholder in the list to the right of the graph, clicking the visibility icon in the Display Options field, then choosing the "OK" button. Note the uniform amplitude of the reflected component on the incident side of the barrier.
3. The reflected wave amplitude is zero at a transmission resonance. To find the lowest such energy, right-click anywhere in the graph and select "Display in Window" from the popup menu. This frees the graph to 'float' in full view as you make adjustments to the energy. Reposition the graph as desired and proceed to Equation View, where the energy of this state is recorded as E₀ = 10.0. Right click anywhere in the equation field and select "Edit parameter.." from the popup menu. Use the slider to change the highlighted digits in the text field, increasing the energy upward from the start value until no reflected amplitude is visible on the graph. [You may wish to zoom in (rotate the mouse wheel) on the reflected wave for a better view.] Fine tune the search by reducing the number of highlighted digits in the text field; the first digit highlighted can be moved left (right) using the up (down) arrows to the right of this field. The energy yielding no visible reflected wave is an approximation to the resonance value.
4. Finally, let's fine tune even more the energy found in the previous instruction. Note that E₀ is the rightmost argument of the refl(..) function that appears in the current view (Equation View), and that the function value changes interactively as we adjust this energy. The refl(..) function calculates directly the reflection coefficient for the given energy. Continue adjusting E₀ until you find one that gives near-zero reflection; this is our most accurate result for the resonance energy. When you are satisfied, finish by selecting "OK" to end the edit session with the current settings. Finally, restore the graph to its rightful place by clicking the close button in the upper right corner of the graph window.
5. Exhibit just the real part of the resonance wave and measure its wavelength in the barrier region. Return to Graphics View and right-click on the resonance waveform, de-select "Color-4-Phase" in the Display Options field, then finish by choosing the "OK" button. Carefully examine the waveform (real part) in the barrier, zooming in (rotate the mouse wheel) and scrolling (right-click and drag the mouse) the display as necessary. Right-click anywhere on the graph, select "Trace" from the popup menu, then record the peak and node locations needed to determine the wavelength of this oscillation. Calculate the ratio of electron wavelength in the barrier to the barrier width.
6. Follow the instructions above to locate the next lowest transmission resonance. Display the resonance waveform and measure its wavelength in the barrier region. Again form the ratio of electron wavelength in the barrier to barrier width. Based on your results from the two lowest resonances, can you identify a trend for this ratio?

Final form of applet...

Technical Notes

• The Numerov method is used to construct the scattering wavefunctions for a given energy. The correct asymptotic boundary conditions at the right endpoint of the interval are derived from the requirement that the wave be a purely transmitted one in this region. The Schrödinger equation is integrated from there to the left endpoint, first for the real part and then again for the imaginary part of the scattering waveform. The resulting wave near the left endpoint of the interval is a mixture of an incident and a reflected wave. The coefficients of each are extracted from the expected asymptotic behavior of the full waveform. These coefficients furnish the needed boundary conditions at the left endpoint to integrate the Schrödinger equation (again using the Numerov method) from there to the right endpoint, thereby generating the incident and/or reflected waves as desired. Further details are reported in Computing in Science & Eng. 8(4), 32 (2006).
• A complex-valued function can be rendered as a single color-for-phase plot rather than two separate plots for the real and imaginary parts. In this display style, the phase of a complex number (the function value) is represented by color which, like phase angle, repeats with a definite period. [The 'colorwheel' follows the rainbow from red to green to violet, then back to red again through magenta.] In computer jargon, the phase is mapped to the hue (the degree of red, green, or blue) component of an hue-saturation-brightness color model.