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 on the Math tab 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 a placeholder for a single scattering (stationary) state of the electron in this environment; 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 on 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. Display the reflected component of ψ₀ by right-clicking on the waveform and choosing "Show Reflected Wave" from the popup menu. Note the uniform amplitude of the reflected component on the left (incident) side of the barrier.
3. The reflected wave amplitude is zero at a transmission resonance. To find the lowest such energy, go to the Math tab, where the energy of this state is recorded as E₀ = 10.0. Right-click anywhere in the value field for E₀ and choose "Edit Parameter..." from the popup menu to bring up the Energy Editor. Return to the Graphics: [x] tab and re-position the editor so as to afford an unobstructed view of the waveform. Now use the slider to manipulate the highlighted digits in the energy field (labeled E) while observing the waveform. Increase the energy upward from the initial value until no reflected amplitude is visible on the graph. Fine tune the search by reducing the number of highlighted digits in the text field; the first digit highlighted can be moved left or right by clicking on the corresponding arrow to the right of this field. The energy yielding no visible reflected wave is an approximation to the resonance value.
   
   
   Extra digits can be added before or after the decimal by typing directly in the editor energy field, then pressing the Enter key. The sum of leading and trailing digits is limited to 9.
   
   
4. Now let's fine tune even more the energy found in the previous instruction. From the Math tab, observe that E₀ is the rightmost argument of the refl(..) function, and that the function value changes interactively as we adjust the energy. The refl(..) function calculates directly the reflection probability at the given energy. Continue adjusting E₀ until you find a value 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.
5. Exhibit just the real part of the resonance wave and measure its wavelength in the barrier region. Return to the Graphics: [x] tab and right-click on the resonance waveform, de-select the "Color-4-Phase" checkbox in the Display Options field, then finish by choosing OK. Carefully examine the waveform (real part) in the barrier, zooming in (press Z) and scrolling (with the arrow keys) the display as necessary. Right-click anywhere on the graph, select "Trace On/Off" 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?

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 amount of red, green, or blue) component of an hue-saturation-brightness color model.