Exercise: Electron Scattering from a Square Barrier

The applet simulates electron scattering from a potential barrier. For this case the barrier is square; 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 a placeholder for a single stationary state of the electron; this is a scattering state with adjustable energy. In this exercise, we will determine the scattering coefficients (reflection, transmission factors) and explore the scattering waveforms in the cases where (1) the electron energy coincides with the top of the barrier, and (2) the electron energy is exactly one-half the barrier height.

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, a value equal to 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. To show separate real and/or imaginary parts of ψ₀, right-click on the waveform and select "Properties..." from the popup menu. Now de-select "Color-4-Phase" in the Display Options field, click the desired visibility icon(s) in the Colors | Visibilities field, then finish by choosing the "OK" button.
3. Carefully examine the waveform (real and imaginary parts) in the barrier region, zooming in (rotate the mouse wheel) and scrolling (left-click and drag the mouse) the display as necessary. What functional form do you observe? Can you account for your observations using Schrödinger's equation?
4. Find the scattering coefficients at this energy. Right-click on the waveform and select "Properties..." from the popup menu, then select "Color-4-Phase" in the Display Options field to restore the color-for-phase plotting style. Right-click on the waveform again and select "Show incident wave" from the popup menu. Now determine the incident and transmitted wave amplitudes by right-clicking anywhere on the graph, selecting "Trace" from the popup menu, and surveying the results. Record values for the incident and transmitted wave amplitudes; squaring the ratio of transmitted to incident amplitude gives the transmission coefficient at this energy. Similarly, the reflection coefficient at this energy can be calculated by measuring the reflected wave amplitude. [To display the reflected wave, right-click the waveform once more and select "Show reflected wave" from the popup menu.] According to classical physics, what would happen to an electron incident on this barrier with energy equal to the barrier height?
5. Next reduce the electron energy to one-half the barrier height. Proceed to Equation View and click in the value field for the parameter E₀, the energy for this stationary state. Position the cursor as appropriate and type in the new value E₀ = 5.0 (one-half the 10.0 eV barrier height), then type Ctrl+Z to register the changed value. Return to Graphics View, and note that the display is updated accordingly. Follow the procedure described in the preceding instruction to find the scattering coefficients at the new energy. Use your measured values to check the sum rule for these coefficients, i.e., the requirement that the transmission and reflection coefficients add to unity.

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.