Exercise: Quantum Scattering Concepts & Methods

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 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]. Also shown is a stationary state of the electron, in the case where the electron energy is E = 7.0 eV. In this exercise, we will explore this stationary state, extracting from it the information needed to calculate the transmission coefficient for the barrier, and compare our numerical result with the theoretical prediction.

1. The stationary states of the electron in this environment are complex-valued; only the real part is shown in the applet above. To show the imaginary part, right-click on the waveform and select "Plot Properties..." from the popup menu. Click the visibility icon beside the "Imaginary" label in the Colors | Visibilities field, then finish by choosing the OK button. Together, the real and imaginary plots portray all the information contained in ψ, but not in the simplest way. In fact, this division into real and imaginary parts is altered if ψ is multiplied by a phase, which does nothing to change the physical state of the electron.
2. A more informative display is afforded by the color-for-phase plotting style, which combines all the information of a complex-valued function into a single plot (see 1st Technical Note below). Right-click on the waveform again and select "Plot Properties..." from the popup menu. Check the "Color-4-phase" box in the Display Options field, and click the visibility icon beside it. Finish by choosing the OK button. Note the uniform amplitude of the wave to the right side of the barrier, signaling that the absolute value |ψ| is constant in this region. This is the transmitted component of the wavefunction, constructed for the case where electrons are incident on the barrier from the left. In contrast, the wave to the left of the barrier is a mixture of incident and reflected components.
3. Show the incident component of this wavefunction. Right-click on the waveform once more and select "Show Incident Wave" from the popup menu. Note the uniform amplitude of the incident wave in the region to the left of the barrier.
4. Find the transmission coefficient at this energy. Right-click anywhere in the graph background, select "Trace On/Off" from the popup menu, and survey the results. Record the values for the incident and transmitted wave amplitudes; squaring the ratio of transmitted to incident amplitude gives the transmission coefficient at this energy. The theoretical value for the transmission coefficient is given by the well-known formula for transmission by a square barrier (cf. Modern Physics 3^rd ed., R. Serway et. al., p.235 (2004)). For electrons with E = 7.0 eV, theory predicts T(E) = 0.4520.

Final form of applet...

Technical Notes

• 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.
• 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).