Problem 1.3: Shown is the probability density for a quantum-mechanical plane wave
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Shown is the probability density corresponding to a quantum-mechanical plane wave traveling to the right. The plane wave experiences constant potential energy step (a finite barrier) between x = 10 and x = 20 where there is a constant potential energy barrier. The probability density of a pure plane wave is always a constant (flat). Therefore, when the probability density is a constant to the left of the potential energy barrier, the wave is 100% transmitted. When there is reflection, the reflected plane wave adds to the incident plane wave, and a standing wave pattern can develop when the amplitude of the reflected wave is the same as the incident wave. Restart.
- Drag the slider to change the plane wave's energy. With what energy do you first notice a transmitted wave? Note: in this situation, transmission is nonzero for any value of the initial plane wave's energy.
- Determine the energies that give you 100% transmission. Note: you will not be able to get the energy of the first resonance exactly since this resonance is too sharp compared to the slider's resolution.
- For the energies that give you 100% transmission, what do you notice about the pattern of the probability density between x = 10 to x = 20?
One-dimensional scattering is covered in detail in Chapter 9.