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Problem 1.3: Shown is the probability density for a quantum-mechanical plane wave

Energy =

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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.

  1. 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.
  2. 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.
  3. 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.

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