Visual Quantum Mechanics

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Plane wave solution and reflection coefficient for a potential well

This is an interactive image. The animation does not show a time-evolution. It just shows the dependence of the solution on the energy E. The movie combines a plot of a plane-wave solution for the rectangular potential well (upper image) with a plot of the reflection coefficient (lower image).

The rectangular potential well (depth V = -16) is indicated by the gray line in the upper image. Here you can see a solution of the stationary Schrödinger equation. This solution is similar to a plane wave in several respects: It has a sharp energy E (which can be adjusted using the slider), it is not square-integrable, and has no probabilistic interpretation. We can form square integrable wave packets by superimposing plane-wave-type solutions from a continuous range of energies.

As you probably know, higher kinetic energies correspond to shorter wave lengths. In the visualization, a short wave length corresponds to a rapid succession of colors. You can see that the solution inside the well always has a shorter wave length: For a given total energy E the kinetic energy is higher inside the well where the potential energy is negative.

To the left of the well this solution is the sum of an incoming and a reflected part. The part of the solution on the right-hand side is called the transmitted plane wave.

The interference of the incoming and the reflected parts cause the ripples in the wave function on the left-hand side. For some energies the reflected part is zero. In this case the amplitudes of the incoming and the transmitted waves are equal and there is no interference in front of the well.

In fact, every single potential jump causes the incoming wave to split into a reflected and a transmitted part.
The vanishing of the reflected part for the scattering at the well can be understood as the destructive interference of the part reflected at the first jump and the part reflected at the second jump.

The amplitude of the reflected plane wave depends on the value of the reflection coefficient R(E). The reflection coefficient depends on the energy E. For this situation, it can be computed explicitly and is shown in the lower part of the animation (as a function of the energy). Here the vertical line indicates the energy of the solution above.

At certain values of the energy, the reflection coefficient is zero. This means that an incoming particle with this energy will not be reflected. Hence there is no reflected wave in the region x < -R (that is, to the left of the well) and nothing can interfere with the incoming wave in this region. Moreover, if nothing gets reflected, then the transmitted plane wave has the same amplitude than the incoming plane wave.