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This Exploration shows the same initial Gaussian wave packet incident on either a step up in potential energy, a potential energy barrier, or a potential energy well. The animations successively increase the step/barrier height and well depth. Shown in the table are the probabilities that the wave packet can be found in the various regions.7 In the animation, ħ = 2m = 1.
For the step up animations, what happens to the packet as E increases? What do you notice about how the wave function behaves in the two regions as a function of E? Make a reasonable estimate for how the transmission coefficient behaves as a function of E.
For the barrier animations, what happens to the packet as E increases? What do you notice about how the wave function behaves in the three regions as a function of E? Make a reasonable estimate for how the transmission coefficient behaves as a function of E. What happens to the part of the wave function in the barrier?
For the well animations, what happens to the packet as E increases? What do you notice about how the wave function behaves in the three regions as a function of E? Make a reasonable estimate for how the transmission coefficient behaves as a function of E. What happens to the part of the wave function in the well?
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7The problem of wave packets incident on a step up in potential energy is discussed by R. Shankar in Principles of Quantum Mechanics, Plenum Press (1994).