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Step Up: Animation 1 | Animation 2 | Animation 3 | Animation 4 | Animation 5 | Animation 6

Barrier: Animation 1 | Animation 2 | Animation 3 | Animation 4 | Animation 5 | Animation 6

Well: Animation 1 | Animation 2 | Animation 3 | Animation 4 | Animation 5 | Animation 6

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.

1. For the step up animations, what happens to the packet as V increases?  What do you notice about how the wave function behaves in the two regions as a function of V? Make a reasonable estimate for how the transmission coefficient behaves as a function of V.
2. For the barrier animations, what happens to the packet as V increases?  What do you notice about how the wave function behaves in the three regions as a function of V?  Make a reasonable estimate for how the transmission coefficient behaves as a function of V. What happens to the part of the wave function in the barrier?
3. For the well animations, what happens to the packet as V increases?  What do you notice about how the wave function behaves in the three regions as a function of V?  Make a reasonable estimate for how the transmission coefficient behaves as a function of V. What happens to the part of the wave function in the well?

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).