## Building States

What happens if the potential energy function changes? The energy eigenfunction will (most likely) be a smoothly varying function without kinks:

• When V(x) is a constant in a region of space, expect the energy eigenfunction to keep its shape (maximum amplitude and curviness) in this region.
• When V(x) smoothly varies in a region of space, expect the energy eigenfunction to smoothly vary its shape in this region.
• When V(x) abruptly changes at a point, expect the energy eigenfunction to abruptly change its shape at this point.  Piece the form for the energy eigenfunction in different regions together such that the resulting function is a smoothly varying function without kinks.

When the potential energy function abruptly jumps and [E V(x)] > 0 on both sides of the jump, there may be an antinode in the energy eigenfunction at this position.  If there is an antinode at the position of the potential energy jump, a smooth eigenfunction results only when the maximum amplitude of the state is uniform across the potential energy jump.  The resulting probability density therefore will not follow the classical expectation from time-spent arguments.

The energy eigenfunction will have a kink if the potential energy function has a spike or an infinite wall (i.e., is very badly behaved).  Finally, Ref. 3 also adds a symmetry statement: if the potential energy function is symmetric about the middle of the well, the energy eigenfunctions are alternately even and odd functions, with the ground state being an even function.

As an example, consider the asymmetric infinite square well: an infinite square well with a finite, and constant, potential energy hump on one side (x > 0) of the well as shown in the animation.  To see the other bound states, simply click-drag in the energy level diagram on the left to select a level. You should be able to find cases in which the potential energy hump height has been tuned to yield one state below the hump energy and one at the hump energy (there are an infinite number above the hump energy, but only some of these states are shown).  Note the possibility of three distinctly different shapes of the energy eigenstates for x > 0 depending on energy.  Also note the number of zero crossings of the energy eigenfunctions and how they relate to the energy of the states.

What happens to the energy eigenfunction as we increase the step height V0?  We begin to notice that the energy eigenfunction once having the same amplitude and curviness over both sides of the well, begins to lose this symmetry.  Given the larger potential energy function in Region II, the wave function there has less curviness. In addition, the amplitude of the energy eigenfunction should increase in Region II because it has a higher probability of being found there.  (By simple time spent arguments: a classical particle would spend more time in Region II due to its reduced speed there.)  In addition, since the added potential energy function is a constant over the entire region, the change in energy eigenfunction curviness and amplitude must be uniform over Region II.

Note that for certain slider values and certain eigenfunctions, you may notice the same amplitude in Region I and Region II, despite the potential energy difference. This is due to the fact that the eigenfunctions happen to match at a antinode.1

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1For more mathematical details see: M. Doncheski and R. Robinett, "Comparing Classical and Quantum Probability Distributions for an Asymmetric Infinite Well," Eur. J. Phys. 21, 217-227 (2000).
2
See for example, A. Bonvalet, J. Nagle, V. Berger, A. Migus, J.-L. Martin, and M. Joffre, "Femtosecond Infrared Emission Resulting from Coherent Charge Oscillations in Quantum Wells," Phys. Rev. Lett. 76, 4392-4395 (1996).