## Building Energy Eigenfunctions

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.  Figure 3 shows one case 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 one of these states is shown).  Note the 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.

See the interactive exercise by following this link: Building States.

Figure 3:  In this exercise, students vary the height of the potential energy “hump” in the right half of the infinite well.  The top graphs show the energy eigenfunction in blue while the bottom graphs show the potential energy function in green and the total energy of the quantum state in orange (the horizontal line). A table with the state’s energy is also shown.  The height of the “hump” is controlled by a slider and the energy state is chosen by clicking on the energy level on the left.