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Solving the Energy Eigenvalue Equation for the Finite Well: Instructor's Guide

Main Ideas

• Solutions for the energy eigenvalue equation
• Matching boundary conditions
• Exponential wave functions
• Continuity

Estimated Time: 40 minutes

The class is broken into thirds and each third is asked solve the energy eigenvalue equation for a region of a finite potential well. Students are then asked to match their solutions at the boundaries.

Prerequisite Knowledge

1. Students should know the eigenvalue equation, $\hat{H}\phi=E\phi$, and that it is satisfied for multiple eigenstates and eigenvalues (see Operators and Functions activity for a good way to introduce this)
2. Students should have some knowledge of solving differential equations, specifically linear, homogeneous second-order differential equations. Students should know the solution for the cases where the linear term is either positive or negative:

$$\frac{d^2\phi}{dx^2}+\phi=0$$ $$\frac{d^2\phi}{dx^2}-\phi=0$$

Activity: Introduction

In a brief mini-lecture, students are reminded of the energy eigenvalue equation (sometimes called the time-independent Schrödinger equation) and that the Hamiltonian can be found by adding the kinetic and potential energy operators. The students are then given the differential form of the kinetic energy operator $-\frac{\hbar^2}{2m}\frac{d^2}{dx^2}$.

As the handout is distributed to the class, the class is broken up into three groups and each group is assigned a region in which to solve the eigenvalue equation. We like to divide the room to mirror the potential energy profile (the left side of the room solves for the leftmost region, etc.). Students are reminded to choose an energy below the top of the well, i.e. a “bound state” (but they don't know this terminology yet).

Activity: Student Conversations

1. Students may not recognize that the shorthand $k$ for $\sqrt{\frac{2mE}{\hbar ^{2}}}$ is merely a convenience. Sometimes they think $k$ is a new parameter that introduces a new unknown.
2. Once the students have solutions for their assigned regions, they should consult the students working on the other regions to match the solutions at the boundaries. It is helpful to divide the students who've considered the center region into half, each half consulting with one of the outer regions.
3. The groups assigned to the center “well” region are often savvy enough to realize there are multiple forms in which to write the solution: either as a linear combination of sinusoidal functions or as a linear combination of (complex) exponential functions. We find that students tend to initially choose the sinusoidal functions because this is the most familiar form.
4. The groups considering the outer regions usually realize they need exponential solutions, but often try to keep the general solution as a linear combination of increasing and decreasing exponentials. It is often instructive to remind them early of the condition that the wave function should be square integrable and shouldn't blow up at infinity.
5. For the sake of understanding how the wave function behaves at the boundaries, it is often helpful to have students draw the wave function on top of a plot energy vs. position, but students should be reminded that the vertical axis on the plot and the amplitude of the wave function do not have the same dimensions.

Activity: Wrap-up

The activity should end with a whole class discussion that lays out the solution on the board, with groups reporting about how the boundary conditions are resolved.

We choose to set the zero energy at the bottom of the well, but it is also common for the question to be posed with the zero energy set at the top of the well. We like the zero at the bottom because it is easy to take the limit at the potential at the top of the well goes to infinity, yielding the familiar infinite well problem. The whole class discussion can include how to “port” a known solution to similar problems (like changing the location of the zero of potential energy, or setting one edge of the well to be at $x=0$).

Another point of interest is the fact that having a complex exponential solution with an imaginary wavenumber yields a exponentially increasing or decreasing function. Few students have previously appreciated this point and have not considered that a wavenumber might not be real.

The wave functions are nicely drawn on the PhET simulations; these can be discussed at length.

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