S2: The Square Shelf Potential

Particle in a Box with One Infinite and One Finite Side

We now want to explore a situation where the particle not only oscillates inside the box, but is (quantum-mechanically) allowed to leak a bit outside the box. This problem is much harder to solve analytically, but we can get the main physical results from our numerical solution.

Change the Potential from InfSquare to SquareShelf. Notice the thin black line, which indicates the square-well potential. The height of the step is shrunk by a factor of ten, so that it fits on the graph: the energy V(x) for X > Xshelf is Vshelf=3eV by default. Notice that we're plotting V(x)-E: the thin black line is not at zero, and is not the y-axis! This way, when the thin black line goes through zero, V(x)-E changes sign, and the behavior of Schrödinger's equation changes.

Try varying the energy. You'll find that the wave-function swings wildly in the range above Xshelf. Investigate energies between two and three for a while. Where V(x)=0, notice that the wave-function oscillates up and down. On the shelf, where V(x) < E, notice that the wave-function tends to go crazy. Increase Distance to Shoot to three: on the shelf, see that Psi either grows exponentially to infinity or to minus infinity.

(S2a) Analytic Solutions for V-E>0

If V(x) is constant and greater than E, we say the electron is in a ``classically forbidden'' region (because it doesn't have enough energy to be there). We can find solutions to Schrödinger's equation in the special case where V(x)=V is a constant. Show, in your lab writeup, that the time-independent Schrödinger equation

in this case has an exponentially growing solution

= A1 exp(B x) and an exponentially decaying solution

= A2 exp(-B x). What is B, when E=2.5eV? (Remember from part 1 that

=26.246 (eV^(-1) nm^(-2)).) Express B in 1/nm (inverse nanometers), and include in your lab writeup.

(S2b) Numerical Solutions for V-E>0

Set Distance to Shoot to 3 nm, and Energy to 2.5 eV (for the Square Shelf potential), and Copy Graph. In the Configure menu of the copied graph, select Y Log. Notice that the graph looks rather like a straight line on the shelf, especially for X>2. (The funny chopped-up look in the well comes from the program trying to take logarithms of negative numbers: don't worry about it). Measure the value of Psi at 2 nm and 3 nm (by using the left mouse button). How close is the ratio of Psi(2) to Psi(3) to that predicted by your exponentially growing solution? You should get agreement to within one percent: include the comparison in your writeup.

Having grow exponentially is not the correct boundary condition! This may be obvious: if the electron doesn't have enough energy to be there, it shouldn't zoom off further and further into the forbidden zone. We can derive this formally by noticing that the wave-function must be normalized. Since the square of the wave-function at X gives the probability density of finding the electron there, and the electron must be someplace (and not more than one place), the integral of must be one:

(Things are more subtle if the electron isn't bound in the well: see S2e below). This implies

as X gets large: we want the exponentially decaying solution.

Even at E=2.5 eV the solution on the shelf is a mixture of exponentially decaying and growing pieces: we had to look at X>2 to see a straight line on the log plot because only there was the exponentially decaying piece so small that it didn't show up. Let's vary the energy until we remove the growing piece! Notice that for E=2.6 eV the wave-function is exponentially growing downward: the constant A1 multiplying A1 exp(B x) is negative. Someplace between E=2.5 eV and 2.6 eV the constant had to go to zero. At that point, just has an exponentially decaying piece, and the solution is physically sensible: we have one of our energy states.

(S2c) Finding an Energy State

By varying the energy carefully, find the point between E=2.5 and 2.6 eV where the exponentially growing piece of the wave-function is smallest. Locate this point to five significant figures (i.e., 2.5XXX), and include in your writeup. (See the help for Energy for hints if you find zeroing in on the correct value tedious.) Draw a picture of the wave-function in your writeup: also draw a picture of the square of the wave function. How many nodes (not including the one at zero, or the one that goes to infinity as you get better and better energies) does this solution have?

(S2d) Finding All the Bound States

Change Distance to Shoot back to 1.5 nm (the exponential growth becomes so fast at lower energies that it's hard to see what's happening), and look for the ground state solution. Raise the energy and the range and find the other solutions. (Make the range (shooting distance) large enough so you can be sure it's asymptoting in the right way, but not so large that you can't control it at all.) How many solutions can you find which are bound (go to zero as X gets large)? You should find a solution with no nodes, one with one node, ... up to the solution you found in part c. Draw each solution in your writeup, and write down its energy. (If your TA wants you to produce a hard-copy, use the ``Copy Graph'' and ``Steal Data'' buttons to generate a single graph with all the bound states, making sure to fill the box for ``Your Name'' first.) Draw an energy level diagram for the solutions you've found. Did the energies shift up or down from those (with the same number of nodes) you found in part 1?

The shelf potential resembles in some ways the potential for electrons near a metal surface. As they leave the metal, the electron potential jumps by the ``work function'' (famous because of its role in the photoelectric effect, which we will likely cover in class). There are lots of surface physics problems for which the exponentially decaying electron wavefunction outside the surface has important consequences... About how many nanometers does the ground-state wave-function extend into the ``vacuum'' beyond Xshelf = 1 nm, before the probability density decays to 1% of its peak value?

(S2e) Continuum States

There are also solutions to Schrödinger's equation which have energies greater than Vshelf=3eV. They are unbound - their wave-functions spread out everywhere, because they have enough energy to escape from the well. (We can't use the normalization condition for these solutions, since they are evenly spread out over everything, but there are ways to fix that...) For V-E<0, what should the wave-function do on the shelf? Change Distance to Shoot to 4 nm, and observe the behavior as you vary E between 3 eV and 5 eV. Draw a picture in your writeup of the wave-function at E=4 eV. Is the wave-length inside the well larger or smaller than outside? Why? (That is, how does the wavelength depend on the size of E-V?) What should the wave look like for energies much larger than 3 eV? Does the program look the same as your prediction? (Note: energies bigger than 100 eV will start causing numerical problems...)

Links Back

Statistical Mechanics: Entropy, Order Parameters, and Complexity, now available at Oxford University Press (USA, Europe).