PH312 Lab 1

An Infinite Square Well With A Non-Zero Bottom

The goal in this lab is to learn the computational techniques needed to model and then analyze (and even visualize) problems in quantum mechanics. In particular, the problem of particles 'confined' to a region in which the kinetic energy of the particle is smaller than the potential energy.

From the results in the example, The 1-D Infinite Square Well With A Non-Zero Bottom, write a function (or a module) that returns a representation of a particle's wave function for all space (i.e., all x values) in the 1-D 'box' problem. Input parameters to this function should include the quantum number, n, the well width, a, and the location in space, x. (Hint: Expect to use a conditional 'If' statement in your function!)

Also, write a function that represents the energy as a function of the quantum number, n, the well width, a, the mass of the particle, m, and the potential energy in the 'bottom' of the well, U_o.

Check to see that the normalization condition is obeyed with your wave function for any given quantum number. Divide and conquer: Among those in the class, assign a different quantum number to each person. This way we should have several confirmation of the normalization condition.

Now, think about this. The normalization condition says that if you add up the probabilities for finding the particle for all locations x, that sum must add up to 1. The probability for finding the particle at location x is P(x). What, then, is P_n(x) in terms of the (input) parameters you have used thus far?

One possible application of a 1-D well of this sort is an electron 'caught' in between two p-type semiconductor layers in a transistor. If the transistor is reverse biased and the base-collector voltage is very large and positive, an electron is trapped between the base and emitter layers of the device. Another application of this is a photon trapped in a laser cavity.

Let's consider the application of an electron trapped in between layers of semiconductors.

You might have to think about this: Compute the energy of the electron versus x and quantum number. The potential energy in the collector (n-type material) which comprises the 'bottom' of the well is the difference in voltage between the base and collector. Typically the voltage in this 'experiment' is 10 Volts, do the potential energy U_o = eV = 1.6 x 10^-18 Joules. The distance between layers is typically 2 x 10^-8 meters. You can look up the mass of an electron (in Kg).

Create three separate plots of the probability of finding the electron, P_n(x), versus x for n =1, 2, and 3. I expect you will want to create a list of values in all space (outside the box as well as inside) and ListPlot the results. Please make your list so that it has actual values of x. That is, it should look something like this:

probvsx = Table[ {x,P[x,n]}, {x, -a, a, a/101}];

In this example, I chose to only include 101 points which should be plenty and only include the range -a to a. We know that P_n(x) outside the box is zero anyhow. Now assign different quantum numbers to the class in the range 4 < n < 10, and plot one P_n(x), versus x.

As a class discuss the findings, make some general conclusions concern the behaviour. What do these plots mean and tell you? Specifically, what should the average position be? If you look at where the particle is at random times, what would you expect? Record the results of your discussion in your lab report.

IF THERE IS TIME, DO THIS SECTION ----------------------------------------------------

Last term, we discovered how to compute the average momentum of a particle from its wave function. The rule is

The momentum, p, is related to x by: , so that the probability of having a momentum is

For the quantum number you used in plotted P_n(x) versus x, compute the average momentum the electron will have.

END OF SECTION ---------------------------------------------------------------------------

Actual experimental measurements on transistors have been done to look for QM behaviour. What would you look for to observe QM behaviour?

You figured out earlier, I hope, that the energy of the electron doesn't depend on its location in the well. Also, there are discrete energies that the electron can have. In the Bohr atom, another QM system (a QM system indicated that something is 'quantized') we computed the energy absorb or emitted when 'jumping' from one discrete energy to another. Of course, the energy need to jump must also be quantized - the reason for discrete spectra or colours in the hydrogen lamp light.

How easy it is to observe the QM behaviour depends on the amount if energy needed to make a transition between QM energies. Compute the energy required to increase the energy of the electron from discrete energy E₁ to E₂, from E₁₀ to E₁₁ and E₁₀₀ to E₁₀₁. Comment on the size of these energies as compared to the thermal energy an electron has at room temperature, which is 1/40 eV = 1/40 * 1.6 x 10^-19 Joules. Do you think physicist who looked for QM behaviour in a transistor can observe it? Assuming one has used the best possible equipment already, suggest possible improvements that could be made in experiments that look for discrete energy transitions in a transistor.

ASIDE: For a moment, consider a similar experiment that is strictly classical. As a class, discuss the classical experiment of a marble trapped in a closed tube (a 1-D well) whose ends are completely elastic. In this case, its speed will remain the same, but its velocity isn't. What should the average momentum and position be if you look the marble at random times?

Now let's explore the classical limit of the QM result for our electron. In the classical regime, there is a lot of energy around so the QM number is large. Find the quantum number of an electron in transistor assuming its kinetic energy is equal to its thermal energy at room temperature. This is probably easiest to do by hand and a calculator.

Plot the probability of finding the electron, P_n(x), versus x for the 'classical limit n'

Compare ideas of where it is likely and unlikely to find the particle if you look at it at random times. (If you did the 'if there is time section' , also comment on what the average momentum would be.) How do these results compare to those in the strictly classical 'marble experiment' in the ASIDE?