Chemistry 3730 Fall 2000 Assignment 2

1.

Suppose that a particle is confined to a two-dimensional square box, i.e. one for which L_x=L_y=L. It is easy to find different sets of quantum numbers which give the same energy. For instance, we get the same energy if n_x=1 and n_y=2 as we do if n_x=2 and n_y=1. Of course, this is a trivial sort of degeneracy. However, there are more interesting degeneracies in which the two sets of quantum numbers are not just permutations of each other. Find one. [5 marks]

Hints: Maple has a function called isolve() which finds integer solutions to equations. Its syntax is similar to that of solve(). There will be a certain amount of trial and error involved in solving this problem. The idea is to set up an equation which will have two or more significantly different and physically significant (i.e. with quantum numbers that make sense for a particle in a box) solutions when the energy is degenerate.

2.

Calculate the expected value of the kinetic energy for the harmonic oscillator for n=0, 1 and 2. Can you detect a pattern in your answers? Relate the average kinetic energy to the total energy of the oscillator. [10 marks]

Hints: Maple will need to know that $\beta$ is positive. Once you have computed the average kinetic energies and picked up the pattern, use the definitions $\beta = \sqrt{\sqrt{km}/\hbar}$ and $\omega_0=\sqrt{k/m}$ to relate the kinetic and total energies. This is easiest to do by hand.

3.

Show that the harmonic oscillator with n=0 obeys the uncertainty principle. [10 marks]

Maple hint: Maple will need to know that $\beta>0$ and that $\hbar>0$.

4.

Work out (by hand is easiest) the coordinates of the classical turning points for the harmonic oscillator as a function of the quantum number n and of the parameter $\beta$. Write a function which returns the probability that the particle is to be found outside of the classical turning points. Then plot your function starting at n=0 and going up to some reasonably large value of the quantum number. What happens to this probability at large n? [15 marks]

Hints and notes: The ability to compute Hermite polynomials must be loaded into Maple by the command

with(orthopoly,H);

Make sure that $\beta$ is undefined before you start this problem. Maple will need to know that $\beta$ is positive. When writing the function that computes the probability, use the fact that the probability that the particle is outside the classical turning points is twice the probability that it is (for instance) to the right of the positive turning point. Furthermore, you will have to use simplify() in your function to get Maple to cancel out the $\beta$'s after performing the integral. The calculations are a little CPU intensive so start by plotting a limited range of values, extending the range a little at a time until you run out of patience.