Chemistry 3730
Fall 1995 Midterm Examination

Answer all questions. There are a total of 30 marks.

1. Rank the following reactions in the order in which you expect tunneling to be most significant. Give your reasoning for the ranking you obtain. You may assume that these reactions all occur in the gas phase and involve distinct molecular and atomic species.
   1. 
   2. 
   3. 
2. Two atoms, A and B, combine to form a molecule AB. Before the reaction, A had a total angular momentum of and B had a total angular momentum . What are the possible values of the total angular momentum of AB?
3. Suppose that F and G are compatible observables and that G and H are compatible observables. Are F and H necessarily compatible? If so, prove it. If not, give a counterexample.
4. Describe in detail two different methods of solution for the following quantum mechanical problem: Find the bound state energies of a particle of mass m in a one-dimensional well whose potential is given by the formula

   

   where and . In other words, the potential is centred on the origin and has the following appearance:

   

   Do not attempt to solve this problem in detail. Someone with a knowledge of mathematics equivalent to your own should be able to solve this problem by simply following your instructions, perhaps with the support of your class notes. In other words, you should do everything required to solve this problem except the tedious bits of mathematics required to obtain a final answer. This question is intended to be open-ended. Do as much as you can in the time you have, but make sure to describe two methods, even if you don't have time to completely fill out both answers. Feel free to make any comments about this problem or its solutions which you feel are interesting or relevant.

Useful information

The general solution of the differential equation

is