Chemistry 3730 Fall 1997 Test 2

Duration: 90 minutes

Aids permitted: calculator, Maple

All answers must be recorded in full in the exam booklets provided. If you use Maple, you should explain what you used it for, although you need not write down every Maple command which you used to solve a problem.

The value of each question is noted. The aggregate value of all questions is 40.

Note the useful information which follows the questions.

Answer all questions.

1. What information do you have if you know the quantum number in a quantum mechanical central-force problem? Be specific. [2 marks]
2. 1. Write down the helium atom Hamiltonian. [3 marks]
   2. Explain how perturbation theory can be used to approximate the energy of the ground state of a helium atom. [5 marks]
3. The hydrogen wavefunction is

   

   Find the value of A. [10 marks]

   Maple hint: You will have to tell Maple that is a positive quantity.

4. Assuming that is small, relate this quantity to , the uncertainty in the kinetic energy. [5 marks]
5. Consider a particle which is free to move from x=0 to x=L (but not outside these limits) subject to the potential energy

   

   Find the best wavefunction of the form

   

   Compute the energy corresponding to this wavefunction. Report this energy as a numerical multiple of (to four significant figures). [15 marks]

   Since you will do all of the calculations using Maple, be very careful to give adequate details of what you are doing in your test paper. You don't need to write down intermediate results (some of which are gross), but you should explain using words and equations what you did.

   Maple hints: You must tell Maple that L is positive before you do any integrals. In Maple, |u| is expressed abs(u). When you invoke solve, store the result in a variable (called, for instance, s and defined by s = solve(...)). If there are multiple solutions, you will then be able to say s[1] to refer to the first solution, s[2] to refer to the second, and so on. Use subs() to substitute these values into earlier expressions. For example, if you wanted to substitute a variable called a by s[1] into an expression stored in a variable called A, you would type subs(a=s[1],A); You will need to use evalf() in the final stage(s) of the computation.