Chemistry 3730 Spring 1997 Test 2

A periodic table is attached.

Aids permitted: calculator, Maple.

Your solutions should explain exactly what computations need to be performed, even if Maple is used to help with the algebra.

Test duration: 90min.

1. Give a qualitative description of the gas-phase IR absorption spectrum of isotopically pure . Briefly describe the quantum mechanical origin of major spectral features. [5 marks]
2. Write down the ground state electronic configuration of germanium. Does the number of unpaired electrons in the ground state correspond to the valence of this atom? Explain. [5 marks]
3. A helium atom is in an excited state. What are the possible values of the total electronic (orbital+spin) angular momentum quantum number? [5 marks]
4. Explain the physical basis of the Born-Oppenheimer approximation. Outline the Born-Oppenheimer procedure for solving the molecular problem in quantum mechanics. [10 marks]
5. Calculate the average kinetic energy of a harmonic oscillator in its ground state:

   

   where is a constant. Report your answer in terms of and . Compare the average kinetic energy to the total energy of the oscillator. [10 marks]

   Maple notes: The number is Pi in Maple. The function is called exp(x) in Maple. It's easier to do the calculation in terms of b and then to substitute the value of b in by hand at the end than it is to do this problem any other way. Use assume(b>0); to tell Maple that b is a positive number. You may find it useful to run your answer through the simplify() function before doing any further work with it.

6. Suppose that, instead of the usual quadratic potential, a pair of particles were bound by the potential

   

   where is a constant and is the displacement of the bond from its equilibrium length. Use the trial wavefunction

   

   Find the best value of b and the best energy obtainable for wavefunctions of this form. Normalize the wavefunction. Express the energy in terms of , and but leave the normalization factor in terms of b. [15 marks]

   Mathematical note: The equation used to find the best value of b may have several solutions. Only solutions whose real part is positive are physically admissible.

   Maple notes: Maple will need to be helped out to solve this problem. Type assume(b>0); into your Maple session before attempting to solve this problem. Use simplify() to get the energy into a simple form after computing it. At one stage of the problem, Maple will probably return a list of solutions (separated by commas). Only one is physically reasonable. It may be necessary for you to do some work (e.g. substitutions) by hand.