Chemistry 3730 Fall 1997 Final Examination

Duration: 3 hours

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.

Note the useful information at the end of this paper.

Answer all questions.

1. How many normal modes would HCN have? [2 marks]
2. 1. Write down the full Hamiltonian of the diatomic molecule AlH. (Aluminium has atomic number 13.) [5 marks]
   2. Write down the electronic Hamiltonian of AlH. [3 marks]
3. Briefly compare the variational and perturbation methods for solving quantum mechanical problems. [5 marks]
4. The ground-state wavefunction for the harmonic oscillator is

   

   where

   

   Calculate the average potential energy for a harmonic oscillator in its ground state. Express your answer in multiples of [10 marks]

   Maple hints: Tell Maple that is positive but don't otherwise define this parameter. Do the substitution by hand in your test paper to complete the solution.

5. Consider the following putative harmonic oscillator wavefunction:

   

   1. Is this a solution of the quantum mechanical harmonic oscillator problem? Explain in detail how you decided this question. If it is a solution, give its quantum number. [8 marks]

      Maple hints: This time, define in terms of k, and before you do any calculations.

   2. Is this wavefunction normalized? Again, explain carefully how you decided this question. [4 marks]

      Maple hints: Enter the following two commands into your Maple session before attempting this part:

      		> beta := 'beta';
      		> assume(beta>0);

      These instructions undefine and instruct Maple to treat it as a positive parameter.
6. Pure rotational spectra can be obtained by working in the microwave region. The selection rule is .
   1. Give an equation for the energy of a photon absorbed during a pure rotational transition of a rigid rotor. [3 marks]
   2. The lines of the rotational spectrum of AlH are spaced apart. What is the equilibrium bond length? Aluminium has a single stable isotope of mass while the mass of is . [10 marks]
7. The current periodic table has s, p, d and f blocks. New elements are being produced in high-energy physics labs more or less routinely however so that we can anticipate eventually adding a g block. What would you predict to be the atomic number of the first g-block element? [3 marks]
8. With which of the Cartesian coordinates (x, y, z), if any, is compatible? [10 marks]
9. A particle confined to the region is subjected to a potential

   

   Using the variational wavefunction

   

   find an approximation to the ground-state energy. [10 marks]

   Maple hints: You will get more than one solution. Store them in a variable called (say) sol. Then sol[1] is the first solution, sol[2] is the second solution, and so on. To substitute these values into another expression, use subs(). For example, if you want to substitute a variable called a by sol[1] into an expression called b, you would type

   	> subs(a=sol[1],b);

   into your Maple session.
10. A positron is the anti-particle of an electron, i.e. it has the same mass but opposite charge.
    1. ``Atoms'' of positronium (one electron and one positron, held together by the Coulomb force) can be produced and (quickly!) studied in high-energy accelerators. Positronium is of course a hydrogenic atom in which the ``nucleus'' and electron have the same mass. What is the ionization energy of positronium? [5 marks]
    2. What is the probability that the electron and positron are within one Bohr radius ( ) of each other in the ground state of positronium? Give an exact algebraic answer. [10 marks]

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

    3. Which is bigger, a hydrogen atom or a positronium atom? Explain how you arrived at your conclusion. [2 marks]
11. Discuss one of the following topics in detail. [10 marks]
    • The Born-Oppenheimer approximation
    • Differences between quantum mechanics and classical mechanics
    • Fourier Transform NMR
    • The MO treatment of the hydrogen molecular ion ( )
    • Refinements to the harmonic-oscillator, rigid-rotor model of a diatomic molecule
    • The self-consistent field method
    • Tunneling