1 Answer all questions in this section.

1. Suppose that electrons had a spin of instead of the usual .
   1. How many valence electrons would lithium have in this alternate Universe? [2 marks]
   2. Name the next element in lithium's group in this alternate Universe. [2 marks]
2. A sample placed in a strong magnetic field develops a macroscopic magnetization. Briefly explain the quantum mechanical basis of this effect. [4 marks]
3. Explain briefly how perturbation theory can be used to calculate the energy levels of atoms. [5 marks]
4. Explain briefly how the effective potential ( ) of Born-Oppenheimer theory is calculated. [5 marks]
5. Both the proton and electron have spin . Show that a hydrogen atom, the composite system made up of one of each of these fermions, can only be a boson. [5 marks]
6. The angular momentum of a two-dimensional system is given by . Given the wavefunction of such a system, how would you calculate the average angular momentum? Use explicit forms for the operators and expand any brackets to show the operations which will have to be carried out. [5 marks]
7. Are kinetic energy and potential energy compatible observables for the one-dimensional harmonic oscillator? [10 marks]
8. For a particle in a one-dimensional box whose walls are at x=0 and x=L, the wavefunction has the general form

   

   Determine the values of A, B and k. [10 marks]

   Useful fact:

   

9. 1. What kind of spectrometer would you need to measure the bond stiffness k for a heteronuclear diatomic molecule? [2 marks]
   2. The fundamental vibrational transition of occurs at . Calculate k for this molecule assuming harmonic-oscillator behavior. The isotopic masses of and are and , respectively. [10 marks]
10. A Hückel MO calculation for ethene gives orbital energies of -1 and 1 while a similar calculation for benzene gives the following orbital energies (and degeneracies): -2, -1 (2), 1 (2), 2.

      
    Figure: 1,3,5-triazine
    

    An extension of HMO theory for heteroatoms gives the orbital energies of methanimine ( ) as -1.2808 and 0.7808 while those of the aromatic molecule 1,3,5-triazine (Fig. 1) are -2.2656, -1.2808 (2), 0.7808 (2), 1.7656. All orbital energies are given relative to the carbon Coulomb integral ( ) in units of the bond (``resonance'') integral .

    One empirical aromaticity scale suggests that benzene and 1,3,5-triazine are roughly equally aromatic, i.e. that they are subject to a similar degree of resonance stabilization. Do the Hückel calculations agree? [10 marks]