Practice Problem Set 3

1. Prove that, at very small values of x, .
2. Give the ground state electronic configuration of a sulfur atom (atomic number 16).
3. Beryllium nuclei have a spin of 3/2.
   1. At a given magnetic field strength, how many energetically different spin states are there?
   2. In NMR, there is a selection rule: Only transitions for which are observed. In the absence of spin-spin coupling, how many lines would the NMR spectrum of beryllium have?
4. In our Universe, electrons have a spin of 1/2 and the periodic table has 8 main groups, plus the transition, lanthanide and actinide series. However, other Universes are conceivable.
   1. Suppose that electrons had a spin of 3/2. How many main groups would there be?
   2. Describe the periodic table of a Universe where the electrons have a spin of 1. Hint: There is a qualitative difference between particles with integer and half-integer spin.
5. Show that if is a solution of Schrödinger's equation, the variational energy computed with this wavefunction is exactly the energy obtained by solving Schrödinger's equation.
6. Suppose that we didn't know how to solve the particle-in-a-box problem. The following trial wavefunction satisfies the boundary conditions:

   

   Determine the best value of the variational parameter and the corresponding variational energy. Show that your result conforms to the variational theorem. Normalize the wavefunction.