Chemistry 3730 assignment 9

Due: Wednesday, Nov. 5, 10:00 a.m.

1. The particle-in-a-box model is a very simple model for the conduction electrons in a metal. However, the electrons are not really free. Because the nuclei are positively charged and form a regular lattice, their effect is to add a periodically repeating potential energy term to the Hamiltonian.

   Suppose that we have a one-dimensional metal of length L. (This might be an appropriate model for a thin wire.) The effect of the nuclei and core electrons is modeled by a perturbing potential

   

   with equal to the lattice spacing in the metal. Compute the energy correction occasioned by this potential for the energy levels occupied by the conduction electrons.

   Maple hint: The answer will be very complicated unless you tell Maple that the quantum number n is an integer. Also, use simplify() to simplify the result.

2. We all know that the potential energy of a pair of particles coupled by the electromagnetic force is

   

   where and are the charges of the particles and r is the distance between them. Suppose that this is not exactly right. Suppose instead that

   

   where k is a constant. The ionization energy of hydrogen calculated from the usual Bohr formula for the energy levels is consistent with the experimental value to at least 1 part in . What is the maximum value of k consistent with this information? Express your answer in numerical form and make sure to get the units right. The 1s wavefunction is