P5870 - Modern Computational Methods in Solids

Spring 2011
Prof. Adrian E. Feiguin

Homework 2

1- In Exercise 2.1 we have seen that the selection rules prevent the Hamiltonian from mixing states with different parity, such that matrix elements with m+n even are non-zero, and zero otherwise. Rewrite the program seen in class to separate the matrix in two blocks, corresponding to (m odd, n odd) and (m even, n even) and diagonalize the two smaller blocks independently.

2- Exercise 2.2: Modify the program for the potential well to solve the hydrogen atom.

3- By using the expressions (46)-(48) in the notes, solve the hydrogen atom analytically for the case of a single gaussian function of the form (41). Write the energy as a function of the variational parameter alpha, and minimize the energy with respect to this parameter to obtain the optimal value. Compare graphically the obtained solution with the exact one (38). Compare the variational energy to the exact one E=-1/2.