==> * Due on Thursday 4/24 in class * <==

We return to the problem of
``electrons on a lattice'' in HW1.
Recall that the 1-D lattice has *N* sites and we impose periodic
boundary conditions. We will fix the spin of each electron, which can
be either 'up' (u) or 'down' (d), and electrons of different spin states are
then distinguishable.
Recall the **single-electron**
Hamiltonian is given by

/ - 1, if i and j are near-neighbors; H_{ij} = | \ 0 , otherwise.

- Let's choose
*N=4*. Suppose we have 3 'up' (u) electrons and 1 'down' (d) electron. Further, in this part, suppose electrons do not interact with each other. What is the total ground-state energy? From the single-particle wave functions, form the necessary determinant and calculate the probability for finding the configuration`ud--u--o--u`in the ground state. (The configuration means 3 'u' electrons on sites 1, 2, and 4 respectively, the 'd' electron on site 1, and site 3 empty.) Explain your numerical result with a simple counting arguement. - Suppose we include a simple form of the electron interaction: when
an 'u' and a 'd' electron are on the same lattice site, there
is an interaction of strength
*U*. This model of electron hopping plus on-site interaction is known as the Hubbard model. It is a fundamental model in quantum many-body physics. If we want to answer the same questions as in (1), we can diagonalize the many-body Hamiltonian directly. What is the dimension of the Hilbert space? Qualitatively, how do you think the answers (the ground-state energy and the probability for`ud--u--u--o`) in (1) will change for a positive*U*? How about a negative*U*?

** (2) Sakurai Problem 6.7.**

** (3a) Interacting fermions in a SHO (part a). **
--- in
pdf file.

** (3b) Interacting fermions in a SHO (part b). **
Obtain the ground-state energies by perturbation
theory, up to second-order. Compare them with the exact results obtained in
(3a). Discuss the reliability of perurbation theory as a function of
the parameter alpha (e.g., by graphs).

** (4) Sakurai Problem 5.23. **

Note that the question in part (b) is
**not** "Can we find higher excited states under *first-order*
perturbation theory?" Rather, it asks if we can find higher excited
states, period.