Physics 880.05: Assignment #2

1. MATLAB Sandbox, Part II.
   1. This is really just an informational problem. You can learn more through the MATLAB help by looking up rand and randn. (Note: MATLAB help is generally really good!) I've learned that the latest MATLAB versions do the seeding differently. Use help rand to learn a new way.
   2. Possible "observations" could be on the general trend for 100 to 1000 to 10000 and in particular on the size of the fluctuations from a true gaussian distribution.
   3. Two of the methods are more-or-less self explanatory (Taylor series and diagonalizing). The Pade approach is not. Be sure to look at the comments; also, a paper describing the approach is linked on the class webpage (at the bottom). You might want to google "Pade Approximant" and look at the Wikipedia and/or MathWorld entries. For part iii, the variable k corresponds to the number of terms. You don't want to print it out until the end of the program.
2. Fun with the One-Particle Stochastic Variational Method in MATLAB.
   1. When thinking about the advantages of a gaussian basis, you might consider how the matrix elements of the Hamiltonian are calculated here and how it would be done with a different basis.
   2. Ill-conditioned matrices arise if the basis is close to being linearly dependent. Recall that if there is linear dependence, the determinant will be zero. If it is close, there will be at least one very small eigenvalue and so a large ratio of eigenvalues. Look at cond under MATLAB help for a discussion of the "condition number", which is an indication of whether your matrix will have numerical problems from ill conditioning.
   3. Think about the wave functions for the ground and first excited states in a hydrogen atom. How do they differ? (Compare to the first two states for a square well, for example.)
3. Stochastic calculation of multidimensional integrals.
   1. If you get stuck on what the limits are, run the code a few times and look at the histogram. How large (positive or negative) can x values become?
   2. The difference in the histogram before and after changing eps to get a 50% acceptance should be significant. You should run the "before" case several times to appreciate the difference (and how much it fluctuates).
   3. You can check whether thermalization is important by changing the value of the number of thermalization steps to a small number (like one).
   4. What does the central limit theorem say about how the standard deviation scales with N?
   5. The action here can be thought of as the integral over a potential. What is the difference between a potential with positive and negative lambda?
4. Continuing with the Partition Function for One Particle.
   1. It may be useful to think of special cases for the matrix A.
   2. What does the index i represent "physically"?
   3. For the MATLAB part, you should think of q, q', k, and j as matrix indices, so the right side of Eq.(3) is just a matrix multiplication. You will have to think about how to construct the matrices (there are short-cut ways using meshgrid, but first think about doing it with loops).
   4. A check of your result is that A^-1A = 1.
   5. What derivatives will "bring down" x_i and x_j in the original Z[f] expression?
5. Directly Solving for the Green's function G₀.
   1. One way to proceed is to substitute for G⁰ and also for the delta function and then to project out a given k. Remember that the non-interacting system is uniform.
   2. If you don't write these down by inspection, you're working too hard. Remember that the delta function is zero in each of those regions.
   3. Try integrating the equation from tau-tau' = -epsilon to +epsilon and then taking epsilon to zero. What survives?