29::196 Computational Physics:
Assignments for week 11
Readings: Yevick: pp 217-226; email 3 questions by Tuesday
11/15/05 NOON.
Optional Readings: Newman and Barkema, MC methods in SM ,
Chapters 1 and 2.
Homework (due Thursday 12/1):
1) Consider the one-dimensional Ising model with N spins (no
magnetic field).
Calculate the partition function, the average
energy and the two spin correlation with
periodic and free boundary conditions.
Show that the low temperature expansion
and the high-temperature expansion yield the same result.
Using Mathematica, try to reproduce these
results using the Metropolis algorithm
2) Consider the quantum hamiltonian
H=
((1/2)*p1^2+(1/2)*x1^2+g*x1^4+(1/2)*p2^2+(1/2)*x2^2+g*x2^4-b*x1*x2;
with g=>0 and 0<=b<1; Discuss the spectrum
in the limits
g->0 or b->0;
When g=0 AND b=0, show that there is a special
degeneracy.
Calculate numerically the first six energy levels
for values of g and b of your choice.
3) Problem 16 in Chapter 9 of Yevick
4) Using C++, create an array of 10,000 random numbers
homogeneously distributed
between two values of your choice. Make a good
looking histogram of the distribution using
a class Histogram in a PLOT.h file (as
in last week exercise in Ch. 8).
Assume that the lowest and largest
values of a GIVEN list are not known before the
histogram is made (do not use the
addValue function).
Follow the general idea of the Mathematica notebook histogram.nb
.