Phys690: Homework Assignment 5
4/2/07
==> Due on Wed 4/11 <==
(1). Particle in a box --- Diffusion quantum Monte Carlo.
We use the diffusion Monte Carlo method to study the ground
state of a simple quantum system -- a particle in a
one-dimensional box.
Here are some notes (pdf file) to
remind you of the background. A Maple program,
with brief documentations,
are provided. While understanding every single detail of
the code is not a requirement, it is essential that you understand its
idea and basic structure. Simply running the code is not the right
way to proceed.
By the time
you do ``production runs'' to write up the homework, you need to have done
enough reading and experimenting so that you are familiar with the code.
The code is written such that, if necessary,
data analysis can be easily separated from the calculation.
After storing the code as file_nm,
start Maple and then type ``read file_nm;''
to get going.
Statistical dependence.
How do you expect the statistical error in the computed
ground-state energy E0 to depend on the number
of walkers, the number of blocks (measurements), and
the number of steps (sweeps) in each block, respectively?
Observe with runs.
Use the parameter choice ``dmc(0.1, 1, 7, 40, 100);''
as a reference; your runs should be comparable to or bigger
than it (i.e., tau smaller, n_wlks
greater, and the combination of n_meas_blks and
n_sweeps
greater). If the number of
steps in each block is made very small, what happens to
the error estimate? Why? Demonstrate by running two calculations
with the same total number of steps,
but different ways to divide them into blocks, e.g.,
dmc(0.1, 1, 7, 40, 100) vs. dmc(0.1, 20, 140, 2, 100).
Comparison of histogram with exact wave function.
Understand how in the code the histogram is prepared and normalized
so as to compare
with the exact ground-state wave function. Do a large calculation,
say, ``dmc(0.05, 1, 10, 100, 200);'', to obtain
a plot.
Dependence on step size tau. As we learned
in class, there is a systematic error associated with a
finite value of the ``time'' step tau.
Remind yourself why. Calculate with a few different
tau values (e.g., from 0.05 to 0.3) and see
how the systematic error in E0
depend on tau. Use at least 200 for n_wlks.
In actual research, either an extrapolation is done with
several values of tau, or it is made certain that
tau is small enough such that the systematic error
is smaller than the statistical error.
Dependence on the trial wave function phiT.
What is the variational energy of the current choice of phiT?
Invent another trial wave function that has a better variational
energy and modify the subroutine energy to use
it.
If phiT happens to be the exact ground-state wave function,
what happens to the
computed E0 and its statistical error?
How does
the mean of our estimate for E0
depend on the quality of phiT? What about its statistical error?
Convergence of the random walk. (Optional)
Compute the energy at each step
as the random walk begins and is converging.
Study how it depends on the propagation time n*tau.
This can
be done by setting n_equil_blks=0 and n_sweeps=1
and printing out blk_energy. The value for n_meas_blks
needs only to be large enough such that the total propagation
time n*tau is ~0.5.
Use a small
tau and a large number of
walkers (>103) in order to get adequate statistics.
Then,
using whatever knowledge necessary about the quantum-mechanical
system,
analytically predict the the computed E at each step
and compare with your ``experiment''.
[Hint: The initial wave function is phi(x) = 1
(-1 < x <1),
which can be expanded in terms of eigenfunctions. As we apply
exp(-tau H) successively to the expansion, what happens?]
Study how the walker distribution evolves
as a function of n.
(2). Regular fractals.
We discussed in class the Koch curve
and fractal dimension. Write a program in a language of your choice
to generate the triadic Koch curve. Show results for a suitable number
(at least 5) of iterations. Then modify the code to generate the
quadric Koch curve. Determine their fractal dimensions.