1
PHZ 5156, Computational Physics
Homework 2 Solution
September 13, 2005
Problem 1 (Garcia 1.23)
(a) v = 2r/T with r = Rcos40 and T=24*3600 s gives v=3.553076903 x102 m/s.
(b) a = v2/r = 2.583870440 x 10-2 m/s2.
(c) a' is the same with R replaced by R+2m=

Major Project: Ground state of the He atom
PHZ 5156
This problem combines what we have learned about the technique of Monte-Carlo
simulation with atomic physics. In particular, we will use Monte-Carlo techniques to
estimate the ground-state wave function

gnuplot Quick Reference
(Copyright(c) Alex Woo 1992 June 1) Updated by Hans-Bernhard Brker, April 2004 o
Graphics Devices
All screen graphics devices are specied by names and options. This information can be read from a startup le (.gnuplot in UNIX). If y

gnuplot Quick Reference
(Copyright(c) Alex Woo 1992 June 1) Updated by Hans-Bernhard Brker, April 2004 o
Graphics Devices
All screen graphics devices are specied by names and options. This information can be read from a startup le (.gnuplot in UNIX). If y

Finite-temperature equation of state
* $ h" # 'F (V , T ) = U 0 + kB T 0 ln,2 sinh& )/ % 2 kB T (. + #
Compute vibrational modes, frequencies Evaluate at a given volume V Compute F at various temperatures T with above
!
Need the Hessian!
Start again with

PHZ 5156 Final project Finite-temperature equation of state For low enough temperatures, the energy of a crystalline lattice can be described within the harmonic approximation. In other words, the energy is given by temperatureindependent constant U0 plus

Computer project 4
PHZ 5156
Results due Tuesday October 24
Please submit your code and plots wherever requested. Results can be handed in
either as a hardcopy, or as an electronic document (e.g. tex, latex, MS word, or even
a .pdf) sent via email.
This as

Computer project 3
PHZ 5156
Results due Tuesday October 10
Please submit your code and plots wherever requested. Results can be handed in
either as a hardcopy, or as an electronic document (e.g. tex, latex, MS word, or even
a .pdf) sent via email.
Conside

Computer project 2
PHZ 5156
Results due Thursday, September 21
Please submit your code and plots wherever requested. Results can be handed in
either as a hardcopy, or as an electronic document (e.g. tex, latex, MS word, or even
a .pdf) sent via email.
1.

Computer project 1
PHZ 5156
Results due Thursday, August 31
Please submit your code and plots wherever requested. Results can be handed in
either as a hardcopy, or as an electronic document (e.g. tex, latex, MS word, or even
a .pdf) sent via email.
1. Loo

Anderson model code
Not tridiagonal! We will use periodic boundary conditions!
Anderson model code, declarations
implicit none INTEGER, PARAMETER : nn=10,nn3=nn*3 ! size of square,number of sites INTEGER, PARAMETER : nbin=1000 ! for dos,part ratio REAL*8,

PHZ 5156 Final project Anderson model This problem is a simple but very powerful model for impurity states in a semiconductor. As impurities (i.e. dopants) increase in concentration, the can form an impurity band. In principle the impurity band might be m

Vi Reference Card
Modes
Vi has two modes: insertion mode, and command mode. The editor begins in command mode, where cursor movement and text deletion and pasting occur. Insertion mode begins upon entering an insertion or change command. [ESC] returns the

PHZ 5156 Final project Ground state of Helium atom This problem combines what we have learned about the technique of Monte-Carlo simulation with atomic physics. In particular, we will use Monte-Carlo techniques to estimate the ground-state wave function a

Quantum Monte-Carlo- Helium atom
12 221 2 H = " (#1 + # 2 ) " " + 2 r1 r2 r12
We assume that we can first of all take the trial wave function,
!
"T ( r1, r2 ) = A exp[#$ ( r1 + r2 )]
If we first ignore electron-electron repulsion, we get
" = 2 / a0
Where

UNIX Introduction and Quick Reference
Quick Reference
Commands may need additional information, such as file or directory names, which are typed immediately following the command name. Clarification and syntax for any command is available by typing man co

Homework 7
PHZ 5156
Due Tuesday, November 2, 2009
1. In class, we discussed the diusion equation and also a bit about Boltzmann
statistics. In this problem, we will explore diusion in the presence of an external
eld.
We saw in class that the mass current

Homework 6
PHZ 5156
Due Thursday, October 15
1. Consider the time-independent Schrodinger equation,
2
d2
+ V (x) (x) = E(x)
2m dx2
where V (x) is a periodic potential with periodicity L such that V (x + N L) = V (x)
where N is any integer.
Blochs theorem

Homework 5
PHZ 5156
Due Thursday, December 4
1. Consider a periodic charge density (r) dened such that
(r + Rn1 n2 n3 ) = (r).
(1)
We dene the vector Rn1 n2 n3 to be
Rn1 n2 n3 = n1 a1 + n2 a2 + n3 a3
(2)
where n1 , n2 , and n3 are integers and the primiti

Homework 4
PHZ 5156
Due Thursday, October 1
1. Consider the diusion equation in one spatial dimension,
2 u(x, t)
u(x, t)
=D
t
x2
with boundary conditions u(x = 0, t) = u(x = L, t) = 0.
Write a code that uses the Crank-Nicholson method, as discussed in cl

Homework 3
PHZ 5156
Due Tuesday, September 22, 2009
1. Consider the diusion or heat ow equation in two spatial dimensions
2
2
+2
x2 y
u=
1 u
2 t
a) Use the method of separation of variables and take u(x, y, t) = F (x, y )T (t)to show
that this equation ca

Homework 2
PHZ 5156
Due Thursday September 10
1. The Lotka-Volterra model is often also referred to as the predator-prey equations. These are two coupled nonlinear dierential equations that can only be solved
numerically. The equations for this model are

Homework 1
PHZ 5156
Due Thursday September 3, 2009
1. Consider the equation of a simple harmonic oscillator, in this case describing a
pendulum of length l in a gravitational eld g ,
d2 g
+ =0
dt2
l
a) Write the second-order linear dierential equation abo

Homework 6
PHZ 5156
Due Thursday, October 26
1. Consider a system with two states with energies E1 = and E2 = , in thermal
equilibrium with an external heat bath at temperature T . Recall that the partition
E
function is given by Z = T r exp ( kB T ) (tra

Homework 4
PHZ 5156
Due Tuesday, October 10
1. Prove that the Rodrigues formula generates the Legendre polynomials (i.e. the
solutions to the Legendre equation). Begin by dening,
v = (x2 1)l
and rst nd
dv
dx
and then multiply by the result by x2 1 to get

Homework 4
PHZ 5156
Due Tuesday, October 3
1. Consider a periodic charge density (r) dened such that
(r + Rn1 n2 n3 ) = (r).
(1)
We dene the vector Rn1 n2 n3 to be
Rn1 n2 n3 = n1 a1 + n2 a2 + n3 a3
(2)
where n1 , n2 , and n3 are integers and the primitive

Homework 3
PHZ 5156
Due Tuesday, September 26
1. In the second computer project, we showed that we could solve the diusion
equation for a point source at t = 0. This is what is known as a Green function.
Then time-dependent Schrodinger equation for a free

Homework 2
PHZ 5156
Due Tuesday, September 12
1
1. a) Show that the basis functions en (x) = 2 einx with n = 0, 1, 2, . forms an
orthonormal set using the denition for the inner product as,
1
e (x)en (x)dx
m
m|n =
1
In other words, show that m|n = m,n .
b