CHE 485: STATISTICAL MECHANICS AND THERMODYNAMICS
ASSIGNMENT # 7 SOLUTIONS
TOPICS: (1) FLUCTUATIONS (2) ISING MODEL
1. (Chandler 3.15) For an open multicomponent system show that
(
)
where
is the fluctuation from the average of number of
to
particles o
Introduction to Statistical Mechanics, David Chandler
Exercise 1.2
Problem Statement
An equation of state for a rubber band is either:
where , , and are constants, is the length of the rubber band, and the other symbols have their
usual meaning. Which of
CHEM 221A Problem Set 3
September 18, 2014
Jon Witte
CTDL 2.9) Given: H |n = En |n .
(a) For an arbitrary operator A, prove n | [A, H] | n = 0.
Recall the Hamiltonian is Hermitian. Thus, we have
n | [A, H] | n = n | AH | n n | HA | n
= En n | A | n n H A
CHEM 221A Problem Set 1
September 4, 2014
Jon Witte
1.) Consider a particle in a box of length L. Using the general form of the eigenfunctions
(with quantum number n), evaluate the following expectation values to show that:
(a) x =
L
2
Recall the eigenfun
Atkins Physical Chemistry 9th Edition
Problem 1.1
Problem Statement
Recent communication with the inhabitants of Neptune has revealed that they have a Celsiustype temperature scale, but based on the melting point (0N) and boiling point (100N) of their
mos
Chemistry 220A Problem Set 2
(due September 11, 2012)
1. The central limit theorem says that the distribution function for a random variable,
X = x1 +x2 +.+xM , is a Gaussian distribution when dierent xi s are uncorrelated
and M . Here, you will consider
QM 1 Homework
Sebastian Requena
Fall 2011
1
Cohen-Tannoudji KI Exercise 2
Consider a particle whose Hamiltonian is given by:
2
H=
d2
(x)
2m dx2
(1)
Where is a positive constant.
a) Integrate the eigenvalue equation of H between and + . Letting approach z
CHEM 221A Problem Set 2
September 7, 2014
Yuezhi Mao
CTDL 2.1
a.
For simplicity, all the s have been omitted (e.g. |m |m ).
U (m, n) = (|m n|)
= ( n|) (|m )
= |n m| = U (n, m)
b.
|m , |n are eigenstates of H.
[H, U (m, n)] = HU (m, n) U (m, n)H
= H |m n|
Introduction to Statistical Mechanics, David Chandler
Exercise 1.4
Problem Statement
Suppose you have two pieces of rubber band, each one obeying the equation of state studied in
Exercise 1.2. The temperature, length per mole, and mole number for the firs
Introduction to Statistical Mechanics, David Chandler
Exercise 1.5
Problem Statement
Consider the system pictured in Fig. 1.7. The piston between subsystems I and II is permeable
to species 1 but not to species 2. It is held in place with the application
Atkins Physical Chemistry 9th Edition
Problem 1.2
Problem Statement
Deduce the relation between the pressure and mass density, , of a perfect gas of molar mass
M. Confirm graphically, using the following data on dimethyl ether at 25C, that perfect
behavio
Atkins Physical Chemistry 9th Edition
Problem 1.4
Problem Statement
The molar mass of a newly synthesized fluorocarbon was measured in a gas microbalance.
This device consists of a glass bulb forming one end of a beam, the whole surrounded by a
closed con
Atkins Physical Chemistry 9th Edition
Problem 1.3
Problem Statement
Charless law is sometimes expressed in the form V=V0(1+), where is the Celsius
temperature, is a constant, and V0 is the volume of the sample at 0C. The following values for
have been rep
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QM 2 Homework
Sebastian Requena
Fall 2011
1
Cohen-Tannoudji HII Exercise 2
In a two dimensional vector space, consider the operator whose matrix, in an
orthonormal basis cfw_|1 , |2 , is written:
0
i
y =
i
0
(1)
a) Is y Hermitian? Calculate its eigenvalue
CHEM221A
Programming #1: Particle in a Box
Due: October 9, 2014
1
Expectations
In addition to providing answers to each of the bolded questions in this problem statement,
you should provide a printed copy of your code.
2
Getting Started
You have been prov
Chemistry 220A Problem Set 3
(due September 20, 2012)
1. Do three or more of the following Exercises from IMSM: 1.5, 1.6, 1.8, 1.14, 1.16 and
1.17.
2. Do two or three of the following Exercises from IMSM: 2.6, 2.7 and 2.9.
3. Show that the following equat
Chemistry 220A Problem Set 5
(due October 4, 2012)
1. Do Exercises 3.15 and 3.16 in IMSM.
2. Do Exercises 3.18 - 3.20 in IMSM.
3. Do Exercise 3.21 in IMSM.
4. In class, we analyzed a phase transition that occurs in a macroscopic model of a
rubber band. Fo
Chemistry 220A Problem Set 5
(due October 9, 2014)
1. Do Exercises 3.22 and 3.23 in IMSM.
2. Consider a one-dimensional harmonic oscillator in a heat bath at temperature T .
The Hamiltonian for the oscillator is
H(x, p) = p2 /2m + m 2 x2 /2
where x is the
Chemistry 220A Problem Set 4
(due September 27, 2012)
1. Do three or more of the following Exercises from IMSM: 2.20 - 2.24 and 2.26.
2. In class, we analyzed a phase transition that occurs in a model of a rubber band.
For small applied tensions, f < f ,
Chemistry 220A Problem Set 1
(due September 4, 2012)
1. Consider a polymer chain with N units pictured in Figure 1. The position of the
ith unit is ri , i = 1, 2, ., N , and the separation between adjacent units is given by
ri+1 ri = ni ,
1 i N 1,
where i
Chemistry 220A Problem Set 6
(due October 18, 2012)
1. Do Exercises 3.22 and 3.23 in IMSM.
2. Consider a one-dimensional harmonic oscillator in a heat bath at temperature T .
The Hamiltonian for the oscillator is
H(x, p) = p2 /2m + m 2 x2 /2
where x is th
Chemistry 220A Problem Set 9
(due November 13, 2012)
1. Do Exercises 5.22 and 5.23 in IMSM.
2. Do Exercise 5.25 in IMSM
3. Do Exercise 5.26 in IMSM
4. In class, we argued that scale invariance near the critical point suggested that the
logarithm of the pa
Chemistry 220A Problem Set 8
(due November 1, 2012)
1. Do Exercises 5.1 and 5.2 in IMSM.
2. (a) Do Exercise 5.3 in IMSM.
(b) For the same d = 1 Ising model considered in Part (a), determine the pair
correlation function, si sj , as a function of the dista
Chemistry 220A Problem Set 7
(due October 25, 2012)
1. Do Exercises 4.22 - 4.26 in IMSM.
2. Consider a volume V with a gas of atoms that form clusters. In other words, N
atoms in the volume V can be partitioned into groups, with N1 monomers, N2
dimers, N3
Chemistry 220A Problem Set 11
(due November 29, 2012)
1. Do Exercise 7.25 in IMSM.
2. Do Exercise 7.28 in IMSM
3. In this problem you will investigate liquid structure and solvation in a lattice gas
model of a liquid. For some guidance, you might consider
Homework 12 (Due Wednesday, October 9th)
1. The chemical potential at T = 0 equals the Fermi energy. Obtain in the low temperature limit
(but for T > 0) by keeping the next order in the expansion for f3/2 (z). Specically, use
f3/2 (z) =
]
[
4
2
3/2
2
(lo
Chemistry 220A Problem Set 12
(The last one! Due anytime before Final Exam day, December 12, 2012.)
1. Work through questions 8.4 to 8.8 in IMSM.
2. 8.21 in IMSM.
3. 8.24 in IMSM.
4. 8.26 in IMSM.
5. 8.27 in IMSM.
6. 8.28 in IMSM.
7. Your professor visite
CHEM 221A Hints for PSet 6
October 7, 2014
Yuezhi Mao
Hints for problem set 6
The last question of problem 3: It is asking for the result of a single measurement rather than expectation
values. We will not require any explicit form of these wavefunctions.