Solutions for problem set 3
1. This is a closed system (T is given, not E) so we use a canonical ensemble. Because the
microsystems are not interacting, we can use the factorization theorem:
Z(T, N, .) =
1
[z(T, .)]N
GN
The dots stand for whatever other m
CPSC 314
Assignment 2 Theory
due: Monday, January 30, 2017, 11:59pm
Worth 5% of your final grade.
Note: this version has corrections to Q2 and Q3, as posted on Piazza.
Answer the questions in the spaces provided on the question sheets.
Name:
Student Numbe
Solutions for problem set 1
1. Problem repeated from last year. See solution there.
2. There are many ways to do this. I will use the formula that probability is the ratio between
the number of desired outcomes and the total number of possible outcomes, a
(1). This is a quantum grandcanonical ensemble for bosons (integer spin). The macrostate is
dened by T, , M (or , = , M if you prefer).
If there is a single atom in the system, the quantum numbers describing its states are = (i, m),
where i = 1, 2, ., M i
Solutions for problem set 2
Problems 1-3 are repeated from last year, please see solutions listed there. The reason for repeating
them is (a) there arent many problems that can be solved with microcanonical ensembles, so I have
limited choices; and (b) th
Solutions for problem set 4
1. This is a canonical ensemble, and the macrostate is dened by N, T, V . The interesting aspect
is that this is a mixed problem, with both classical and quantum degrees of freedom. This shouldnt
bother us too much, though. To
Solutions for problem set 1
1. Problem repeated from two years ago, see solution there.
2. Let nF be the number of steps taken forward, and nB the number of steps taken backwards.
Since the person takes N total steps in this time, we must have nF + nB = N
Solutions for problem set 2
1. (i) Because they are simple atoms, each one has f = 3 degrees of freedom, therefore we have
a 6(N1 + N2 )-dimensional phase space. For simplicity of notation, let me use ri , pi , i = 1, N1 for the
coordinates and momenta of
Solutions for problem set 3
1. Since T is known this is a canonical ensemble, moreover it is a classical problem. The macrostate
is dened by T, V, N . For the microstate, we already know from the prev. hmw. that each molecule
has f = 5 degrees of freedom
(1). This problem is repeated from hmw5 of last year, please see solution there (and look at the
other problems, too).
2. This is a grandcanonical classical ensemble for H. A microstate is characterized by the number
N of H in the system, of which N1 = 0,
Solutions for problem set 1
1. (i)
(a) By denition,
(1) =
dxex = ex |x= = 1
x=0
0
By denition,
1
=
2
0
1
dx ex
x
2
Switch variables to x = y dx = 2ydy :
1
=
2
0
1
2ydy ey =
y
2
dyey =
where we use the fact that this is an even function, and the known resu
Solutions for problem set 2
Problems 1-3 are repeated from last year, please see solutions listed there. The reason for repeating
them is (a) there arent many problems that can be solved with microcanonical ensembles, so I have
limited choices; and (b) th
Solutions for problem set 3
1. This is repeated from last year, see solution there (and have a look at the other problems, too).
2. This is a classical canonical ensemble. If you remember various useful tricks, this problem can
be solved in under 5 mins i
CPSC 314
Assignment 2 Coding
Due 23:59:59 on February 10, 2017,
1. (20 points) Coding with Transformations
Your task will be to create and animate a model of a horse. The main goal of the
assignment is to practice rotation, translation, and scaling transf
1
2. Microstates and Macrostates
Goal: To apply the fundamental hypothesis of statistical mechanics in simple
systems.
Consider a macroscopic system with a large number (N) particles. The macroscopic state
or marcostate can be specified by relatively few
1
P403 Problem Assignment #1: Due Monday Jan 16 10:00 am
1. Estimate the multiplicity ( N , n ) and probability P( N , n ) for a spin magnet where
x n / N is a small number and n N N . See page 3 of the notes on Microstates
and Macrostates.
2. Consider a
1
3. Entropy, Absolute Temperature and Free Energy
Goal: Apply the fundamental assumption to derive the condition for thermal
equilibrium and a definition for entropy, absolute temperature and free energy.
It follows from the fundamental assumption that i
1
6. Chemical Potential and the Grand Partition Function
Some Math Facts (see Appendix E for details)
If F(V,T) is an analytic function of state variables V and T such that
dF = SdT pdV
then it follows:
F
F
S =
, p =
T V
V T
Also since
2F
2F
S
1
4. Canonical Distribution
Consider a single microsystem in contact with a large system 0 . The total system
t 0 is isolated from the surroundings. Assume has just two possible energies
E1 and E2 separated by E .
Let p1 and p2 be the probabilities that
1
5. Continuous Energy Levels
In order to deal with translational degrees of freedom e.g. a gas of particles, where there
is dense semi-continuous spectrum of possible energies for a single particle it is
convenient to introduce the concept of density of
1
Problem Set #2 ; due Monday Jan 30 10:00 AM
1. Suppose a particle has a single (non degenerate) ground state and a two fold
degenerate excited state at an energy above the ground state energy.
(a) What is the partition function for N such particles assu
1. Introduction (Chapters 1 and 2 )
Goal: Review the empirical laws of TD so that we can contrast them with the more
fundamental approach in SM
In thermodynamics (TD) one attempts to understand the properties of macroscopic
objects such as a liter of gas
Solutions for problem set 4
1. The ensemble is canonical and quantum. The macrostate is characterized by T, N, B. The
microstate is characterized by the z-axis spin projections m1 , ., mN of all spins.
Because the spins are non-interacting and locked-in i
Solutions for problem set 5
1. Clearly this is a grand-canonical system characterized by T, V, 1 , 2 . Each atom has f = 3
degrees of freedom, and the density of probability to be in a microstate is given to us.
(a) From the normalization condition, we nd
Solutions for problem set 4
1. This is repeated from last year, please see solution listed there.
2. Let us enumerate the microstates. Clearly, generically we expect some atoms to be single and
some to be paired in molecules. Let n be the number of molecu
1 4. Canonical Distribution Consider a single microsystem in contact with a large system 0 . The total system t = + 0 is isolated from the surroundings. Assume has just two possible energies
E1 and E2 separated by E .
Let p1 and p2 be the probabilities th
1 3. Entropy, Absolute Temperature and Free Energy Goal: Apply the fundamental assumption to derive the condition for thermal equilibrium and a definition for entropy, absolute temperature and free energy. It follows from the fundamental assumption that i
1 2. Microstates and Macrostates Goal: To understand and be able to apply the fundamental hypothesis of statistical mechanics in simple systems. Consider a macroscopic system with a large number (N) particles. The macroscopic state or marcostate can be sp
1
6. Chemical Potential and the Grand Partition Function
Some Math Facts (see Appendix E for details) If F(V,T) is an analytic function of state variables V and T such that
dF = - SdT - pdV then it follows: F F S = - , p = - V T T V Also since 2F 2F S p ,
1
7. Distributions
A perfect gas is one in which the energy of a particle (microscopic constituent) is independent of the presence of other particles. The total energy is then just sum of all the single particle energies. Of course this is an approximatio