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
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
(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 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
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 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 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 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
(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 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
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
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
8. Ideal Gases
Free Energy for an Ideal Gas of N Identical Particles Recall the chemical potential is the change in free energy when a particle is added to the system at constant volume. Thus if we know how varies with particle number r we can determine
1 11. Degenerate Fermi Gas. Temperature variation of the chemical potential We now consider the case of a perfect gas of fermions which is degenerate (i.e. not ideal). In other words the density exceeds the quantum concentration q . Recall the Fermi Dirac
12. Phase Transitions.
Example of a first order phase transition: the Gas-Solid phase transition: Real gases are not perfect but interact through a weak Van der wals interaction often called a 6-12 potential of the form:
V ( R ) = V0 [( R / R0 ) -12 - 2(
1 10. Bose Einstein Condensation (Degenerate Bose Gas) Now consider a system of N perfect (non-interacting) bosons under conditions where the mean occupancy for low energy levels is not much less than 1. Recall this occurs when
m k T q = B B2 or greater.
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
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 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 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 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 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. 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
1
9. Photon Gas
Consider a cavity with dimensions L L L with conducting walls in thermal contact with a heat bath at temperature T. The space of the cavity is evacuated but there will still be energy in the cavity in the form of standing electromagnetic w