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Phase space in quantum mechanics an

EP 332(M)
Statistical Physics
Lecture 3: Introduction to
Thermodynamics
Thermodynamics is a funny subject. The
first time you go through it, you don't
understand it at all. The second time you
go through it, you think you understand it,
except for one or

EP 332(M)
Statistical Physics
Lecture 5: The Microcanonical
Ensemble
Microcanonical Ensemble
System
Energy E
Adiabatically isolated
Macrostate M=M(E,x)=M(E,V,N)
The set of corresponding microstates form the
microcanonical ensemble.
Constant Energy Surface

EP 332(M)
Statistical Physics
Lecture 4: Phase Space and
Liouville Theorem
Microstate of a system
Complete specification of system: Positions and
velocities of all particles: 6N coordinates
Quantum mechanical microstate: Wavefunction
Macrostate of a syst

Grand-canonical ensembles
As we know, we are at the point where we can deal with almost any classical problem (see below),
but for quantum systems we still cannot deal with problems where the translational degrees of
freedom are described quantum mechanic

Prof. Dr. Roland Netz
Julius Schulz, Klaus Rinne
Exam Solutions: Advanced Statistical Physics
Part II: Problems (75P)
1 Lenoir Cycle (25P)
Consider 1 mol of an ideal gas, which initially has a volume V1 and temperature T1 at pressure p1 . The gas
undergoe

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Density matrix
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See also: Quantum statistical mechanics
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A density mat

Statistical Physics
Section 4: Assemblies of Weakly Interacting Constituents
Remember that an assembly comprises N microscopic constituents (atoms, spins etc) which
we generally refer to as particles.
By weakly interacting particles we mean that although

The Equations of the Canonical Ensemble
(N, V, T independent variables)
Full Canonical
Ensemble
= 1 k BT
e
Distinguishable
Particles
(e.g., atoms in a crystal
lattice
e
Q ( N ,V , T ) =
Full partition
function
Q ( N ,V , T ) =
Log partition
function
ln

EP 332(M): Assign. 5 (Quantum Statistical Mechanics)
1. The Hamiltonian matrix for a quantum system can be written as
0 1 0
B
H = g 1 0 1 g > 0
2 0 1 0
Compute the canonical partition function and the average energy as a
function of the temperature.
2. Co

EP 332(M): Assignment 4 (Grand Canonical Ensembles)
1. Consider a system of non-interacting, identical but distinguishable particles. Using
both the canonical and grand-canonical ensembles, nd the partition function, and
the thermodynamic functions of Int

EP 332(M): Assignment 3 (Canonical Ensemble)
1. Consider a single quantum-mechanical particle in an innite one-dimensinal well of
width L. From elementary quantum mechanics, we know that the spectrum of allowed
energies is given by
E(n) =
n2 h 2 2
n = 1,

EP 332(M): Assignment 2
1. Consider two microcanonical ensembles of identical gases separated by a partition kept
at the same temperature T, the rst with N1 molecules in a volume V1 , and the second
with N2 molecules in a volume V2 , such that N1 /V1 = N2

EP 332(M): Assignment 1
1. i) An honest die is rolled 4 times. What is the probability of having at least one 1?
ii) If two honest die are rolled 24 times, what is the probability of having at least one
double 1?
2. A population starts with a single amoeb