1. Lecture 19 — The Canonical Ensemble Chapter 6, Friday February 22nd
Finish discussion of equipartition theorem
Identical particles and quantum statistics
Symmetry and antisymmetry
Bosons
Fermions
Implications for statistics
Reading: All of chapters 5 and 6 (pages )
Assigned problems, Ch. 5: 8, 14, 16, 18, 22
Homework 5 due today
Assigned problems, Ch. 6: 2, 4, 6, 8, (+1)
Homework 6 due next Friday

2. More on the equipartition theorem
Classical uncertainty:
Where is the particle?
V(x)
V = ∞
V = ∞
V = 0
W = ∞
S = ∞ x
x = L

3. More on the equipartition theorem: phase space
Area h
Cell:
(x,px)
dpx px dx x

4. Examples of degrees of freedom:

5. Quantum statistics and identical particles
Indistinguishable events 1. 1.
Heisenberg
uncertainty
principle h 2. 2.
The indistinguishability of identical particles has a profound effect on statistics. Furthermore, there are two fundamentally different types of particle in nature: bosons and fermions. The statistical rules for each type of particle differ!

6. Bosons This wave function is symmetric with respect to exchange.
This wave function is also symmetric with respect to exchange, but it is not normalized. Note that there are 3! terms (permutations), i.e. 3P1.
In general, for N particles, there will be N! (or NP1) terms in the wave function, i.e. A LOT!

7. Bosons Easier way to describe N particle system:
The set of numbers, ni, represent the occupation numbers associated with each single-particle state with wave function fi.
For bosons, these occupation numbers can be zero or ANY positive integer.

8. Fermions This wave function is antisymmetric with respect to exchange.
It turns out that there is an alternative way to write down this wave function which is far more intuitive:

9. Fermions This wave function is antisymmetric with respect to exchange.
It turns out that there is an alternative way to write down this wave function which is far more intuitive:
The determinant of such a matrix has certain crucial properties:
It changes sign if you switch any two labels, i.e. any two rows.
It is ZERO if any two columns are the same.
Thus, you cannot put two Fermions in the same single-particle state!

10. Fermions This wave function is antisymmetric with respect to exchange.
As with bosons, there is an easier way to describe N particle system:
The set of numbers, ni, represent the occupation numbers associated with each single-particle state with wave function fi.
For Fermions, these occupation numbers can be ONLY zero or one.

11. Fermions
As with bosons, there is an easier way to describe N particle system:
The set of numbers, ni, represent the occupation numbers associated with each single-particle state with wave function fi.
For Fermions, these occupation numbers can be ONLY zero or one. 2e e