1. Lecture 26 — Review for Exam II Chapters 5-7, Monday March 17th
Canonical ensemble and Boltzmann probability
The bridge to thermodynamics through Z
Equipartition of energy & example quantum systems
Identical particles and quantum statistics
Spin and symmetry
Density of states
The Maxwell distribution
Reading: All of chapters 5 to 7 (pages )
Exam 2 on Wednesday, in class
Next homework due next week

2. Review of main results from lecture 15
Canonical ensemble leads to Boltzmann distribution function:
Partition function:
Degeneracy: gj

3. Entropy in the Canonical Ensemble
M systems
ni in state yi
Entropy per system:

4. The bridge to thermodynamics through Z
js represent different configurations

5. A single particle in a one-dimensional box
V(x)
V = ∞
V = ∞
V = 0 x
x = L

6. A single particle in a three-dimensional box
The three-dimensional, time-independent Schrödinger equation:

7. Factorizing the partition function

8. Equipartition theorem
If the energy can be written as a sum of independent terms, then the partition function can be written as a product of the partition functions due to each contribution to the energy.
free energy may be written as a sum. It is in this way that each degree of freedom ends up contributing 1/2kB to the heat capacity.
Also,

9. Rotational energy levels for diatomic molecules
l = 0, 1, 2... is angular momentum quantum number
I = moment
of inertia
CO2 I2 HI HCl H2
qR(K)

10. Vibrational energy levels for diatomic molecules
n = 0, 1, 2... (harmonic quantum number) w
w = natural
frequency of
vibration
I2 F2 HCl H2
qV(K)

11. Specific heat at constant pressure for H2
CP = CV + nR
H2 boils w
CP (J.mol-1.K-1)
Translation

12. Examples of degrees of freedom:

13. Bosons
Wavefunction symmetric with respect to exchange. There are N! terms.
Another way to describe an N particle system:
The set of numbers, ni, represent the occupation numbers associated with each single-particle state with wavefunction fi.
For bosons, occupation numbers can be zero or ANY positive integer.

14. Fermions
Alternatively the N particle wavefunction can be written as the determinant of a matrix, e.g.:
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 antisymmetric with respect to exchange
It is ZERO if any two columns are the same.
Thus, you cannot put two Fermions in the same single-particle state!

15. Fermions
As with bosons, there is another way to describe N particle system:
For Fermions, these occupation numbers can be ONLY zero or one. e 2e

16. Bosons
For bosons, these occupation numbers can be zero or ANY positive integer.

17. A more general expression for Z
What if we divide by 2 (actually, 2!):
Terms due to double occupancy – under counted.
Terms due to single occupancy – correctly counted.
SO: we fixed one problem, but created another. Which is worse?
Consider the relative importance of these terms....

18. Dense versus dilute gases
Dilute: classical, particle-like
Dense: quantum, wave-like lD
Either low-density, high temperature or high mass
de Broglie wave-length
Low probability of multiple occupancy
Either high-density, low temperature or low mass
de Broglie wave-length
High probability of multiple occupancy
lD  (mT )-1/2
lD  (mT )-1/2

19. A more general expression for Z
Therefore, for N particles in a dilute gas:
and
VERY IMPORTANT: this is completely incorrect if the gas is dense.
If the gas is dense, then it matters whether the particles are bosonic or fermionic, and we must fix the error associated with the doubly occupied terms in the expression for the partition function.
Problem 8 and Chapter 10.

20. Identical particles on a lattice
Localized → Distinguishable
Delocalized → Indistinguishable

21. Spin
Symmetric
Antisymmetric }
Fermions:

22. Particle (standing wave) in a boxLx Ly Lz
Boltzmann probability:

23. Density of states in k-spacekz ky kx

24. The Maxwell distribution
In 3D: V/p 3 is the density of states in k space
density of states per unit k interval
D(k)dk gives the # of states in the range k to k + dk
Number of occupied states in the range k to k + dk
Distribution function f (k):

25. Maxwell speed distribution function

26. Density of states in lower dimensions
In 2D: A/p 2 is the density of states in k space
density of states per unit k interval
D(k)dk gives the # of states in the range k to k + dk
In 1D: L/p is the density of states per unit k interval

27. Density of states in energy
In 3D:

28. Useful relations involving f (k)

29. The molecular speed distribution function

30. Molecular Flux
Flux: number of molecules striking a unit area of the container walls per unit time.

31. The Maxwell velocity distribution function