1. Lecture 9 Energy Levels Translations, rotations, harmonic oscillator
Independence of energy storage
Many particle systems
Independent distinguishable particles
Independent indistinguishable particles

2. Energy levels from quantum mechanics
Need to calculate the partition functions to have any use of statistical mechanics
For a single particle independent modes of motion will contribute to energy and thus the partition function.
For independent particles the energy is the sum of the individual particles energies
All this will provide a rather simple way of calculating Q for many physical systems

3. Translation Consider a single diatomic molecule is the box
M - total mass of the molecule
a,b,c - dimensions of the box containing the molecule
h - Planck constant

4. Rotation and vibrations
I = ml2 - moment of inertia, m mass of one atom, l atom to atom distance
Vibrations - harmonic oscillator
 - frequency

5. Single gas diatomic molecule particle partition function
When energy is additive the partition function is a product of partition function
Therefore free energy is the sum

6. Many independent particle
Total energy is the sum of single particle energies
So the partition function is the product of single particle functions
For N independent, identical, distinguishable particles

7. Indistinguishable particles
States such as:
a) first particle in state one and second particle in state 2
and
b) second particle in state one and first particle in state 2
are the same “total” state since particle are not distinguishable
This leads to over counting of the partition function
That has to be corrected by

8. Bosons and Fermions
1/N! correction is not quite right. If particles 1 and 2 are in the same state there was no over counting. This is possible for bosons. But typically number of states is much larger than number of particles so this situation is very rare and does not contribute in the thermodynamic limit to the Q, such as to affect intensive properties.
For fermions two particles can not be in the same state - we over counted even with N! correction. Again for classical, not quantum systems, we are fine.

9. Problem 10.6
N distinguishable particles, each having two energy levels e1=1/2kT and e2=3/2kT. Calculate energy, entropy and heat capacity at constant volume. Particles are bosons.