1. Classical Statistical Mechanics: 1 dimensional Simple Harmonic Oscillator

2. Recall the Equipartition
Theorem:
In the Classical Cannonical
Ensemble it is easy to show that
The thermal average energy of a particle per independent degree of freedom is
(½)kBT.

3. The Boltzmann Distribution
Canonical Probability Function P(E):
This is defined so that P(E)dE  probability
to find a particular molecule between E & E + dE Z
Define: The Energy Distribution Function
(Number Density) nV(E):
This is defined so that nV(E)dE  number of
molecules per unit volume with energy between E
& E + dE

4. Examples: Equipartition of Energy in Classical Statistical Mechanics
Free Particle Z

5. Other Examples of the Equipartion Theorem
LC Circuit
Harmonic
Oscillator
Free Particle
in 3 D
Rotating
Rigid Body

6. 1 D Simple Harmonic Oscillator

7. Quantum Statistical Mechanics: 1 dimensional Simple Harmonic Oscillator

8. Quantum Mechanical Simple Harmonic Oscillator
Quantum Mechanical results for a simple harmonic oscillator with classical frequency ω:
n = 0,1,2,3,..
The Energy is
quantized! E
Energy levels are equally spaced!

9. Thermal Average Energy for a Quantum Simple Harmonic Oscillator
We just discussed the fact that the Quantized
Energy solution to the Schrodinger Equation for a
single oscillator is:
n = 0,1,2,3,..
Now, let this oscillator interact with a heat
reservoir at absolute temperature T, & use the
Canonical Ensemble to calculate the thermal
average energy:
<E> or <>

10. Quantized Energy of a Single Oscillator:
On interaction with a heat reservoir at T, & using
the Canonical Ensemble, the probability Pn of
the oscillator being in level n is proportional to:
In the Canonical Ensemble, the average energy
of the harmonic oscillator of angular frequency ω
at temperature T is:

11. Denominator = Partition Function Z.
Now, straightforward but tedious math
manipulation!
Thermal average energy:
Putting in the explicit form:
Denominator = Partition Function Z.

12. Denominator = Partition Function Z.
Evaluate using Binomial expansion for x << 1:

13. ε can be rewritten:
Final Result:

14. The Zero Point Energy = minimum energy of the system.
(1)
This is the Thermal Average Energy for a Single Harmonic Oscillator.
The first term is called
“The Zero-Point Energy”.
It’s physical interpretation is that, even at
T = 0 K the oscillator will vibrate & have a
non-zero energy.
The Zero Point Energy = minimum energy of the system.

15. Thermal Average Oscillator Energy:
(1)
The first term in (1) is the Zero Point Energy.
The denominator of second term in (1) is often written:
(2)
(2) is interpreted as thermal average of quantum
number n at temperature T & frequency ω.
In modern terminology, (2) is called
The Bose-Einstein Distribution:
or The Planck Distribution.