Classical Statistical Mechanics: 1 dimensional Simple Harmonic Oscillator

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.

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

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

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

1 D Simple Harmonic Oscillator

Quantum Statistical Mechanics: 1 dimensional Simple Harmonic Oscillator

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!

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 <>

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:

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

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

ε can be rewritten: Final Result:

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.

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.