Classical Statistical Mechanics 1 dimensional Simple Harmonic Oscillator

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)k. BT.

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

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 E quantized! Energy levels are equally spaced! ħω ħω ħω

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: ħω n = 0, 1, 2, 3, . . • Now, let this oscillator interact with a heat reservoir at absolute temperature T, & use the Canonical Ensemble to calculate thermal average energy: <E> or < >

Quantized Energy of a Single Oscillator: ħω n = 0, 1, 2, 3, . . • 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:

• 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:

(1) This is the Thermal Average Energy for a Single One Dimensional 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.