Lecture 29 — The Planck Distribution Lecture 29 — The Planck Distribution Chapter 8, Wednesday March 26Chapter 8, Wednesday March 26thth

•The Planck distribution

•The Free energy of a photon gas

•Radiation pressure

•Quiz (if time)

Reading: Reading: All of chapter 8 (pages 160 - 185)All of chapter 8 (pages 160 - 185)Homework 8 due Mon. Mar. 31stHomework 8 due Mon. Mar. 31stAssigned problems, Assigned problems, Ch. 8Ch. 8: 2, 6, 8, 10, : 2, 6, 8, 10,

1212

20 40 60 80 1000

2

4

6

84 perfect scores

Q1 - 7.9Q2 - 7.3Q3 - 6.0

Num

ber

of s

tude

nts

Score (%)

Exam 2 statisticsExam 2 statistics

Planck's law (quantization of light energy)Planck's law (quantization of light energy)

N distinguishable oscillators in the walls of the cavityM indistinguishable energy elements (quanta) hso that UN = M

1 1 1 11 !; 1 ln 1 ln

! 1 ! B

N M U U U UW S k

M N

1 1is the average energy of a single oscillator, i.e. NU U NU

1 1 1 11 ln 1 ln/ / / /B

U U U US k

hc hc hc hc

1 1/ /4 5

/ 8 8gives ;

1 1B Bhc k T hc k T

hc hcU u U

e e

(energy quantization)hc

h

Maxwell-Boltzmann statisticsMaxwell-Boltzmann statistics

Define energy distribution function: 0

exp / ,such that 1Bf A k T f d

Then,

0 0exp( / )B Bf d A k T d k T

This is simply the result that Rayleigh and others used, i.e. the average energy of a classical harmonic oscillator is kBT, regardless of its frequency.

Planck Planck postulatedpostulated that the energies of harmonic oscillators could that the energies of harmonic oscillators could only take on discrete values equal to multiples of a fundamental only take on discrete values equal to multiples of a fundamental energy energy = = hh, where , where is the frequency of the harmonic oscillator, is the frequency of the harmonic oscillator, i.e.i.e. 0, 0, , 2, 2, 3, 3, , etc.etc.......

Then,Then, UUnn = = nnnhnh = = 0, 1, 2...0, 1, 2...

Where Where nn is the number of modes excited with frequency is the number of modes excited with frequency . Although . Although Planck knew of no physical reason for doing this, he is credited with Planck knew of no physical reason for doing this, he is credited with the birth of quantum mechanics.the birth of quantum mechanics.

The new quantum statisticsThe new quantum statistics

exp / exp /n n B Bf A U k T A nh k T

Replace the continuous integrals with a discrete sums:

0 0

exp /n n Bn n

U U f nh A nh k T

0 0

exp / 1n Bn n

f A nh k T

Solving these equations together, one obtains:

/

exp / 1 exp / 1 exp / 1B B B

h hcU

k T h k T hc k T

Multiplying by D(), to give....

5

8( )

exp / 1B

hcu

hc k T

This is Planck's lawThis is Planck's law

12n n

n = 0, 1, 2... (quantum number)

= naturalfrequency ofvibration

Bk T

Vibrational energy levels for diatomic Vibrational energy levels for diatomic moleculesmolecules

En

ergy

En

ergy