Lecture 28 — The Planck Distribution Chapter 8, Monday March 24 th

1. Lecture 28 — The Planck Distribution Chapter 8, Monday March 24th • Before Planck: Wien and Rayleigh-Jeans • The ultraviolet catastrophe • The Planck distribution Reading: All of chapter 8 (pages 160 - 185) Homework 8 due Mon. Mar. 31st Assigned problems, Ch. 8: 2, 6, 8, 10, 12

2. Black-body spectrum Rayleigh-Jeans equation Stefan-Boltzmann Law Define spectral energy distribution such that u(l)dl is the fraction of energy per unit volume in the cavity with wavelengths in the range l to l + dl. Power per unit area radiated by black-body R = s T4 Found empirically by Joseph Stefan (1879); later calculated by Boltzmann s = 5.6705 × 10-8 W.m-2.K-4. u(l)dl = (# modes in cavity in range dl) × (average energy of modes) Density of modes: Wien's displacement Law lmT = constant = 2.898 × 10-3 m.K, or lmT-1

3. Wien, Rayleigh-Jeans and Planck distributions Wilhelm Carl Werner Otto Fritz Franz Wien

4. The ultraviolet catastrophe There are serious flaws in the reasoning by Rayleigh and Jeans Furthermore, the result does not agree with experiment Even worse, it predicts an infinite energy density as l 0! (This was termed the ultraviolet catastrophe at the time by Paul Ehrenfest) Agreement between theory and experiment is only to be found at very long wavelengths. The problem is that statistics predict an infinite number of modes as l0; classical kinetic theory ascribes an energy kBT to each of these modes!

5. Planck's law (quantization of light energy) Planck found an empirical formula that fit the data, and then made appropriate changes to the classical calculation so as to obtain the desired result, which was non-classical. The problem is clearly connected with u(l) , as l 0 In fact, no classical physical law could have accounted for measured blackbody spectra Max Planck, and others, had no way of knowing whether the calculation of the number of modes in the cavity, or the average energy per mode (i.e. kinetic theory), was the problem. It turned out to be the latter. The problem boils down to the fact that there is no connection between the energy and the frequency of an oscillator in classical physics, i.e. there exists a continuum of energy states that are available for a harmonic oscillator of any given frequency. Classically, one can think of such an oscillator as performing larger and larger amplitude oscillations as its energy increases.

6. Planck's law (quantization of light energy) WN,M N = 1  1, 1 N = 2   1, 2, 1 N = 3    1, 3, 3, 1 N = 4     1, 4, 6, 4, 1 Pascal’s triangle N distinguishable oscillators in the walls of the cavity M indistinguishable energy elements (quanta) e, so that UN = Me

7. Planck's law (quantization of light energy) N distinguishable oscillators in the walls of the cavity M indistinguishable energy elements (quanta) e, so that UN = Me

8. Maxwell-Boltzmann statistics Planck postulated that the energies of harmonic oscillators could only take on discrete values equal to multiples of a fundamental energy e = hn, where n is the frequency of the harmonic oscillator, i.e. 0, e, 2e, 3e, etc.... Then, Un = ne = nhn = 0, 1, 2... Where n is the number of modes excited with frequency n. Although Planck knew of no physical reason for doing this, he is credited with the birth of quantum mechanics. Define energy distribution function: Then, 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.

9. The new quantum statistics Solving these equations together, one obtains: Multiplying by D(l), to give.... This is Planck's law Replace the continuous integrals with a discrete sums:

10. Vibrational energy levels for diatomic molecules n = 0, 1, 2... (quantum number) w w = natural frequency of vibration Energy