1 Quantum Theory 1

2 Topics l Discovery of the Electron l Millikan’s Experiment l Blackbody Radiation l An Act of Desperation l Summary

3 The Discovery of the Electron l 1838 – M. Faraday l Discovery of arc light in dilute gas l 1857 – H. Geissler l Discovery of fluorescence in dilute gas l 1880s – W. Crookes l Development of cathode ray tube l 1895 – J. Perrin l Cathode rays found to be negatively charged

4 The Discovery of the Electron l 1896 – P. Zeeman l First evidence of atomic particles with a specific charge-to-mass ratio l 1897 – J.J. Thomson l Measurement of charge-to-mass ratio of cathode ray particles. l In effect, this was the discovery of the electron and the dawn of modern physics, in particular, particle physics (also known as high energy physics)

5 The Discovery of the Electron C cathode A,Bslits D,Edeflection plates

6 The Discovery of the Electron C cathode A,Bslits D,Edeflection plates Thomson found that whatever gas was used he always got the same q/m ratio of about 0.7 x C/Kg. The cathode particles were called corpuscles by Thomson and later electrons by Lorentz

7 Millikan’s Experiment Millikan began experiments in 1909 He measured e to be e = x C Charged oil drops Suspended oil drops

8 Blackbody Radiation Josef Stefan 1835 – 1893 In 1879, Josef Stefan found an empirical formula to describe the power radiated by an ideal black body σ = x W/m 2 K 4 Five years later, Ludwig Boltzmann was able to derive this formula from thermodynamics

9 Blackbody Radiation The radiation energy density per unit wavelength u(λ) was found to depend only on the absolute temperature T Wikemedia Commons

10 Blackbody Radiation In the late 19 th century, the race was on to derive, from first principles, the black body spectrum The task was to compute the energy density of radiation within a cavity

11 Rayleigh-Jeans Law Lord Rayleigh 1842 – 1919 In 1900, Lord Rayleigh showed that the radiation energy per unit volume per unit wavelength in a cavity must have the form In 1905, Sir James Jeans showed that a = 8πk, where k is Boltzmann’s constant

12 Rayleigh-Jeans Law Lord Rayleigh 1842 – 1919 Unfortunately, when integrated over all wavelengths, this law predicted an absurdity: the electromagnetic energy density in a cavity is infinite! This result, called the ultraviolet catastrophe, revealed a serious flaw in classical physics

13 An Act Of Desperation

14 The October Revolution In early October 1900, Max Planck found through trial an error that the formula could reproduce the experimental results An Act of Desperation

15 An Act of Desperation

16 Planck was to make a presentation at the German Physical Society meeting on October 19, But since he had arrived at his formula through inspired guesswork, he was rather anxious to derive it in a more respectable way! An Act of Desperation

17 Planck’s Model l A cavity is modeled as a system of N oscillators (presumably, atoms) that can emit and absorb electromagnetic radiation l The radiation can be distributed amongst the oscillators in a large number of ways Ω l Ludwig Boltzmann had earlier introduced the formula S = k ln Ω, for the total entropy of a system An Act of Desperation

18 Planck’s Model l Planck computed the entropy per oscillator assuming that the radiation is absorbed and emitted in discrete amounts l He regarded this assumption as nothing more than a mathematical trick, an “act of desperation”. But, try as he might, he could not make headway without this crazy assumption An Act of Desperation

19 Planck’s Model l For a given , Planck assumed that the energy in the cavity is in the form of M indistinguishable quanta of energy  distributed over N indistinguishable oscillators l The total energy of the radiation is E = M . So the average energy per oscillator is e = E/N = M  /N l Planck derived s, the entropy per oscillator, i.e., s = S / N An Act of Desperation

20 Planck’s Model l To compute the entropy per oscillator Planck had to compute the number of ways to distribute M indistinguishable quanta amongst the N indistinguishable oscillators l He then had to determine the value of the quantum of energy  An Act of Desperation

21 The Reluctant Revolutionary, which relates the To determine , Planck started with his formula for the spectral density u(λ) and computed the from it the entropy per oscillator, using the thermodynamic relation 1/T = ∂S/∂E, which relates the absolute temperature T, entropy S and average energy E of N objects. l When he compared his two expressions for the entropy he found they agreed only if he assumed each quantum had energy An Act of Desperation

22 An Act of Desperation And thus did this reluctant revolutionary start, in an “act of desperation”, the quantum revolution The constant, h, which does not appear in classical physics is called Planck’s constant in his honor

23 Extra Credit (5) 1. Using Planck’s model, derive Planck’s formula for the entropy per oscillator 2. Then use 1/T = ∂S/∂E = ∂s/∂e to show that the average energy per oscillator is given by e =  /(e  /kT - 1) Hint: for large N, ln N! ~ N ln N – N and recall that e = εM / N Due 10/06/09

24 Planck’s Model The number of ways of distributing M indistinguishable things amongst N indistinguishable boxes is Extra Credit (5)

25 Summary l Discovery of Electron l J.J. Thomson measured e/m ratio in 1897 l Millikan measured e in 1909 l Ultraviolet Catastrophe l Rigorous application of the laws of thermodynamics, Newton and Maxwell by Lord Rayleigh led to an absurd prediction: a hot oven should emit an infinite amount of energy!

26 Summary l The Birth of Quantum Physics l In 1900, Planck derived the correct formula for the blackbody spectral density under the assumption that electromagnetic energy changes in discrete amounts given by  = hc/ , an assumption he regarded as “an act of desperation” l He spent many years, thereafter, trying to evade this assumption; but failed!