Case Study 6 Quantisation and Quanta After 1905

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Case Study 6 Quantisation and Quanta after 1905 1 Case Study 6 Quantisation and Quanta after 1905 In Case Study 6,we took the story of quantisation and quanta up to the publication of Einstein’s great paper of 1905. To recapitulate the story so far: • Planck derived the Planck formula for black-body radiation in 1900 using dodgy arguments which no-one could understand. Planck had quantised the oscillators and regarded this as no more than a “formal procedure”. • There were no papers on quanta until Einstein’s paper of 1905. • Einstein quantised the radiation field rather than the oscillators which are in equilibrium with the radiation. • Among the predictions of the theory was an explanation for the photoelectric effect. 2 Einstein’s Derivation of Planck’s Law (1) In 1906, Einstein realised that he could reformulate Planck’s ideas using Boltzmann’s statistical mechanics properly. The system of oscillators can take only discrete energy levels with energies, 0, ǫ, 2ǫ, 3ǫ, ... Boltzmann’s expression for the probability that an oscillator is in a state with energy E = nǫ in thermal equilibrium at temperature T where n is an integer is p(E) ∝ e−E/kT . Suppose there are N0 oscillators in the ground state, E = 0. Then, the number in the −ǫ/kT −2ǫ/kT n = 1 level is N0 e , the number in the n = 2 level is N0 e , and so on. Therefore, the average energy of the oscillator is found in the usual way: N · 0+ ǫ · N e−ǫ/kT + 2ǫ · N e−2ǫ/kT + ... E = 0 0 0 , −ǫ/kT −2ǫ/kT N0 + N0e + N0e + ... N ǫ e−ǫ/kT [1 + 2(e−ǫ/kT ) + 3(e−ǫ/kT )2 + ... = 0 . −ǫ/kT −ǫ/kT 2 N0[1 + (e )+(e ) + ... ] 3 Einstein’s Derivation of Planck’s Law (2) We recall the following series expansions: 1 1 =1+ x + x2 + x3 + ... =1+2x + 3x2 + ... (1 − x) (1 − x)2 Hence, setting x =e−ǫ/kT , the mean energy of the oscillator is ǫ e−ǫ/kT ǫ E¯ = = 1 − e−ǫ/kT eǫ/kT − 1 If we now follow the classical Boltzmann procedure, we allow the energy elements ǫ to tend to zero and then, expanding eǫ/kT − 1 for small values of ǫ, ǫ 1 ǫ 2 1 ǫ 3 eǫ/kT − 1=1+ + + + ···− 1 kT 2! kT 3! kT 4 Einstein’s Derivation of Planck’s Law (3) Thus, for small values of ǫ/kT , ǫ eǫ/kT − 1= kT and so ǫ ǫ E = = = kT eǫ/kT − 1 ǫ/kT Thus, if we take the correct classical limit, we recover Maxwell’s equipartition theorem for an oscillator, namely, the average energy of an oscillator in thermal equilibrium is E = kT . In order to obtain Planck’s law, however, we must not allow ǫ to go to zero. According to this simple model, the only allowable energy states are those corresponding to 0, ǫ, 2ǫ, 3ǫ,... Einstein recognised immediately the relation between this proper formulation of the quantisation of the oscillators and his previous analysis of the quantisation of the radiation itself. 5 Einstein’s Derivation of Planck’s Law (4) In his own words, ‘We must assume that, for ions which can vibrate at a definite frequency and which make possible the exchange of energy between radiation and matter, the manifold of possible states must be narrower than it is for bodies in our direct experience. We must in fact assume that the mechanism of energy transfer is such that the energy can assume only the values 0, ǫ, 2ǫ, 3ǫ, ... ’ Thus, the expression for the energy density of radiation per unit frequency interval becomes: 8πν2 8πν2 ǫ 8πhν3 u(ν)= E = = , c3 c3 eǫ/kT − 1 c3(ehν/kT − 1) if we identify the energy elements with the energy of the quanta of light ǫ = hν. 6 Einstein’s Theory of Specific Heats In the same paper, Einstein also addressed the problem of the specific heats of solids at low temperatures. At room temperature, the specific heat of many solids is described by Dulong and Petit’s Law according to which the heat capacity per mole of a solid is roughly 3R where R is molar gas constant, R = 8.3145 J k−1 mol−1. We suppose that the solid consists of N0 atoms per mole and that they all vibrate in three independent directions. Since, in equilibrium, each mode of vibration has energy kT , the total internal vibrational energy of the solid is 3N0kT . Therefore, the total internal energy is U = 3N0kT and so the heat capacity is dU C = = 3N0k = 3R dT 7 Einstein’s Theory of Specific Heats Some solids, such as beryllium, boron and carbon, had smaller heat capacities than 3R. In addition, the specific heats of solids decrease with decreasing temperature, 3R only being attained at high temperatures. Einstein proposed that his quantum formula for the average energy of an oscillator at temperature T should be used, ǫ E = eǫ/kT − 1 rather than the classical formula. For simplicity, he assumed that solids vibrate at a frequency νE, known as the Einstein frequency. Then, the internal energy of the solid is hνE U = 3N0E = 3N0 . ehνE/kT − 1 The heat capacity is found by differentiation dU hν 2 ehνE/kT C = = 3R E . hν /kT 2 dT kT (e E − 1) 8 Comparison of Einstein’s Theory with Measurements of the Heat Capacity of Diamond In his paper of 1906/7, Einstein compared the theory with measurements of the heat capacity of diamond and found encouraging, if not perfect, agreement. This was an important prediction because Walther Nernst was beginning to measure the heat capacities of solids at low temperatures. 9 Einstein on Fluctuations in Black-body Radiation One of Einstein’s most beautiful papers on quanta was written in 1909 and concerned the random fluctuations in black-body radiation. Einstein began by reversing Boltzmann’s relation between entropy and probability. W =eS/k Now divide up the volume V into a large number of cells and suppose that ∆ǫi is the energy fluctuation in the ith cell. Then, the entropy of the cell is ∂S ∂2S = (0) + ∆ + 1 (∆ )2 + Si Si ǫi 2 2 ǫi ... ∂E ∂E ! But, averaged over all cells, we know that there is no net fluctuation, ∆ǫi = 0, and therefore P 2 1 ∂ S 2 S = Si = S(0) + 2 (∆ǫi) + ... ∂E2! X X 10 Einstein (1909) Therefore, the probability distribution of the fluctuations is ∂2S (∆ǫ )2 = (const) exp 1 i W 2 2 . " ∂E ! P k # This is the sum over a set of normal distributions and so, for any single cell, we can write 2 1 (∆ǫi) Wi = (const) exp −2 " σ2 # where k σ2 = . −∂2S/∂E2 We can now go through the usual process of inverting the Planck distribution to express 1/T in terms of the energy density of the radiation and identifying it with (∂S/∂E) where the energy can be taken to be E = Vu(ν) ∆ν. ∂S 1 k 8πhν = = ln + 1 3 . ∂E T hν c u(ν) ! 11 Fluctuations in Black-body Radiation It is now straight-forward to find σ k c3 σ2 = − = hνE + E2 (∂2S/∂E2) 8πν2V ∆ν ! In terms of the fraction fluctuations, we can write σ2 hν c3 = + E2 E 8πν2V ∆ν ! This is a remarkable formula. The first term arises from the Wien region of the spectrum and is the statistical fluctuation expected if there are N = E/hν photons present, recalling that the fractional fluctuation is expected to be ∆N/N ≈ 1/N 1/2. 12 Fluctuations in Black-body Radiation The second term arises from the Rayleigh-Jeans region of the spectrum. It represents the fluctuations due to the random superposition of waves. As was shown earlier, the number of modes in the frequency range ν to ν + ∆ν is 8πν2 ∆νV Nmode = . c3 When we superimpose waves of a single mode with random phases, the fluctuations in the energy correspond to (∆E/E)2 = 1 (see TCP2, Chapter 15, if necessary) and so the second term tells us that ∆E2 1 c3 = = 2 2 . E Nmode 8πν ∆νV I find this an amazing result. It says that we should add together statistically the wave and particle aspects of the radiation field to find the total fluctuation in the radiation intensity. 13 The 1911 Solvay Conference In 1910, Nernst found good agreement between his measurements of the heat capacity of materials at low temperatures and Einstein’s theory of the specific heats of solids. He then persuaded the Belgian industrialist Solvay to sponsor the First Solvay Conference in 1911. This meeting made the issues concerning of quantum physics widely known in the physics community. 14 The Reception of the Concepts of Quanta Although the community in general found the ideas of quanta hard to swallow, by the 1911 Solvay conference, the mood was swinging in favour of quanta. Those in favour included Lorentz, Nernst, Planck, Rubens, Sommerfeld, Wien, Warburg. Langevin, Hasenohrl, Onnes. Those against included Poincare´ The solid circles show the number of authors and Jeans. who published on quanta each year. Open Rutherford, Brillouin, Marie Curie, circles - black body radiation. Perrin and Knudsen were neutral. 15 Bohr’s Theory of the Hydrogen Atom At this point, we need to change gears and review a number of the other great discoveries and problems which had arisen in the understanding of atomic physics over the period 1895 to 1911. The key events of this part of the story were: • Thomson and the discovery of the Electron.
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