Case Study 6 Quantisation and Quanta After 1905

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 . ∂E T hν c3u(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.

• Rutherford and the discovery of the nucleus.

• The structure of atoms and the Bohr model of the hydrogen atom.

16 Thomson and the Discovery of the Electron

The discovery of the electron in 1897 is traditionally attributed to J.J. Thomson. He showed that the charge-to-mass ratio, e/me, of cathode rays is about two thousand times that of hydrogen ions. But others were hard on his heels.

• In 1896, Pieter Zeeman discovered the broadening of spectral lines when a sodium ﬂame is placed between the poles of a strong electromagnet. Lorentz interpreted this result as the splitting of the spectral lines due to the motion of the ‘ions’ in the atoms about the magnetic ﬁeld direction – a lower limit of 1000 was found for the value of e/me.

17 The Discovery of the Electron

• In January 1897, Emil Wiechert used the magnetic deﬂection technique to obtain a measurement of e/me for cathode rays and concluded that these particles had mass between 2000 and 4000 times smaller than that of hydrogen, assuming their electric charge was the same as that of hydrogen ions. He obtained only an upper limit to the speed of the particles since it was assumed that the kinetic energy of the cathode rays was Ekin = eV , where V is the accelerating voltage of the discharge tube.

• Walter Kaufmann’ experiment was similar to Thomson’s. He found the same values of e/me, no matter which gas ﬁlled the discharge tube, a result which puzzled him. He found a value of e/me 1000 times greater than that of hydrogen ions. He concluded ‘...that the hypothesis of cathode rays as emitted particles is by itself inadequate for a satisfactory explanation of the regularities I have observed.’

18 Thomson’s Contributions (1)

J.J.Thomson was the ﬁrst to interpret the experiments in terms of a sub-atomic particle. In his words, cathode rays constituted

‘...a new state, in which the subdivision of matter is carried very much further than in the ordinary gaseous state.’

• In 1899, Thomson used one of C.T.R.Wilson’s early cloud chambers to measure the charge of the electron. He counted the total number of droplets formed and their total charge. From these, he estimated e = 2.2 × 10−19 C, compared with the present standard value of 1.602 × 10−19 C. This experiment was the precursor of the famous Millikan oil drop experiment, in which the water-vapour droplets were replaced by ﬁne drops of a heavy oil, which did not evaporate during the course of the experiment.

19 Thomson’s Contributions (2)

• Thomson also demonstrated that the β-particles emitted in radioactive decays and those ejected in the photoelectric effect had the same charge-to-mass ratio as the cathode rays.

• Thomson pursued a much more sustained and detailed campaign than the other physicists in establishing the universality of what became known as electrons, the name coined for cathode rays by Johnstone Stoney in 1891.

• It seems fair to regard Thomson as the discoverer of the ﬁrst sub-atomic particle.

20 How Many Electrons Are There in the Atom?

• There might have been as many as 2000 electrons, if they were to make up the mass of the hydrogen atom. The answer was provided by a brilliant series of experiments carried out by Thomson and his colleagues, who studied the scattering of X-rays by Thomson scattering in thin ﬁlms.

• Thomson worked out the scattering rate using the classical expression for the radiation of an accelerated electron. The cross-section for the scattering of a beam of incident radiation by an electron is

e4 8πr2 = = e = 6 653 10−29 m2 σT 2 2 4 . × , 6πǫ0mec 3 2 2 the Thomson cross-section, where re = e /4πǫ0mec is the classical electron radius.

21 How Many Electrons Are There in the Atom?

• In collaboration with Charles Barkla, Thomson showed that, except for hydrogen, the number of electrons was roughly half the atomic weight. The number of electrons and, consequently, the amount of positive charge in the atom increased by units of the electronic charge.

• The key question was, ‘How are the electrons and the positive charge distributed inside atoms?’ In the picture favoured by Thomson, the positive charge was distributed throughout the atom and, within this sphere, the negatively-charged electrons were placed on carefully chosen orbits – the rather subtle ‘plum-pudding’ model of the atom (see later).

22 The Discovery of the Nucleus

The discovery of the nucleus resulted from a brilliant series of experiments carried out by Rutherford and his colleagues Hans Geiger and Ernest Marsden in the period 1909-12 while Rutherford was at Manchester University. α-particles pass through thin films rather easily, suggesting that much of the volume of atoms is empty space, although there was evidence of small-angle scattering. Rutherford persuaded Marsden to investigate whether α-particles were deflected through large angles on being fired at a thin gold foil. A very small number of particles almost returned along the direction of incidence. In Rutherford’s words:

‘It was quite the most incredible event that has ever happened to me in my life. It was almost as incredible as if you ﬁred a 15-inch shell at a piece of tissue paper and it came back and hit you.’

Typically, the α-particles were travelling at 10,000 km s−1.

23 The Discovery of the Atomic Nucleus

In 1911, Rutherford hit upon the idea that, if all the positive charge were concentrated in a compact nucleus, the scattering could be attributed to the repulsive electrostatic force between the incoming α-particle and the positive nucleus. He used his knowledge of central orbits in inverse-square law ﬁelds of force to work out the properties of what became known as Rutherford scattering. The angle of deﬂection φ is φ 4πǫ m cot = 0 α 2 2 p0v0, 2 2Ze where p0 is the collision parameter, v0 is the initial velocity of the α-particle and Z the nuclear charge. The probability that the α-particle is scattered through an angle φ is

1 4 φ p(φ) ∝ 4 cosec , v0 2 the famous cosec4(φ/2) law derived by Rutherford, which was found to explain precisely the observed distribution of scattering angles of the α-particles.

24 The Discovery of the Atomic Nucleus

The fact that the scattering law was obeyed so precisely, even for large angles of scattering, meant that the inverse-square law of electrostatic repulsion held good to very small distances indeed. The nucleus had to have size less than about 10−14 m, very much less than the sizes of atoms, which are typically about 10−10 m.

The papers by Rutherford, Geiger and Marsden from 1909 to 1913 are classics of 20th century physics. Rutherford attended the ﬁrst Solvay Conference in 1911, but made no mention of his remarkable experiments, which led directly to his nuclear model of the atom. Remarkably, this key result for understanding the nature of atoms made little impact upon the physics community at the time, and it was not until 1914 that Rutherford was thoroughly convinced of the necessity of adopting the nuclear model of the atom. Before that time, however, someone else did – Niels Bohr, the ﬁrst theorist to apply successfully quantum concepts to the structure of atoms.

25 Problems of Atomic Models

• The electrons in the atom cannot be stationary because of Earnshaw’s theorem, which states that any static distribution of electric charges is mechanically unstable. They either collapse and disperse to inﬁnity under the action of electrostatic forces.

• The alternative is to place the electrons in orbits, the ‘Saturnian’ model of the atom. The most famous of these early models was that due to the Japanese physicist Nagaoka, who attempted to associate the spectral lines of atoms with small vibrational perturbations of the electrons about their equilibrium orbits. The problem with this model was that perturbations in the plane of the electron’s orbit are unstable, leading to instability of the atom as a whole.

26 Radiative Instability

Suppose the electron is in a circular orbit of radius a. Equating the centripetal force to the electrostatic force of attraction between the electron and the nucleus of charge Ze, 2 2 Ze mev = = ¨r 2 me| | 4πǫ0a a where |¨r| is the centripetal acceleration. The time it takes the electron to lose all its kinetic energy by radiation is

E 2πa3 T = = |dE/dt| σTc Taking the radius of the atom to be a = 10−10 m, the time it takes the electron to lose all its energy is about 3 × 10−10 s. Something is profoundly wrong. As the electron loses energy, it moves into an orbit of smaller radius, loses energy more rapidly and spirals into the centre.

27 The Plum-Pudding Model

The solution was to place the electrons in orbits such that there is no net acceleration when the acceleration vectors of all the electrons in the atom are added together so that the electrons had to be very well ordered in their orbits. If there are two electrons in the atom, they can be placed in the same circular orbit on opposite sides of the nucleus and so, to first order, there is no net dipole moment as observed at infinity, and hence no dipole radiation. There is, however, a finite electric quadrupole moment and hence radiation at the level (λ/a)2, relative to the intensity of dipole radiation. Since λ/a ∼ 10−3, the radiation problem can be significantly relieved. By adding more electrons to the orbit, the quadrupole moment can be cancelled out as well and so, by adding sufficient electrons to each orbit, the radiation problem can be reduced to manageable proportions. This was the basis of Thomson’s plum-pudding model of the atom.

28 Niels Bohr

• Niels Bohr completed his doctorate on the electron theory of metals in 1911. He convinced himself that this theory was seriously incomplete and required further mechanical constraints on the motion of electrons at the microscopic level. • He spent the following year in England, working for 7 months with J.J.Thomson at the Cavendish Laboratory in Cambridge, and four months with Ernest Rutherford in Manchester. Bohr was immediately struck by the signiﬁcance of Rutherford’s model of the nuclear structure of the atom and began to devote all his energies to understanding atomic structure on that basis.

29 Bohr and the Quantum Theory

• He appreciated the distinction between the chemical properties of atoms, associated with the orbiting electrons, and radioactive processes associated with activity in the nucleus. On this basis, he could understand the nature of the isotopes of a particular chemical species.

• Bohr realised that the structure of atoms could not be understood on the basis of classical physics. The obvious way forward was to incorporate the quantum concepts of Planck and Einstein into the models of atoms. Recall Einstein’s statement,

‘...for ions which can vibrate with a deﬁnite frequency, ... the manifold of possible states must be narrower than it is for bodies in our direct experience.’

This was precisely the type of constraint which Bohr was seeking.

30 The Earliest Attempts

In 1910, a Viennese doctoral student, A.E.Haas, realised that, if Thomson’s sphere of positive charge were uniform, an electron would perform simple harmonic motion through the centre of the sphere. For a hydrogen atom, for which Q = e, the frequency of oscillation of the electron is 1/2 1 e2 = ν 3 . 2π 4πǫ0mea ! 2 Haas argued that the energy of oscillation of the electron, E = e /4πǫ0a, should be quantised and set equal to hν. Therefore, 2 πmee a h2 = . ǫ0 According to Haas’s approach, Planck’s constant was simply a property of atoms, whereas those already converted to quanta preferred to believe that h had much deeper signiﬁcance.

31 Bohr’s First Attempt (1)

In the summer of 1912, Bohr wrote an unpublished Memorandum for Rutherford, in which he made his ﬁrst attempt at quantising the energy levels of the electrons in atoms. He proposed relating the kinetic energy T of the electron to the frequency ν′ = v/2πa of its orbit about the nucleus through the relation

1 2 ′ T = 2mev = Kν , where K is a constant which he expected would be of the same order of magnitude as Planck’s constant h.

This criterion ﬁxed the kinetic energy of the electron about the nucleus. For a bound circular orbit, mv2 Ze2 = 2 a 4πǫ0a where Z is the positive charge of the nucleus in units of the charge of the electron e.

32 Bohr’s First Attempt (2)

The binding energy of the electron is 2 1 2 Ze U E = T + U = 2mev − = −T = , 4πǫ0a 2 where U is the electrostatic potential energy. The quantisation condition enables both v and a to be eliminated from the expression for the kinetic energy of the electron, so that

mZ2e2 = T 2 2. 32ǫ0K which was to prove to be of great signiﬁcance for Bohr.

33 John William Nicholson

In 1912, the Cambridge physicist John William Nicholson showed that, although the Saturnian model of the atom is unstable for perturbations in the plane of the orbit, perturbations perpendicular to the plane are stable for orbits containing up to ﬁve electrons. The frequencies of the stable oscillations were multiples of the orbital frequency and he compared these with the frequencies of the lines observed in the spectra of bright nebulae, particularly with the ‘nebulium’ and ‘coronium’ lines.

Performing the same exercise for ionised atoms with one less orbiting electron, further matches to the astronomical spectra were obtained. The frequency of the orbiting electrons remained a free parameter. When he worked out the angular momentum associated with them, Nicolson found that they were multiples of h/2π. This work perplexed Bohr.

34 Bohr (1913)

The breakthrough came in early 1913, when H.M.Hansen told Bohr about the Balmer formula for the wavelengths, or frequencies, of the spectral lines in the spectrum of hydrogen, 1 ν 1 1 = = (1) R 2 − 2 , λ c 2 n where R = 1.097 × 107 m−1 is the Rydberg constant and n = 3, 4, 5,... . As Bohr recalled much later,

‘As soon as I saw Balmer’s formula, the whole thing was clear to me.’

Bohr could determine the value of his constant K from the running term in 1/n2 in the Balmer formula which can be associated with the energy of the orbit. For hydrogen with Z = 1, 2 mee = T 2 2 32ǫ0n K

35 How Bohr Derived the Quantisation Condition

Then, when the electron changes from an orbit with quantum number n to that with n = 2, the energy of the emitted radiation would be the difference in kinetic energies of the two states. Applying Einstein’s quantum hypothesis, this energy should be equal to hν. Bohr found that the constant K was exactly h/2. Therefore, the energy of the state with quantum number n is 2 mee = T 2 2 8ǫ0n h The angular momentum of the state could then be found by writing 1 ′2 2 2 ′2 T = 2Iω = 2π mea ν . It immediately follows that nh J = Iω′ = 2π This is how Bohr arrived at the quantisation of angular momentum according to the ‘old’ quantum theory.

36 The Pickering Series

In the ﬁrst paper of his trilogy of 1913, Bohr noted that a similar formula could account for the Pickering series, which had been discovered in 1896 by Edward Pickering in the spectra of stars. In 1912, Alfred Fowler discovered the same series in laboratory experiments. Bohr argued that singly-ionised helium atoms would have exactly the same spectrum as hydrogen, but the wavelengths of the corresponding lines would be four times shorter, as observed in the Pickering series. Fowler objected, however, that the ratio of the Rydberg constants for singly-ionised helium and hydrogen was not 4, but 4.00163.

Bohr realised that the problem arose from neglecting the contribution of the mass of the nucleus to the computation of the moments of inertia of the hydrogen atom and the helium ion.

37 The Rydberg Constants for H and He+

If the angular velocity of the electron and the nucleus about their centre of mass is ω, the condition for the quantisation of angular momentum is nh = µωR2 2π where µ = memN/(me + mN) is the reduced mass of the atom, or ion, which takes account of the contributions of both the electron and the nucleus to the angular momentum; R is their separation. Therefore, the ratio of Rydberg constants for ionised helium and hydrogen should be

me 1+ RHe+ M = 4  me  = 4.00160, RH 1+    4M  where M is the mass of the hydrogen atom. Thus, precise agreement was found between the theoretical and laboratory estimates of the ratio of Rydberg constants for hydrogen and ionised helium.

38 Einstein’s Reaction

The Bohr theory of the hydrogen atom was the ﬁrst convincing application of the quantum theory to atoms. Bohr’s results were persuasive evidence that Einstein’s quantum theory had to be taken really seriously for processes occurring on the atomic scale. The ‘old’ quantum theory was, however, seriously incomplete and constitutes an uneasy mixture of classical and quantum ideas.

The results provided further strong support for Einstein’s quantum picture of elementary processes. When Einstein heard of Bohr’s analysis of the Balmer series of hydrogen in September 1914, Einstein remarked cautiously that Bohr’s work was very interesting, and important if right. When Hevesy told him about the helium results, Einstein responded,

‘This is an enormous achievement. The theory of Bohr must then be right.’

39 Einstein (1916) ‘On the Quantum Theory of Radiation’

By 1916, the pendulum of scientiﬁc opinion was beginning to swing in favour of the quantum theory, particularly following the success of the Bohr’s theory of the hydrogen atom. These ideas fed back into Einstein’s thinking about the problems of the emission and absorption of radiation and resulted in his famous derivation of the Planck spectrum through the introduction of what are now called Einstein’s A and B coefﬁcients.

Einstein showed how the Maxwell and Planck distributions can be reconciled through a derivation of the Planck spectrum, which gives insight into what he refers to as the ‘still unclear processes of emission and absorption of radiation by matter.’

40 Quantum Emission and Absorption of Radiation

The paper begins with a description of a quantum system consisting of a large number of molecules which can occupy a discrete set of states Z1,Z2,Z3,... with corresponding energies ǫ1,ǫ2,ǫ3,... . The relative probabilities Wn of these states being occupied in thermodynamic equilibrium at temperature T are given by Boltzmann’s relation ǫn Wn = gn exp − , kT where the gn are the statistical weights, or degeneracies, of the states Zn. As Einstein remarks forcefully in his paper

‘[this equation] expresses the farthest-reaching generalisation of Maxwell’s velocity distribution law.’

41 Quantum Emission and Absorption

Consider two quantum states of the gas molecules, Zm and Zn with energies ǫm and ǫn respectively, such that ǫm >ǫn. Following the precepts of the Bohr model, it is assumed that a quantum of radiation is emitted if the molecule changes from the state Zm to Zn, the energy of the quantum being hν = ǫm − ǫn. Similarly, when a photon of energy hν is absorbed, the molecule changes from the state Zn to Zm.

The quantum description of these processes follows by analogy with the classical processes of the emission and absorption of radiation.

Spontaneous emission Einstein notes that a classical oscillator emits radiation in the absence of excitation by an external ﬁeld. The corresponding process at the quantum level is called spontaneous emission, and the probability of it taking place in the time interval dt is n dW = Am dt, similar to the law of radioactive decay.

42 Induced Emission and Absorption

Induced emission and absorption By analogy with the classical case, if the oscillator is excited by waves of the same frequency as the oscillator, it either gains or loses energy, depending upon the phase of the wave relative to that of the oscillator, that is, the work done on the oscillator can be either positive or negative. The magnitude of the positive or negative work done is proportional to the energy density of the incident waves.

The quantum mechanical equivalents of these processes are those of induced absorption, in which the molecule is excited from the state Zn to Zm, and induced emission, in which the molecule emits a photon under the inﬂuence of the incident radiation ﬁeld. The probabilities of these processes are written:

m Induced absorption dW = Bn ̺ dt, n Induced emission dW = Bm̺ dt.

43 Balancing Absorption and Emission

The lower indices refer to the initial state and the upper indices the ﬁnal state. ̺ is the m n energy density of radiation with frequency ν. Bn and Bm are constants for a particular physical processes, and are referred to as ‘changes of state by induced emission’.

We now seek the spectrum of the energy density of radiation ̺(ν) in thermal equilibrium. The relative numbers of molecules with energies ǫm and ǫn in thermal equilibrium are given by the Boltzmann relation and so, in order to leave the equilibrium distribution unchanged under the processes of spontaneous and induced emission and induced absorption of radiation, the probabilities must balance, that is,

−ǫn/kT m −ǫm/kT n n gne Bn ̺ = gme (Bm̺ + Am) . absorption emission | {z } | {z }

44 Derivation of Planck’s Radiation Law

In the limit T →∞, the radiation energy density ̺ →∞, and the induced processes n dominate the equilibrium. Therefore, allowing T →∞ and Am = 0, m n gnBn = gmBm. The equilibrium spectrum ̺ can therefore be written An /Bn ̺ = m m . ǫm − ǫn exp − 1 kT But, this is Planck’s radiation law. Suppose we consider only Wien’s law, which is known to be the correct expression in the frequency range in which light can be considered to consist of photons. Then, in the limit ǫm − ǫn/kT ≫ 1, n A ǫm − ǫn hν ̺ = m exp − ∝ ν3 exp − . n Bm kT kT

45 The Values of Einstein’s A and B Coefﬁcients

Therefore, we find the following ‘thermodynamic’ relations An m ∝ ν3, ǫ − ǫ = hν. n m n Bm The value of the constant can be found from the Rayleigh-Jeans limit of the black-body spectrum, ǫm − ǫn/kT ≪ 1. 8πν2 An kT ̺(ν)= kT = m 3 n c Bm hν and so An 8πhν3 m = . n 3 Bm c The importance of these relations between the A and B coefficients is that they are n n m associated with atomic processes at the microscopic level. Once Am or Bm or Bn is known, the other coefficients can be found immediately.

46 Einstein’s Real Motivation

Einstein now used these results to determine how the motions of molecules would be affected by the emission and absorption of quanta. The analysis was similar to his earlier studies of Brownian motion, but now applied to the case of quanta interacting with molecules. The quantum nature of the processes of emission and absorption were essential features of his argument.

He found the key result that, when a molecule emits or absorbs a quantum hν, there must be a positive or negative change in the momentum of the molecule of magnitude |hν/c|, even in the case of spontaneous emission. In Einstein’s words,

‘There is no radiation of spherical waves. In the spontaneous emission process, the molecule suffers a recoil of magnitude hν/c in a direction that, in the present state of the theory, is determined only by ‘chance’.’

47 Millikan (1916)

“We are confronted, however, by the astonishing situation that these facts were correctly and exactly predicted nine years ago by a form of quantum theory which has now been generally abandoned.” He refers to Einstein’s ‘bold, not to say reckless, hypothesis of an In 1916, Millikan published his results of electromagnetic light corpuscle of measurements of the dependence of the energy hν, which ﬂies in the face of photoelectric effect upon the frequency of the thoroughly established facts of the incident radiation. interference.’

48 The End of the Story

• In 1923, Arthur Holly Compton discovered the Compton effect, the scattering of X-rays by electrons, which could only be explained by assuming that the X-rays are quanta, or photons, with energy E = hν and momentum p = hν/c.

• In 1924-6, Erwin Schodinger¨ and Werner Heisenberg discover wave and matrix mechanics respectively, which soon became recognised as the foundations of quantum mechanics.

49 Einstein’s Achievement