Müller I. A history of thermodynamics. The doctrine of energy and entropy (1185104), страница 50
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7.3. Max Planck (1858–1947)27Ibidem.28This is the somewhat shortened quotation from M. Planck: “A Scientific autobiographyand other papers.” Williams and Norgate, London (1950).Brush writes: I suppose that most people who read (or repeat) this quotation think Planckis referring to his quantum theory, but in fact he was talking about his struggle toconvince scientists in the 1880’s and 1890’s that the second law of thermodynamicsinvolves a principle of irreversibility, and that the flow of energy from hot to cold is notanalogous to the flow of water from a high level to a low one, as Ostwald and theenergeticists claimed.
Cf. S.G. Brush: “The kind of motion we call heat, ...” loc. cit. p.640.Photoelectric Effect and Light Quanta211Photoelectric Effect and Light QuantaHeinrich Hertz had noticed that light falling upon metals stimulates theemission of electrons. This became known as the photoelectric effect, orsimply the photo-effect. Philipp Eduard Anton von Lenard (1862–1947)investigated the effect systematically in 1902 and he found that the energyof the emitted electrons does not depend on the intensity of the incidentlight. A brighter light just produces more electrons, not more energeticones.
Instead, light of a higher frequency creates more energetic electrons.There was no explanation until Einstein stepped forward with anextrapolation of Planck’s energy quanta.29Einstein argued that, if an oscillator could only exchange quanta ofenergy hȞ with the surrounding radiation field, the emitted radiation itselfshould appear as quanta; they came to be called light quanta at first andcould, perhaps, be considered as little particles of light with the energy hȞ.If such a light quantum hits an electron, bound to a metal with less energythan hȞ, the light may kick the electron loose and make it move off with thesurplus. The higher the frequency, the higher the surplus and the quicker theelectron moves. On the other hand, if the light quantum – for lowfrequency– carries less than the binding energy of the electron to the metal,there is no emission of electrons. The threshold frequency, when emissionstarted, was found to be a characteristic property of the metal.This is all simple enough except that one has to accept the idea of lightquanta.
Since the idea was based on Planck’s theory of energy quanta, itssuccess was a first confirmation of that theory other than radiation itself.Einstein’s hypothesis of the photo-effect went a long way, perhaps even allthe way toward establishing the new quantum theory.30 Einstein receivedthe Nobel prize for this in 1921. However, among the scientists whoremained sceptical, was Planck.31Simple as the explanation of the photo-effect may be, it had a truly farreaching consequence on natural philosophy. Indeed, Einstein thuscancelled out the luminiferous ether as unnecessary by assuming that lighttravelled in quanta and therefore had particle-like properties and was notmerely a wave that required some material [the ether] to do the waving.32So the question of absolute space, in which the ether was at rest was finallydone away with.29A.
Einstein: “Über einen die Erzeugung und Verwandlung des Lichtes betreffendenheuristischen Standpunkt.” [On a heuristic point of view concerning the creation andreaction of light.] Annalen der Physik (4) 17 (1905).30 I. Asimov: “Biographies ...” loc.cit. p. 517.31 A. Hermann (ed): “Deutsche Nobelpreisträger.” loc.cit. p. 91.32 Asimov: “Biographies ...” loc.cit.
p. 589.212 7 Radiation ThermodynamicsRadiation and AtomsTime went on and Planck’s concept of energy quanta of hypotheticaloscillators in cavity walls found its way into the atom. Niels Henrik DavidBohr (1885–1962) constructed a model of the atom in 1913, whose essentialfeature is quantized energy levels for electrons in the electric field of thenucleus. That model prevailed with slight modifications to this day and bynow it is taught in elementary schools.Thus it became possible to think about atoms in equilibrium with aradiation field and – not surprisingly – Einstein was first and foremost todevelop the idea.33 He introduced the novel concept of stimulated emissionand derived Planck’s radiation formula without Planck’s interpolation. Thematter is simple enough so that we can replay it here in an understandableform on less than one page.We are interested in radiation with frequency Ȟ and spectral energydensity eȞ(Ȟ,T).
If the frequency is such that hȞ = İn–İm holds, the radiationmay be emitted and absorbed when the electron moves between the levelswith İn and İm. The emission and absorption probabilities are respectivelyR P oO# $GQ (Q ,6 )andR O oP%GQ (Q ,6 ) .Two of the three terms – those with A and C – represent spontaneousemission and absorption. They are eminently plausible.
But the third term –the one with B – is not. It represents what Einstein called induced orstimulated emission and at the end, upon reflection, we shall recognize thatthat concept was introduced ad hoc so that the argument leads to the Planckdistribution. Einstein expresses this by saying:In order for the desired result to come out we need to extend our hypotheses.The probabilities of finding atoms with energies İn and İm are proportionalto the Boltzmann factors exp(–İn/kT) and exp(–İm/kT).
Therefore theexpectation values for emission and absorption areε( A Beν (ν , T ))eε kTnÇeε kTiand Ceν (ν , T )e kTmÇeε kTi.In equilibrium both expressions must be equal so that the equilibriumspectral energy density has the form33A. Einstein: “Strahlungsemission und –absorption nach der Quantentheorie.” Deutschephysikalische Gesellschaft, Verhandlungen 18 pp. 318–323 (1916).A. Einstein: “Quantentheorie der Strahlung.” Physikalische Gesellschaft Zürich,Mitteilungen 16 pp. 47–62 (1916).A.
Einstein: “Quantentheorie der Strahlung.” [Quantum theory of radiation] PhysikalischeZeitschrift 18 pp. 121–128 (1917).Radiation and AtomsGQ (Q ,6 )213# 1.JQ% G M6 %$Since eȞ (Ȟ,) may be expected to be infinite, B must be equal to C and,since for small Ȟ the Raleigh-Jeans formula ought to hold, we maydetermine A/C and obtaineQ (Q , T )8SQ 2 hQ,hQc 3 e kT 1which is the Planck distribution.The new and original feature in Einstein’s argument is stimulatedemission.
Thus he envisages a process by which the radiation energy eȞamplifies itself by shaking a quantum hȞ loose from the atom and theprobability for this amplification is proportional to the extant value of eȞ, sothat a run-away amplification is conceivable.In the 1917-paper there is a thoughtful but inconclusive discussion aboutthe momentum exchange between matter and radiation, and about the recoilof size hcQ , or actually hcQ2 c of an atom that emits a light quantum hȞ.Although momentum is much on his mind, Einstein seems to shy awayfrom definitely assigning the momentum JQE n to a light quantum moving inthe direction n.Thus, although he came close, Einstein missed the full import of stimulatedemission, which amplifies the energy of the emission-stimulating ray of radiationby a light quantum that moves in the direction of the ray.
This fact was later – inthe 1920’s and 1930’s – recognized and incorporated into the treatment of thephoton gas by astrophysicists, see below. But then Einstein did not look back andso he – and everybody else – failed to recognize the potential applicability of thephenomenon for the creation of coherent, unidirectional, and monochromatic light.The result lay dormant for 50 years, before some clever electrical engineers used itin the 1960’s to construct an amplifier that became known by the acronym maser =microwave amplifier by stimulated emission of radiation.
Shortly afterwards thesame was done for light in the laser.Still, Einstein’s improved derivation of the Planck formula was eagerlyaccepted. Bose34 comments on the argument and calls it a remarkablyelegant derivation.35 And yet, Bose had some reservations, essentially basedon the fact that Einstein’s final result needs to refer to the Rayleigh-Jeansformula which is purely classical.
Bose’s own argument avoids this. Bose3435S.N. Bose: “Plancks Gesetz ...” loc. cit.Actually it is Einstein who calls Einstein’s argument bemerkenswert elegant [remarkablyelegant], because he translated Bose’s paper. However, we may assume that Bose’sunpublished original English version used words to that extent.214 7 Radiation Thermodynamicswas the first to take the cells of phase space seriously. We recall thatBoltzmann had previously introduced cells as the smallest elements that canaccommodate a point (x,c), or (x,p); Boltzmann had considered this – cf.Chap.
4 – as a conceptual artifact introduced for mathematical convenience,and he did not need to speak about the cell-size, because it dropped out ofhis final results. For Bose that size had to be equal to h3, if he wished toobtain the Planck distribution. Also Bose introduced the new way ofcharacterizing a distribution of light quanta and counting the number ofrealizations. We review Bose’s paper in the briefest possible manner inInsert 7.4.Photons, A New Name for Light QuantaEinstein’s hypothetical light quanta had the energy hȞ, but they could notreally be considered particles until they were firmly endowed with amomentum.
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