Müller I. A history of thermodynamics. The doctrine of energy and entropy (1185104), страница 49
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Apart from the one cited above they areM. Planck: “Über eine Verbesserung der Wien’schen Spektralgleichung.” [On animprovement of Wien’s spectral equation] Verhandlungen der deutschen physikalischenGesellschaft 2 (1900) pp.
202–204.M. Planck: “Über das Gesetz der Energieverteilung im Normalspektrum.” [On the law ofenergy distribution in the normal spectrum.] Annalen der Physik (4) 4 (1901) pp. 553–563.14 These researches were published in 1901:H. Rubens, F. Kurlbaum: Annalen der Physik 4 (1901) p. 649.O. Lummer, E. Pringsheim: Annalen der Physik 6 (1901) p.
210.15 M. Planck: “Die Entstehung und bisherige Entwicklung der Quantentheorie.” [The originand subsequent development of quantum theory] Nobel lecture to the Royal SwedishAcademy of Sciences in Stockholm, held on June 2nd, 1920.206 7 Radiation ThermodynamicsNothing was then more plausible than to set [the reciprocal of] thisexpression equal to the sum of a term with the first power and a term withthe second power of the energy.Of course, it was trial and error both ways, but a little less so in the secondmanner. Obviously Planck did not quite remember his arguments after 20 years.Maybe this is the place to quote a thoughtful remark by Einstein:16 Everyreminiscence is coloured by today’s being what it is, and therefore by a deceptivepoint of view.Planck’s derivation of the radiation formula1Planck, steeped in thermodynamics, as he was, replaced /T in Rayleigh-Jeans’swsQweQand Wien’s laws byusing the Gibbs equation for the spectral entropy densitysȞ.
Thus he obtained respectivelysνeνsν8πν 2 k 1and3eνeνcekln ν 3 .4hν c BνDifferentiation with respect to eȞ provides2 sνe28πν k 122νc3e2andν sνe2νk 1hν eν,and it was between those two algebraic functions that Planck interpolated to obtainw 2 sQk1.hQ e Q c 3 e 2we 2Q8ShQ3 Q1Integration provides /T again on the left hand side and thus1Tc3 eShQ 3 Q ,8lnhQ 1 c3 e8ShQ 3 Qkif one fixes the constant of integration by requiring that eȞ ĺ for Tĺ .Solving for eȞ one obtains the Planck distribution.Insert 7.216P.A.
Schilpp (ed.): “Albert Einstein:philosophers, New York (1949).Philosopher – Scientist.” Library of livingEnergy Quanta207However, Sir James Hopwood Jeans (1877–1946), a mathematicianmuch interested in astronomy, was not convinced that the Rayleigh formulawas wrong for high frequencies. He kept a campaign going till the end ofthe first decade of the 20th century in which he criticizes the cavity modeland maintains that no stationary state can prevail in such a cavity.17 Hisarguments faded away with the growing confidence in the Planckdistribution. But the battle leaves its traces in the textbooks, because theviolet catastrophe is a handy tool for the illumination of the scientific terrainof classical physics before quantum physics prevailed.
As late as 1910Planck was moved to refute Jeans’s arguments.18 He says:The radiation theory of J.H. Jeans is the most satisfactory one according tothe present state of physics; however, it must be rejected, because it leadsto a contradiction with observations.Note that Planck, even in 1910, ten years after his radiation formula, doesnot consider his own contribution as belonging to the present state ofphysics.Note also that the low-frequency limit of the Planck distribution – the RaleighJeans formula – provides a possibility to determine the Boltzmann constant k. Wemay recall here Loschmidt’s complicated and inaccurate argument for thecalculation of k, in order to determine the molecular mass µ, cf. Chap.
4. Thisargument can now be considered obsolete and indeed Einstein in his reminiscencesspeaks of …Planck’s determination of the true size of the atom from the law ofradiation.19 On the other hand, in his work on Brownian motion in 1905 Einsteinproposes to measure k by observation of a Brownian particle, see Chap. 9; thatwould be a cumbersome method in comparison.Energy QuantaFrom the above we conclude that according to Planck’s interpolation themean energy İ of the oscillator must be equal toεhνhνkTe 1kT 2T1 ØÈ.ÉÊ ln hν Ù1 e kT ÚIf that is compared with the generic expression for İ derived from theBoltzmann factor, cf.
Insert. 7.1, namely17J.H. Jeans. Philosophical Magazine, February 1909 p. 229.J.H. Jeans: Ibidem, July 1909 p. 20918 M. Planck: “Zur Theorie der Wärmestrahlung.” [On the theory of heat radiation] Annalender Physik (4) 31 (1910) pp. 758–768.19 P.A. Schilpp (ed.): In: “Albert Einstein: Philosopher – Scientist.” “Autobiographicalnotes.” loc. cit.208 7 Radiation ThermodynamicsεÇεnεnexp[ kT] È ε ØkTln É Ç exp[ kTn ]Ù ,ÚT Ê n 02n 0Ç exp[ εnkT]n 0we obtainfHexp[ M6]¦PP01JQ1 G M6.Obviously the equation represents the summation of an infinite geometricseries provided that İn = nhȞ holds.Thus one may conclude – or must conclude – that the oscillator is notable to accommodate all energies, but only equidistant energies 0, hȞ,2hȞ,… The oscillator can absorb – and emit – only energy quanta of size hȞand, if the eigen-frequency grows, those quanta become ever bigger. Forlarge eigen-frequencies the quanta are so big that the thermal motion of theparticles of the wall of the cavity cannot provide them.
Therefore highfrequency oscillators are inactive, i.e. they remain at rest, – at least that wasthe idea at first. It is because of that, that the spectral energy density eȞ ofthe radiation is concentrated at relatively low frequencies. However, whenthe temperature grows, the range of accessible frequencies becomes biggerand the bulk of the area below eȞ(Ȟ,T) shifts to the right, as observed, cf.Fig. 7.2, and as expressed by Wien’s displacement law.It is this – formally, and in retrospect – fairly straightforward argumentby which Planck has introduced the concept of quantized energy levels ofan oscillator.20 Of course, the argument was totally at odds with classicalthinking. Therefore physicists – foremost Planck himself – suspected thatthe whole thing might be a piece of mathematical jugglery without anycorrespondence to anything real in nature.
[Planck] struggled for years tofind a way around his own discovery.21At some time during this struggle Planck came up with the idea thatmaybe the emission of radiation from the oscillator indeed happened insteps of size hȞ, but that absorption was continuous.22 According to the newhypothesis the oscillator was supposed to accumulate absorbed radiationbetween two steps so that on average it would be found half-way betweennhȞ and (n + 1)hȞ. This led Planck to an alternative equation for theexpectation value İ, namely20Since molecules usually represent high frequency oscillators, their vibrational degrees offreedom do not contribute to the specific heat at normal temperatures. The same is true forthe rotation of a two-atomic molecule about the axis that links the atoms.
Thus quantummechanics finally explained that puzzling observation about specific heats.21 According to I. Asimov: “Biographies” loc.cit. p. 506.22 M. Planck: “Eine neue Strahlungshypothese.” [A new hypothesis about radiation]Verhandlungen der deutschen physikalischen Gesellschaft, February 3, 1911.Max Karl Ernst Ludwig Planck (1858–1947)εhν2hνhνkT209.e 1Accordingly, in effect the oscillator had to have energy levelsİn = (n+1/2)hȞ – instead of İn = nhȞ – so that it could never be quite withoutenergy; even for T = 0 there had to be a zero point energy.Miraculously this equation – and the concept of zero point energy – waslater confirmed by proper quantum mechanics, based on the Schrödingerequation, although continuous absorption was never taken seriously, – ornot to my knowledge.
The zero point energy is nowadays taken to be areflection of Heisenberg’s uncertainty relation applied to the oscillator.Max Karl Ernst Ludwig Planck (1858–1947)Max Planck was 42 years old when he derived the radiation formula. Hehad studied under Helmholtz, Kirchhoff and Weierstraß. His doctoralthesis23 is a rehash of Clausius’s ideas which Planck admired greatly. Heclaimed that Helmholtz had not read his work.
Kirchhoff read it anddisapproved, while Clausius was not interested.Planck’s great achievement is the formulation of the correct radiationformula and – in consequence – the realization that the formula requiredquantized energy levels of an oscillator. Of course, Planck sent the paperaround. Boltzmann received a copy and, according to Planck,24 he expressedhis interest and basic agreement with my reasoning. As there is noreflection of this reaction in Boltzmann’s work, it was probably no morethan politeness. Indeed, according to Lindley25 Boltzmann had never hadmuch time for Planck.
The two scientists had been in contact over Planck’sidea that the explanation of irreversibility required electro-magneticradiation damping and could not be explained by the kinetic theory.Boltzmann won this argument hands down. And then there was the Zermelocontroversy, see Chap. 4, which must have soured relations.Planck himself remained sceptical for many years of his own discovery,calling it an act of desperation.26 When Einstein went ahead and tookquanta seriously, Planck did not wish to follow. Instead he continued tosearch for a way to reconcile the new concept with classical physics.
Hesays: My vain efforts to incorporate the quantum of action somehow into theclassical theory took several years and much work. Some of my colleagues23M. Planck: “Über den zweiten Hauptsatz der mechanischen Wärmetheorie.” [On thesecond law of the mechanical theory of heat] Dissertation, Universität München (1879).24 Planck: Nobel lecture. loc. cit.25 D. Lindley: “Boltzmann’s atom.” loc.cit. p.
212.26 A. Hermann (ed.): “Deutsche Nobelpreisträger” loc.cit. p. 91.210 7 Radiation Thermodynamicshave seen this as tragic. But I disagree...27 Ironically Planck’s well-knownand oft-quoted dictum about the non-acceptance of new ideas, cf. Fig. 7.3,is therefore primarily applicable to himself.Planck’s own achievement, along with his partisanship of the works ofhis colleagues Nernst and Einstein, and his soft-spoken but steadfastrectitude in politically turbulent times made Planck one of the mostrenowned physicist of his time, second only to Einstein. Thus it happened atthe end of the second world war, – when Planck was fleeing the rampagingRussian army, and was picked up at the roadside by an American passportchecking patrol – that his name was recognized and he was given VIPtransport to Göttingen in a jeep.
There at the age of nearly ninety years, hebecame acting head of the Kaiser Wilhelm Institute, – the last one, because,when a worthy younger director was appointed, the institute was renamedMax Planck Institute.Planck’s head was used on early 2 deutsch-mark coins, – not for longthough, because soon a more deserving politician was found to replace him.The only way to get revolutionaryadvances in science accepted is towait for all old scientists to die.28Fig.
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