Müller I. A history of thermodynamics. The doctrine of energy and entropy (1185104), страница 48
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As long as there is something somewhere to absorb the radiationand reemit it, the intermediate reflections are irrelevant. In fact, a singlecharge e with mass m connected to the wall by a linearly elastic springcapable of motion in the x-direction (say) should be sufficient. The springmust only be in thermal contact with the wall so that the oscillating masshas the mean energy İ = kT, cf.
Insert 7.1. And there must be one spring ofeigen-frequency Ȟ for every frequency of radiation.Now, if physicists know anything very well, it is the harmonic oscillator;so they were on home ground with the one-oscillator model of a cavity. It istrue that in the present case the oscillating mass m has a charge e so thatthere is radiation damping, but that was no difficulty for the top scientists inthe field. Actually, as early as 1895, Planck had written a long article6 inwhich he showed that the equation of motion of a one-dimensionaloscillator with mass m, charge e, and eigen-frequency Ȟ in an electric fieldE(t) reads approximately, i.e. for weak damping7x 8S 2 e 2 2Q x 4S 2Q 2 x3mc 3eE (t ) .mIt is true that E(t) is a strongly and irregularly varying function in thecavity, but only the Fourier component will appreciably interact with theoscillator which has its eigen-frequency Ȟ.
Let the energy density residing inthat component be 1/2İ0EȞ2, see Chap. 2. This represents 1/6 of the spectralenergy density eȞ of the cavity radiation, because the y- and z-componentsof the electric field also contribute to the energy density, and so do thecomponents of the magnetic field; all of them contribute equal amounts.Thus it turns out – from the solution of the equation of motion – that themean kinetic and potential energy İ of the oscillator is related to theradiative energy density eȞ , or the energy flux density JȞ = c/4 eȞ by,Q67E 8SQ4 E32H.M.
Planck: “Über elektrische Schwingungen, welche durch Resonanz erregt und durchStrahlung gedämpft werden.” [On electrical oscillations excited by resonance and dampedby radiation] Sitzungsberichte der königlichen Akademie der Wissenschaften in Berlin,,mathematisch-physikalische Klasse, 21.3.1895.
Wiedemann s Annalen 57 (1896) p. 1.Planck was much interested in radiation; primarily because he believed for a long time thatradiation damping is the essential mechanism of irreversibility. Boltzmann opposed theidea and eventually Planck disabused himself of it.This equation and the following argument are too complex to be derived here, even as anInsert. However, they are replayed in all good books on electrodynamics. I found aparticularly clear presentation in R.
Becker, F. Sauter: “Theorie der Elektrizität.” [Theoryof electricity.] Vol. 2 Teubner Verlag, Stuttgart (1959).Violet Catastrophe203Therefore, all that John William Strutt (1842–1919) – Lord Rayleighsince 1873 – had to do was to insert the mean energy İ of the oscillator inorder to come up with JȞ(Ȟ,T), the spectral energy flux density of the blackbody radiation. According to the best of Rayleigh’s – or anybody else’s –knowledge at the time, that mean energy is kT, cf. Insert 7.1, so thatRayleigh obtained8J Q (Q , T )c 8SQ 2kT4 c3( Rayleigh - Jeans formula) .9The formula fits the observed curve well for small frequencies, but it is adisaster for large ones: To begin with, the expression is not even integrableand, besides, it increases monotonically. These circumstances becameknown as the violet catastrophe, – or ultraviolet catastrophe10 – because thehigh frequencies, beyond the violet in the visible spectrum, were very badlyrepresented by the formula indeed.Obviously, in order to agree with observations, cf.
Fig. 7.2, oscillatorswith high eigen-frequencies Ȟ must get less than their classical share İ = kTof energy. And the share must depend on the value of the eigen-frequencyand decrease with it. Planck asked the question: How much do theoscillators get? How much in Latin is quantum – with plural quanta– and soPlanck’s answer to the question, and all it entailed, became eventuallyknown as quantum mechanics.11The violet catastrophe of cavity radiation heralded the fall of classicalphysics which amounted to a scientific revolution. It started in 1900 withPlanck’s paper: “Zur Theorie des Gesetzes zur Energieverteilung imNormalspektrum.”12 Ironically nobody at the time noticed the full signifycance of what had begun, certainly not Planck himself, – and not for manyyears.
We proceed to consider this.8Lord Rayleigh: Philosophical Magazine 49 (1900) p. 539.We shall discuss Jeans’s contribution below.10 So named by Paul Ehrenfest in 1910, – posthumously says S.G. Brush: “The kind ofmotion we call heat …” loc. cit. p. 306.
And indeed, by that time, to all intents andpurposes, the Raleigh-Jeans theory was dead.11 Of course, Planck did not write Latin, but the Latin word Quantum is routinely used in theGerman language meaning portion, or share, or ration.12 [On the theory of the law of energy distribution in the normal spectrum] M. Planck:Verhandlungen der deutschen physikalischen Gesellschaft 2 (1900) p. 202.Normal spectrum is Planck’s word for the black body spectrum.9204 7 Radiation ThermodynamicsExpectation value of the energy of a classical oscillatorWe recall the Boltzmann factor, by which the probability of a body to have anÈε Øenergy İn (n = 0,…) is proportional to exp ÉÊ n ÙÚ .
Therefore the expectation valuekTİ of the energy is given byÇε eε kTnnεn 0ÇekTT2ε kTnεn ØÈ kTÉÊ ln nÇ0 eÙÚ .n 022 2If İn is the energy m (x + ν x ) of an oscillator of mass m, and eigen-freq-2uency Ȟ, the index n is a double indexε ØÈ nln É Ç e kT ÙÉÊn 0ÙÚ( x, x ) and we may writeËÛÝln Ì Ç exp m ( x 2 ν 2 x 2 ) Ü2 kTÌ ÜËÍÈ2π kT ØÙmν ÚÍ x, xÛln ÌY 1/ 3 Ô exp m ( x 2 ν 2 x 2 ) dxdx Ü2 kTÌÜln ÉÊ Y 1/ 3.ÝHence follows İ=kT by insertion. [The summation over ( Z , Z ) was convertedhere into an integration by virtue of the measure factor Y used before, cf.
Chaps. 4and 6. Since that factor does not influence the result, the conversion – from sum tointegral – might be considered as an auxiliary mathematical tool. CertainlyBoltzmann considered it so, as we have discussed in Chap. 4.]Insert 7.1Planck DistributionThe revolution started as an interpolation project between the Wien ansatzand the Rayleigh-Jeans formula which were good for high and lowfrequencies respectively. Actually given the task, a student can do theinterpolation, – and identify the coefficient B of the Wien ansatz –, simplyby studying the two relations given above like the pieces of a puzzle. Heobtains the following formula after a little time which, admittedly, may beshortened by hindsight.J Q (Q , T )c 8SQ 2 hQ.hQ4 c 3 e kT 1Planck Distribution205This is Planck’s radiation formula, or the Planck distribution.
Planckapparently could not see how easy it was to get. Therefore he proceededalong a cumbersome route which I replay in Insert 7.2, for historicalcorrectness, as it were.The value of h may be determined by fitting the function to the observedcurves. Thus h turns out to be equal to 6.55·10-34Js. This is sometimes calledthe action quantum, because it has the dimension of an action.
More often itis called the Planck constant.I believe that the true history of the interpolation that led to the Planck radiationformula will never be known. Planck himself gave slightly conflicting accounts. Tobe sure, textbook folklore has it that there was an interpolation between Wien’sansatz and the Rayleigh-Jeans formula.
I have so argued myself above. However, inthe relevant papers by Planck in 1900/01 13 there is no mention of Rayleigh, letalone Jeans. So maybe Planck did not know Rayleigh’s work which, after all, hadappeared only in the same year 1900. Planck says that he was convinced of thedeficiency in Wien’s formula by the results of low-frequency experiments madeknown to him by the experimentalists F. Kurlbaum and H. Rubens who confirmedearlier measurements by O.
Lummer and E. Prings heim.14 And then he says,referring to the arguments reported in Insert 7.2Pursuing this idea I came to construct arbitrary expressions for the entropywhich were more complicated than those of Wien … but acceptable.Among those expressions my attention was caught byw 2 sQDwe 2Qe Q (E e Q )which comes closest to Wien’s in simplicity and … deserves to befurther investigated.On the other hand, in his Nobel lecture of 192015 Planck says that the measurements of Kurlbaum, Lummer et al. convinced him that for low frequencies theexpression should read13∂ 2 sν∂eν2~1eν2:There were three such papers.
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