Thermodynamic s, Thsnr.pl Radiation and the Beginning* or luantum Phvsics, P. 3.0. Bruskiewich Mathematical-Physics, University ©f British Columbia, Vancouver B.C. The beginnings of Quantum Theory is introduced in such a manner as to present the continuity of effort between the "classical" and "modern" physics. Emphasis will be on the specific problem of the energy spectrum of a blackbody source in thermal equilibrium, and the transfer of energy on the atom- ic scale. In the year 1900»Max Planck presented a paper to the Grerras.n Physical Society in Berlin(l) that helped to change the intrinsicJchara- cter of science, Eis concern m this paper was with the' mechanical ,thermo dynamic and elect rodynamic processes specifically associated with " ideal" resonators inside an "ideal" black- body enclosure. In a sense, his work represented the consistent extension of the endeavours of many other"clasa ical" physicists like Kirchoff ,Clausius > Maxwell, St ©fan ,Wi en and Bolt smarm, but it also represented a personal daring (encompassed in the introduction of the concept of discrete action ) that was later to blossom into what we now call quantum physics, Needless to say, at the time his approach did not follow that of others ( James Jeans, for instance), but the conclusions, though originally heuristic and highly theoretical, that can be drawn from his work have shown, beyond any doubt, that Planck's concept of discrete action is very fundamental to the intrinsic understanding of the physical universe. There are many ways one could approach Planck's achievement, however, to appreciate the fundamental importance of his work it would be best to follow Planck' s own formulation of his blackbody spectral law. Max Planck' s career as a physic ist began with over fifteen years ©f study in the field of thermodynamics, mere specifically the second law, thermodynamic potentiality, and irreversibility, Because of his beginnings (as a diadple of Rudolf Olauaius, the formulator of the second law ) , Planck considered the second lav; to be universally valid, namely processes in which the total entrop«> f a system decreg sed over extended time were to be considered srictly impossible. '. originally did not consider Ludwig Boltsmann' s reformulation of the second lav/ (into a statistical law) as being acceptable because Boltzmann's statistical mechanics did not make - the increase of entropy absolutely certain, only highly probable. Planck often expressed the deep, esoteric conviction that the loftiest goal of any if not all science was the search for the absolute.This convict ion is clearly reflected in his appreciation of the general physical laws, but beyond the formulation of fundamental laws, he attempted, as others attempted before him (e.g. Ludwig Boltzmann ) , to relate the unexplained physical processes and paradoxes of his time, to fundament ally universal laws.Flanck believed that the Principle of Increasing Entropy could bo preserved intack as a rigorous theorem in a comprehensive world view .When it came to the unexp lained phenomenon of blackbody radia tion, it was this belief that acted as the touchstone to Planck's attempt at expl aination* In March I895 Planck presented a paper(2) in which he discussed the problem of resonant scattering by an oscillating dlpole(of dimension comp atible with the wavelength) , of a pi " plane elctromagnetic wave(5) .'What this scattering process offered to him was a way of relating the equilibrium ctate(4) radiation in a blackbody enclosure to the states of hypothetically ideal re son at or s } which Planck introduced as making up the walls of an ideal blackbody enclo sure(5) • Planck studied the mechanisms of emi s si on/ab sorption end of damping of these hypothetical resonators ;He realised that by disregarding mechanical damping by concerning himself uniquely with radi ation damping, and by taking the mechanism for the system's irreversibility as being the conversion of the incident plane waves into spherical waves, you could more fully begin to understand the dynamics of the radiation equilibrium in the enclosure. However, a very fundamental flaw in his approach was brought up by Boltzmann(it was a flaw that forced Planck to change his attitude towards Boltzmann 1 s statist! cal mechanics ) • The equations of electro dynamics do not lead to a monotonic conver gence to a state of thermal equilibrium, Nothing in the laws of electrodynamics prevent the inverse of the scattering processes from occuring^This important fact forced Planck to admit that statist! cal descriptions were necessary(6).It also forced him to consider energy exchange by .discrete means as opposed to continuous energy exchange. Backtracking alittle bit, we can now begin to approach the mathematics of Planck's concept. Experiments by Josef Stefan in 1879 j s -nd theoretical work by Ludwig Boltzmann in 1884, established that JD(1T.- law. Wien* a spectral law was be a reasonable representation of the blackbody spectral distribution known up to 1899* In 1899 > Planck, continuing hi 3 work begun in 1895 on ^n e analogy of the" ideal" resonators, proved (7) that the distribution function must be of the form where 3(v,T) represents the average energy of the "ideal" resonators at frequency v, at cavity temperature. Though this equati on applies to a hypothetical wall comp osed of "ideal" resonators, because of Kirchoff ' s 1&59 proof regarding the constituents of the walls of a black* body enclosure, once a valid frequency- temperature relation is found for the average resonator energy E(v»T), it can be applied to a cavity formed by any real material. Because of" his belief in the universality of the second law* Planck attempted to discern the form of 3(v,T) by looking at the fundamental relationship between the energy and entropy of a system in a state of thermal equilibrium. He began with dfc T , and by differentiating again he came up with the equally important equation J — „' proved rigorously that the distribution function p(v,T) must be of the form The derivative dS/dT could be evaluated by combining Wien 1 s spectral law with Planck's distribution In 189 2 ) Wilhelm Wien function so that « „, <trt* ' p(at,t) - \r S-iy/r) In I896, ";;ien, using a somewhat questionable theoretical present ation, proposed that f(v,T) was of the form U«/t) " A £ «>r 1 . anc The entropy - energy derivative then becomes ent constants, giving p(v,T} ■= Av^e, * where A & B rep res Rv |5irE The simplicity of this entropy- energy derivative greatly impressed Planck, but spectral measurements taken Wien 1 s spectral (at the same time that Planck did his work (2) in 1899) &t high temperatures and low frequencies mads it apparent that T >fien' s spectral lav; had serious empirical limit ations. In 1900 , working with quite new data ^Planck derived a simple dependence, which agreed with the high temperature data (i.e. p(v,T)c*iT, 2(v,T><r, dE/dT is constant ) .Planck realized that these two \* <js cases were limiting cases, one for relati VS^y?" ^ high energy and the other for low. ' $-* ' Almost within the same few days he formul o, ' ■ atc-a a single equation which sufficed to "' X 6 «- exoress both limiting cases, namely « -. - \e (a +& )} -I this single equation yields Integration of 41 CA-tG)] 4-i with d serving as the constant of integrstion.Replacing dS/dE by l/T, the equation becomes is\ e A a — rr; "valuation of the constant of integration(i.e. looking at the equation as l/T •> ) gives d - 0. Solving the last equation yielded an appr opriate equation for the resonant energy, namely (l%~ giving the expression A r 3 (*>~0 and p(v,T) the form Q-v Ji P<W7 = 7—^ which can be evaluated for B and thenceforth G, giving " ~F av» /4 ^ - SjLvfk h being a new universal constant now known as Planck's constant (h-6. 6?,6 \10"^ T J-sec). Planck, himself a theoretician, was not satisfied with just fitting a formulation to empirical data and began to put his concepts into a rigorous form. As I pointed out before, criticism By Ludwig Boltzmann forced Planck to turn towards a statistic;! interpretation of entropy .This statistical approach yielded dramatic results. Planck began by looking at the N ideal" resonators making up the walls of his hypothetical blackbody enclosure .Each resonator has an average energy<E>and an average entropy^S^ giving a total energy E t of 11(E) and a total entropy S^of l$>.It is here that Flanck introduced Boltzmann' s famous expression pfL ^i^-wv-a^wtl ^<J\j^>Jn{(J3\ n«Hw»l ,and follow&% the^Bolztmann devised previously to deal with the entropy divided amongst many oscillators(8) . By proposing that the total energy E. be made up of n discrete units of energy, each of amount e,he was able to equate 1^2) to ne and open up the approach to the fundamental problem of W; How to calculate the number of ways n discrete units of energy can- distributed amongst for the distribu tion function. Returning to the limiting case encompassed in Wien' s spectral law, Planck showed that A, a (supposive) constant* really was a function depending on v, so that A = F' v% with P representing a(real } constant .Thus f(v,T) takes on the form N resonators. More fundamentally, Planck realized that previous limitations in his reasoning, as well as in ioltzmann's work, could be removed by considering the energy transfer to actually be occuring by discrete means. Flanck now had a way of relating W to N and n. Beginning with \r I -+K -0 lAt (ti-i and making the (5) approximation { on the assumption that both N and. n are largo ) expression .•■„ ■v\)l M- and by using Stirling" s ftyyiitU - l a a-j^^/je/Ci ha <*wk_ y\ -ti\ ~ ~t{v\-L--t he wag able to express klnW as jUfi-fhe entropy of single "ideal" resonator is a s - c' He transformed this entropy relation into an energy relation ££r the taking jj^the derivative X> - d£ \e •\ l ? A E and by (again) equating this to 1/T, so then j(/\ e, v G \ e Solving for 2(v,T), he ended up with his previously derived expression eKt q, " x By returning to Wien's distribution function PKtV nrHOf and his own he - was ablo- te reasoned fl^fe- that e«w or (again) returning to Planck 1 ' a prf ri i OTn fnrrmi 3 m ti l <m J thfiti" 5 hv. Thus he was able t function 1 1 . * ' ■*■ his distribution (a 5 * -i; quite rigorously. TMsenew radiation formulae brought a great deal of pleasure^' rSei. they were in agreement with the then new spectral data of Kurlbsum and Rubens, who at the time were doing precise spectral measurements at high temperature and low frequencies ) , but also a great deal of distress in that, if one accepte d his formulation^ th on"^he.rgy„tran sfer appear ed^ to be occuring b y discrete steps as opposed to the cTxLssicaliBSEIc] continuous transfer , Planck, even though he was the formulator of this important discovery, refused for many years, to accept fully the consequences of discrete energy transfers It was left to someone as bold and ira-gina tive as Albert Sinstien to expand Planck's original work -IwtJfW luiU maim^ JL ^Mx/SW^ ^ ************ pLfiiw, (1) Zur Theorie des Gesetzes der 3ner<tie~ vertoilung in Normalspecrtu m, M.Planck, Verb. d.Deutsch.Phys.Ges. 2, 202,(1900) , ( english translation! Planck's Original Papers ,Taylor and Francis, 1972) ., (2)M.Planck,Ann.Hrys.(5) 57,1,(1396) . (5) The intensity of the plane polarised light is independent of the orientation of the plane of polarization as well as the position and direction of the light. Inside the enclosure the radiation is isotropic and homogeneous. (4) The state of thermodynamic equilibrium is that state in which the entropy of the system has the maximum value compatible with the total energy as fixed by the initial conditions of the system, (5) Planck was able to use the analogy of discrete "ideal" resonators as constituting the enclosure's walls because Gustav Kirchoff had previously provelft (in 1©?9) that the energy spectrum of a large blackbody enclosure is entirely independent of the constituents, shapes and sizes of the materials making up its walls. (4) (6) In thia resolution lay dormant the complete theoretical description of the mechanism of radiation. (7) How Planck aid this is best seen in the englieh translation of his book Waermestrahlung (191?) » by M.Masiua, Blakiston's Son & Qo, 19l4 f titled Planck's Heat P.adiation « (8) Boltzmann originally introduced this statistical description as a computational aid, however ; later he became quite satisfied to consider its application as being more iAott~ w Mew general. fc f^^A >(**<- L l P . fc° Dfl Qoihf*** Kii^ , FHp'i6Q Tcte^ ^t* resell