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Is There a Connection Between Planck's Constant, Boltzmann's

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Is There a Connection Between Planck's Constant, Boltzmann's National Library BWbthBque nationale of Canacla - du.Canada Canadian Theses Service' Service pes theses canadiennes --I - icroform is heavily dependent upon thep ~arqualit6de cette microformed6pend grandement de la submitted for micpfilming. qualit6 de la these soumise au microfilmage Nous avons to ensure the highest quality of tout fait pour assurer une quit6 sup6rieure de reproduc- tion? IC . page de missing. contact the university which .S'd manque des pages, veuillez communrquer avec uee. I'universit6 qui a conf6r6 le grade Some pages may have'indistinct print especially if the La qualit6 d'impression de certaines pages peut larsser A " orginat pages werelyped with a poor typewriter ribbon or dksirer, surtout si les pages originalesont 6t6 dactylogra- - if the university sent us an inferior photocopy. ghi6es B I'aide &XI ruban us6 ou si I'universit6 nous a fart parvenir une photocopie de qualrt6 inferieure \ - ~e~rOductionin full or in part of this microform is La reproduction, meme-partielk, de cette microlorme est, by the Canadian Copyright Act, R.S.C. 1970. c gjb:: . soumise B la Lgi canadienne sur le droit d'auteur, SAC subsequent amendments. 1970, c. C-30. et ses amendements subsbquents THERE A CONNECTION BETWEEN PLANCK'S CONSTANT, BOLTZMANN ' S CONSTANT AND THE SPEED OF LIGHT? -9 Peter Danenhower . B.Sc. (Hons. Firs: Class), Simon Fraser University, 1977 0 - 1 d SUBHITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE - in the ~e~artment of - r"dazh~matlcs anu St_a>istics SIHOE FRASER UNIVERSITY ,?xi?, 1987 a-- ni- rl~krsreserves. This work may not be . reFrs5:ce- :c whole 3r in part, by photocopy tez-s, wirno~tpermission of-the author. B1Mroth6quenationaie . ' du Canada Theses Servlce Sewce +s thffes m-nes . g. Ottawa. Canada KIAON4 + U The author has- granted an irrevocable non- t'auteur a accord6 une licence in4vocable et usive licence alt6wing the National Library non exclusive pwmettant B la BiMioth4que of to reproduce, ban, distribute or sell / nationale du Canada de reprMuire, preter, .' iri "'-es of hislher thesis by any means and in , distribuer ou vendre des copies de sa these 3form or format, making this thesis avaibbk de quelque manibre et sous quelque forme ; to interested persons. que ce soit pour mettre des exemplaires de cette these a la disposition des personnes interessks . The a&or retains ownership of the copyright t'auteur consewe h pcopribtb du droit d'auteur in hidher thesis. Neither the thesis nor qui prot&e sa tMse. Ni b: tMse ni des extraii substantial extracts from it may be mtedor substanW de ceb-ci ne doivent Btre otherwise reproduced without hislher per- irnprirnbs ou autrement reproduits sans son rn@sion. autorisation. r ISBN 0-315-59300-8 . 1 e APPROVAL \ Name: Peter Dancnhower Degree: Master oef Science Title of thesis: Is There a Connection '~etweenPlank's ionstant ;' x Boltzmann's Cbnstant and the ~peed'afLight? i Examining Committee: - :' i Chairman: -, C. Villegas <' \ t @ , E. Pechlaner -- Senior Supervisor External ~zamine Department of Mat cs and Statistics - Simcz Fraser Date Ap?roved: July 27, 1987 h CI,- ,. .. d 0. PARTIAL COPYRIGHT LICLNSE - -- d * i -- -- f I hereby grant to Simon Fraser University the right to lend e - * my thesis, project or extended essay (the title of which is shown below) - to users of the S.imn Fraser University ~ibrsry, and to make partial or 'single copies only for such users or response to a request from ttre I in - i i I i brary of any other university. or other educat iona I ) nst itvion, on>- t A i is own beha l f or for one of i ts users. I further aGree tha-t perm i ss ion :" for mu1 tiple,$opying of this work for scholarly purposes may be granted rt by me or the Dean of Graduate Studies. It is understood that copying . or pObllcation of this work for firancial gain shall not be allowed -3 without my writte~permission. Title of ThesislFrs~ect/Extended Essay V i date) ABSTRACT * 0 :An'attempt.is made to find a connection between Planck's constant';h, Boltzspann's constant, k, and the speed of light, c. a=- , The method used i b to 'study blackbody radiation without: quantum / mechanics. #sical thehodynamics and statistical mechanics C s -a?e reviewed. The problem of finding aWtisfactory relativistic 1- , generalization of these theories is discussed and the i 1 ' canonical approach due to Balescu i$ presented. A discussion of the standard treatment ,of blackbody radiation (including quantum 11 results) follows. The quantum result (~lancklaw with zero po_int a derived, as first done by Einstein,3 " Hopf.and Boyer, using the , techniques of stochastic electrodynamics (non-quantum ,derivation). Unfprtunately, P nckl'sconsta& must be introduced as a scale factor in this treatment. Hence, there are still too - s 2 - many free choices for a relationship between h, c, and k to be a C 3 necessity. Accordingly, two attempts arb made to study this problem in more detail: 1. A "classical Fermi-Dirac" statistics is cieveloped, t% =reat the walls oc the blachbody cau-ity as a Fersi gas. 2. Adjastments t~'~~~errnod~namics,required by the n~n-qaanturnderivation of the spect~raldensity, are subjected to E~P~elativisti~ rn~rmod~~am&s previously developed. These two prablems are'very difficult and little progress is made on . - e1~2er.Eence, WP zre ief; wi'th 'no concfusion'about definite ir.3ependence cr deperBence of h, k and c. iii -1 Many thanks to my advisor, Dr. Edgar Pechlaner, for I patiently ea&ing my impractical inclinations, my stubborness," and my refusal to-accept his frequent suggestions to switch to a more tractable topic.-f am also indebted to him for many helpful * conversations, for meticuoiousl~ proof reading several drafts of < this thesis, and for gen~r'busassistance from his research grants. Thanks to the others Pn my committee, Dr. A. Das and ?# Dr. S. Kloster, for sev;ral useful discussions and for proof t. :+> reading'the thesis, I am grateful to my one time office mate, 4- Mr. Ted Biech, for many long conversations, which usually,. at least began with some aspect of mathematics or relativity. I am + also indebted to Dr.'~ichaelPlischke of the S.F.U. Physics Department, who made se a1 valuable suggestions and who kindly d P bore with me, even thou is advice to switch topics. - - Thanks to the Departmen f Mathematics and Statistics f%r financial assistance in the form of graduate *search ,. -+-- scholarships, And finally, a special thanks to the Departmental Secretary for Graduate Students, ~k..Sylvia Holmes, fsr taking -- care cf ail-sorts of scids d ends %o efficiently tha.t I am maware of what most of :hem were! ." Approval .................................................... -'ii A stract ......y..........................-................'iii 9 I I. Introduction ............................................ -1 . .--, 11; Review of Thermodynamics and Statistic&.'Mechanics ...... 9 Classical Thermodynamics ................................ 9 Classical Statistical Mechanics 21 = - ........................ &< 6. 111. Relativistic Thermodynamics and Statistical Mechanics .. 27 i l i/- General Considerations ................................. 27 - Relativistic Thermodynamics 34 - ............................q- ~elativistic Statistical Mechanics ..................... 38 ._ IV. Blackbody Radiation .................................... 59 Early Results ............................................ 59 Wien's Displacement Law ...........,................,,,. 63 The Rayleigh - Jeans and Planck Laws ................... 66 V. Blackbody Spectrum Using Stochastic Elec%rodynamics .... 71 Derivation of the Zero Point Spectrum .................. 72 Some Theoretical Considerations in More Detail-.......-.. 99 r... t T --I" ~xtensionsof Boyer's Analysis ........................ 106 Classical Ferml - Dirac Statistics .................... ,106 Modification of the Rayleigh - Jeans Method ........... 110 - - 3e:ativistic ~her~~dynamics........................... 119 "- Q' C~nclusi~n........................................... 119 -- Figure Page ' 2.1 . Carnot Cycle ........................................... 18 2.2 Pressure - Volume Diagram for Carnot Cycle &....... ,. .. 20 4.1 Carnot Cycle for -=** , / A'' CHAPTER I - - INTRODUCTION --- ,a An ongoing problem in the development I - physics is deciding just how many degrees .,*. - available for choosing units and dimensio - parameters. While the literature and man4 texts include brief overviews of this topic, - detailed discussions are not too commoff. For an in depth review, see L6vydLeBlond [I]. We can % illustrate the problem with a simple example: If we choose a .! ?k a time scale arbitrarily, we then appear to be able to choose a r length scale arbitrarily, thereby fixing the speed of light, We can introduce a mass scale an& determine the dimension and scale of force (if we choose dimensions for the constant in Newton's 9 / second law - normally set equal to 1). We seem to be able to '- '-contipue this chain of defining parameters and constants / -- - (subject to physical relevance) as we please. For example, if we choose electric charge and temperature as independent then we e get two more fundamental constants; Q,e permitivity constant (of t free space) and Boltzmann's constant. The 'questioniis: Do we really have all this freedom? Is it possible that by devehping ma~hematicalphysics in this manner, we have made arbitrary choices which are in fact incompatible? If sP , then unification of diverse fields of physics would be impossible. In this thesis we attempt to gain more insight into this situation by searching for a physical connection between three f Andamental physical constants: Planck' s constant, h, Boltzmann's cdnstant$'k, and the speed of light, c. We wish to . -- make it clear at the onCset that we are not referring to A - i "mathematical rel&ionshipsn. A search for mathematical relationskiips involves looking for clusters of constants that I .. have the same dimension. For example, if the unit of'length is defined as h 2----- where m and e are respectively the mass and * 4x2'me k' fl charge on an electron, and the unit of time is defined as h3 then that leaves c proportional to - This iort-'of"' 8r3rne.$' .
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