Revealing the Essence of Planck s Constant. Koshun Suto

Transcription

1 Rvaling t Essnc of Planck s onstant Kosun Suto Ky words: Planck s constant, Univrsal constants, Fundamntal pysical constants PAS No: Sq, Ta Abstract According to traditional classical quantum mcanics tory, du to t prior xistnc of Planck s constant, considrd a univrsal constant, it is tougt tat t nrgy of a poton can b dtrmind if its frquncy is known, and t wavlngt of a quantum can b dtrmind if its momntum is known (E =ν and λ=/p). In tis papr, owvr, t cas is mad tat logically, sinc t product of t momntum and wavlngt of any poton can b xprssd by t constant pλ, Planck s constant only coms into xistnc wn pλ is rplacd wit. In tis papr, w sow tat Planck s constant is not a univrsal constant but is instad just a usual fundamntal pysical constant. Abstract Slon l intrprtation traditionll d la téori d la mécaniqu quantiqu classiqu, du à l'xistnc antériur d la constant d Planck considéré comm un constant univrsll, il st accpté qu l'énrgi d'un poton put êtr détrminé si sa fréqunc st connu, t la longuur d'ond d'un quantum put êtr détrminé si sa quantité d mouvmnt st connu (avc E =ν ou λ= /p). pndant, dans ctt discussion, l cas st fait qu, logiqumnt puisqu la quantité d mouvmnt t l produit d la quantité d movmnt t d la longuur d'ond d n'import qul poton puvnt êtr xprimés par la constant pλ, la constant d Planck xist sulmnt lorsqu st xprimé comm étant égal à pλ. Dans ctt étud, nous montrons qu la constant d Planck n'st pas un constant univrsll, mais qu ll st just un constant pysiqu ordinair. 1

2 I. Introduction In 1900, wn driving a formula tat drivd an xprimntal valu of black-body radiation, M. Planck ( ) proposd t quantum ypotsis stating tat t nrgy of a armonic oscillator wit oscillation frquncy ν would quantiz at t intgral multipl of ν. Tis was t first tim tat Planck s constant appard in pysics tory [1]. Planck s constant is tus tougt to b a fundamntal pysical constant dfind in t ralm of quantum tory, but t ssnc of tis constant is gnrally not wll undrstood. Blow is Einstin s formula xprssing t quality of nrgy and mass []. E = mc (I.1) Hr, m is t mass of a particl and c is t spd of ligt in a vacuum. T spd of ligt may also b xprssd troug t following formula. c = λν (I.) Manwil, Einstin s rlational xprssions rgarding t nrgy of a poton ar xprssd by t following formulas: E E = ν (I.3)[3] = pc (I.4) Hr, ν is t poton s frquncy and p is t poton s momntum. T formula on wic (I.4) is basd is Einstin s nrgy-momntum rlationsip as blow. ( mc ) = ( m c ) + p c (I.5) 0 Bcaus a poton s mass is zro, w obtain (I.4) by substituting m 0 =0 in (I.5). In otr words, Formulas (I.1) and (I.4) ar logically drivd formulas.

3 Formula (I.3), on t otr and, xplains an obsrvd valu and for tis rason it is an xprimntal formula, not a logically drivd formula. If w nxt combin quals from Formulas (I.1) and (I.4), w obtain t following formula dscribing t momntum of a poton. p = mc (I.6) If w combin quals from Formulas (I.1) and (I.3), wn considring Formula (I.6), w obtain Einstin s following formula [4]. p = (I.7) λ Tis formula xprsss t rlationsip btwn a poton s momntum and wavlngt. L. d Brogli ( ) usd t abov logic of t drivd Formula (I.7) in prdicting tat t rlation btwn an lctron s momntum and wavlngt can b xprssd, as in (I.7) [5]. Manwil, A. Einstin ( ) sowd tat t momntum of ligt quanta can b rprsntd by t following formula [6]. p = ν (I.8) c Einstin sowd tat wn considring Formula (I.), Formula (I.7) can b drivd from Formula (I.8) and Formula (I.4) can b drivd from Formulas (I.3) and (I.8). Incidntally, w obtain t following wn xprssd in matmatically invrs proportions. xy = a (I.9) omparing Formulas (I.7) and (I.9), x and y corrspond to p and λ and a corrspond to. Tis supports t concpt tat Planck s constant is no mor tan a simpl constant. 3

4 Tus, using non-istoric rasoning, t tru ssnc of tis constant is rvald in tis papr. Bforand, lt us vrify t following points rgarding fundamntal pysical constants and Plank s constant. Fundamntal pysical constants play an ssntial part in lmntary formulas tat dscrib natural pnomna and can b largly dividd into univrsal constants and matrial constants. Also, pysical quantitis and constants ar includd in fundamntal pysical constants tat blong to on catgory. Pysical quantitis blonging to micro matrial constants includ lctron mass m, lmntary carg, and lctron ompton wavlngt λ, and includ suc constants as t fin-structur constant α and t Rydbrg constant R. T Boltzmann constant k and t Avogadro constant N A ar xampls of macro matrial constants. Howvr, Planck s constant is tougt to b a fundamntal constant rprsntativ of quantum mcanics and considrd to b a univrsal constant. Bcaus Plank s constant as an action quantity dimnsion, it was at first calld an action quantum wn quantum tory originally mrgd. appars in t inquality wn W. Hisnbrg ( ) discovrd t uncrtainty principl in 197. ΔΔ x px / Planck s constant, along wit t spd of ligt in a vacuum c and t Nwtonian constant of gravitation G, also plays an important rol wn assmbling planck units from univrsal constants. From t abov, Planck s constant is a constant by nam, but it as com to b strongly rgardd as bing t smallst unit of angular momntum. Planck s constant is currntly masurd by t watt balanc mtod. Among otrs, tis rsarc is bing prformd by NPL and NIST [7], [8]. T stimatd valu of Planck s constant according to 1998 ODATA was dtrmind basd on data masurd using t watt balanc mtod. A mtod tat dtrmins from only dirctly masurd quanta, tis xprimnt was tougt to b abl to obtain t bst valu for. Anotr way to masur Planck s constant is t X-ray crystal dnsity (XRD) mtod. Tis mtod masurs t Avogadro constant N A using a virtually pur singl crystal suc as 4

5 silicon. NMIJ and PTB av rcntly rportd xtrmly accurat masurmnt rsults [9], [10]. Bcaus Planck s constant and t Avogadro constant ar closly rlatd to ac otr, a mor accurat masurmnt of is possibl if a mor accurat masurmnt of N A is obtaind. T ODATA Rcommndd Valus of t Fundamntal Pysical onstants (00) usd ts masurmnt rsults and t masurmnt rsults of t abovmntiond watt balanc mtod as a basis to dtrmin Planck s constant [11]. Similar progrss is mad vry yar in masuring Planck s constant, but littl progrss as bn mad in undrstanding t ssnc of tis constant. Trfor, in tis papr w attmpt to rval t ssnc of tis constant. II. Driving Planck s onstant from Fundamntal Pysical onstant If m is t mass of an lctron, an lctron s mass nrgy E 0 can b rprsntd by t following formula. E 0 = mc (II.1) Manwil, if ν is t frquncy of a poton carrying an amount of nrgy quivalnt to E 0, t following is tru (S Appndix). E0 = ν (II.) ombining quals from Formulas (II.1) and (II.), w obtain: mc = ν (II.3) Fundamntally ts two typs of nrgy av diffrnt caractristics, but from a quantitativ prspctiv, it is possibl to combin tm as quals. Tus, a poton s frquncy ν is xprssd as follows. 5

6 mc ν = (II.4) Nxt, a poton s wavlngt λ bcoms: c λ = ν = mc (II.5) Now, an lctron s ompton wavlngt λ is rprsntd by t following formula. λ = (II.6) mc T wavlngt of a poton wit nrgy E 0 is t sam as t ompton s wavlngt λ of an lctron. Tus, (II.1) can b transformd as follows. E = m c 0 = mcλ ν (II.7) In (II.7), λ is t wavlngt of a poton, not an lctron. Howvr, bcaus t rigt sids of (II.7) and (II.3) matc, t following rlationsip olds tru in t cas of a poton as wll. mcλ = (II.8) Nxt w apply t sam logic on protons tat was applid on lctrons. Rpating t sam logic on protons as was applid on lctrons, w finally driv t following formula. 6

7 mcλ = (II.9) p,p Hr, m p is t proton s mass, λ,p is t proton s ompton s wavlngt as wll as t poton s wavlngt, wic as t sam valu III. Planck s onstant Drivd from t Various Enrgis of a Poton T spcific nrgy ld by a poton was considrd in t prvious captr. Tis captr is a mor gnralizd discussion basd on a poton aving various typs of nrgy. First, by gnralizing (II.3) w obtain t following: mc = ν (III.1) Hr, m is not ncssarily t ntir mass of t lctron. T mass of t lctron is bing gradually rducd du to t mission of potons, and m corrsponds to t rducd mass of tat lctron. (wn 0 < m) In otr words, (III.1) is saying tat t rduction in lctron nrgy is qual to t nrgy of t mittd potons. T currnt rducd mass m is dfind as follows. m= a m (wn 0< a) (III.) T momntum of a poton mittd from t lctron at tis tim is xprssd as follows. p = mc = a mc (III.3) Also, sinc tr is an invrs proportional rlationsip btwn a poton s momntum and wavlngt, t wavlngt of tis poton is can b xprssd by t following formula. 7

8 λ a λ = (III.4) Tus, t product pλ of an mittd poton s momntum and wavlngt is: pλ = mcλ λ = (a mc )( ) (III.5) a = mcλ W can s tat ultimatly, t product pλ of t momntum and wavlngt of any poton is t sam as t constant m cλ. Tat is, pλ = mcλ = (III.6) onsidring (III.6), it is possibl to logically driv (I.3) from (I.1). Tus, E = mc = mcλν (III.7) = ν IV. Discussion W nxt substitut t following valus for pysical quantitis in m cλ [11]. 31 m kg = (IV.1) c = λ m s (IV.) 1 = m (IV.3) By doing so, m cλ bcoms as follows. 8

9 34 = J s mcλ (IV.4) Manwil, Planck s constant as t following valu [11]. 34 = J s (IV.5) m cλ and ar a prfct matc. T currntly known valus for m or λ wr not dtrmind troug xprimntation. m was dtrmind troug prcis calculations from Rydbrg constant formulas, and λ was obtaind by substituting m in t formula λ =/m c. Basd on masurd data from tortical formulas or xprimnts dsignd to rprsnt t fundamntal laws of pysics, many fundamntal pysical constants ar bing adjustd to avoid conflicts from arising btwn ts constants. Bcaus t formula to dtrmin an lctron s ompton wavlngt is λ =/m c, naturally t modifid vrsion of tis Formula (II.8) is tru. Howvr, λ in (II.8) is not t wavlngt of an lctron, but is t wavlngt of a poton wit t sam valu as t lctron s ompton wavlngt. Also, it is important to not tat m c is not t momntum of an lctron but t momntum of a poton. Logically, owvr, Planck s constant sould tougt of as a constant tat only coms into xistnc onc Formula (I.1) is rwrittn into Formula (I.3) to includ a poton s frquncy, and t subsqunt rcognition tat t non-frquncy componnts mcλ form a constant wic can b rplacd by. In otr words, (II.8) can b intrprtd to man not m cλ and ar idntical but instad to man m cλ is. Trfor, tis papr dos not claim t discovry of any rlationsip in (II.8). It simply dfins tat bcaus t valus of m cλ and m p cλ, p ar t sam in (II.8) and (II.9), tis valu sould raftr b calld Planck s constant. Ratr tan naming tis constant as Planck s constant, w can simply rgard it as m cλ = m p cλ, p =pλ=const. 9

10 Howvr, bcaus tis constant as bn istorically usd in otrs of Planck s rsarc, it as takn on t imag of bing a discovrd univrsal constant. V. onclusion 1. Planck s constant only cam into xistnc onc it was dfind According to xisting tory, Formulas (I.1) and (I.3) av bn tougt to av similar importanc. Howvr, according to our discussion, (I.1) is t mor fundamntal of t two. Formula (I.3) is Formula (I.1) rwrittn to also includ frquncy. T rigt sid of (III.7), t product of t pysical quantitis mcλ xcpt for frquncy, is a stady valu. Rgardlss of wtr m cλ or m p cλ, p ar calld Planck s constant, in tis papr w conclud tat Planck s constant only cam into xistnc onc it was dfind. Howvr, not bing awar of wat sould av bn dfind, tis task was skippd, and tus Planck s constant was blivd to b a discovrd univrsal constant. Tus, it is valid to rgard Planck s constant not as a univrsal constant but as a pysical constant on par wit t fin structur constant α or t Rydbrg constant R.. Formula (I.1) is t mor fundamntal tan Formula (I.3) As mntiond in t Introduction, t tougt procss bind Einstin s drivation of Formula (I.7) is as follows. ν (I.3) E = ν p = (I.8) c c = λν (I.4) E = pc, p= (I.7) λ (I.) (V.1) Manwil, d Brogli combind quals from Formulas (I.1) and (I.3) and furtr drivd Formula (I.7) wn considring Formula (I.6). Namly, 10

11 (I.1) E = mc E = ν (I.3) (I.6) p= mc pc = pλν = ν p = (I.7) λ (V.) Howvr, Formula (I.3) was drivd according to t following procss in tis papr. ( mc ) = ( m c ) + p c (I.5) (I.1) E = mc 0 m = 0 E = pc (I.4) p= mc (I.6) pλ = mcλ = mcλ = const = p= λ λν E = mc = mc = ν 0 (I.3) (I.7) (V.3) Wn comparing Formula (I.3), Formulas (I.7) and (I.8), istorically, Formulas (I.3) and (I.8) wr obtaind bfor Formula (I.7). Tus, Einstin and d Brogli prsumd Formulas (I.3) and (I.8) wn driving Formula (I.7). Howvr, t supposition of tis papr is tat bcaus logically pλ= is tru, Formula (I.3) can b drivd. Formulas (I.1) and (I.3) ar traditionally tougt to b rprsntativ formulas of t toris of spcial rlativity and quantum mcanics, t roots of modrn pysics, and ts two formulas av bn tougt to av similar importanc. Howvr, according to our discussion, Formula (I.1) is t mor fundamntal of t two. 11

12 Appndix T rviwr of tis papr mad t following commnt. T proof of t papr is basd on t assumption tat an lctron at rst can dcay into a singl poton, wic is in violation of consrvation of momntum. Furtr, consrvation of lctric carg proibits dcay of an lctron into any numbr of potons. Tus, t dcay t autor sks to invstigat cannot occur. T autor is sufficintly mindful of t point bing mad in t rviwr s commnts rgarding t currnt rvision of tis papr, but a sligt supplmnt is addd to avoid any doubt or confusion. T autor is not assrting in tis papr tat t rst mass nrgy of an lctron dcays into a singl poton. Wil t nrgy of naturally xisting potons carry a varity of valus, tis papr appns to us an xampl of wat would appn to t wavlngt of a poton if it ad t sam nrgy as E 0. Acknowldgmnts Dr. K. Fujii s Japans paprs and wb articls and Dr. H. Ebiara s Units and onstants commntary wr rfrncd in prparing t introduction. I would lik to xprss my tanks to bot Dr. Fujii and Dr. Ebiara. I would also lik to xprss my tanks to t staff at AN Translation Srvics for tir translation assistanc. 1

13 Rfrncs [1] M. Plank, Pys. Gs., 37 (1900). [] A. Einstin, Ann. Pys. 18, 639 (1905). [3] A. Einstin, Ann. Pys. 17, 13 (1905). [4] A. Einstin, Pys. ZS. 18, 11 (1917). [5] L. d. Brogli, Pil. Mag. 47, 446 (194). [6] L. d. Brogli, ompt.rnd. 179, 39 (194). [7] B. P. Kibbl, I. A. Robinson, and J. H. Blliss, Mtrologia. 7, 173 (1990). [8] E. R.Williams, R.L.Stinr, D.B.Nwll, and P.T.Olsn, Pys. Rv. Ltt. 81, 404 (1998). [9] K. Fujii, A. Wasda, N. Kuramoto, S. Mizusima, M. Tanaka, S. Valkirs, P. Taylor, R. Kssl, and P. D Bièvr, IEEE Trans. Instrum. Mas. 5, No., 646 (003). [10] P. Bckr, H. Bttin, H-U Danzbrink, M. Gläsr, U. Kutgns, A. Nicolaus, D. Scil, P. D Bièvr, S. Valkirs, and P. Taylor, Mtrologia, 40, 71 (003). [11] P. J. Mor and B. N. Taylor, ODATA rcommndd valus of t fundamntal pysical constants: 00, Rv. Mod. Pys.77, No.1, 1 (005). 13