Revealing the Essence of Planck s Constant. Koshun Suto

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1 Rvaling th Essnc of Planck s onstant Koshun Suto Ky words: Planck s constant, Univrsal constants, Fundamntal physical constants PAS No: Sq, Ta Abstract According to traditional classical quantum hanics thory, du to th prior xistnc of Planck s constant, considrd a univrsal constant, it is thought that th nrgy of a photon can b dtrmind if its frquncy is known, and th wavlngth of a quantum can b dtrmind if its momntum is known (E =hν and λ=h/p). In this papr, howvr, th cas is mad that logically, sinc th product of th momntum and wavlngth of any photon can b xprssd by th constant pλ, Planck s constant only coms into xistnc whn pλ is rplacd with h. In this papr, w show that Planck s constant is not a univrsal constant but is instad just a usual fundamntal physical constant. Abstract Slon l intrprtation traditionll d la théori 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 photon 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 =hν ou λ= h/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 photon puvnt êtr xprimés par la constant pλ, la constant d Planck xist sulmnt lorsqu h 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 physiqu ordinair. 1

2 I. Introduction In 1900, whn driving a formula that drivd an xprimntal valu of black-body radiation, M. Planck ( ) proposd th quantum hypothsis stating that th nrgy of a harmonic oscillator with oscillation frquncy ν would quantiz at th intgral multipl of hν. This was th first tim that Planck s constant h appard in physics thory [1]. Planck s constant is thus thought to b a fundamntal physical constant dfind in th ralm of quantum thory, but th ssnc of this constant is gnrally not wll undrstood. In this papr, using non-historic rasoning, th tru ssnc of this constant is rvald. Bforhand, lt us vrify th following points rgarding fundamntal physical constants and Plank s constant. Fundamntal physical constants play an ssntial part in lmntary formulas that dscrib natural phnomna and can b largly dividd into univrsal constants and matrial constants. Also, physical quantitis and constants ar includd in fundamntal physical constants that blong to on catgory. Physical quantitis blonging to micro matrial constants includ lctron mass m, lmntary charg, and lctron ompton wavlngth λ, and includ such constants as th fin-structur constant α and th Rydbrg constant R. Th Boltzmann constant k and th Avogadro constant N A ar xampls of macro matrial constants. Howvr, Planck s constant h is thought to b a univrsal constant rprsntativ of quantum hanics. Bcaus Plank s constant has an action quantity dimnsion, it was at first calld an action quantum whn quantum thory originally mrgd. h appars in th inquality whn W. Hisnbrg ( ) discovrd th uncrtainty principl in 197. ΔΔ x px / Planck s constant h, along with th spd of light in a vacuum c and th Nwtonian constant of gravitation G, also plays an important rol whn assmbling planck units from univrsal constants. From th abov, Planck s constant is a constant by nam, but it has com to b strongly rgardd as bing th smallst unit of angular momntum.

3 II. Planck s onstant Drivd from Fundamntal Physical onstant Blow is Einstin s formula xprssing th quality of nrgy and mass []. E = (II.1) Hr, m is th mass of a particl and c is th spd of light in a vacuum. Manwhil, Einstin s rlational xprssion rgarding light quanta is as follows [3]. E (II.) Th photon s nrgy E is proportional to its frquncy ν, and this constant of proportionality is known as Planck s constant. Formula (II.1) and Formula (II.) ar traditionally thought to b rprsntativ formulas of th thoris of spcial rlativity and quantum hanics, th roots of modrn physics, and ths two formulas hav bn thought to hav similar importanc. If m is th mass of an lctron, an lctron s mass nrgy E 0 can b rprsntd by th following formula. E 0 = (II.3) Manwhil, if ν is th frquncy of a photon carrying an amount of nrgy quivalnt to E 0, th following is tru. E0 (II.4) Th quation in (II.4) is not basd on th assumption that an lctron at rst can dcay into a singl photon, which is in violation of consrvation of momntum. Th dcay cannot occur. Whil th nrgy of naturally xisting photons carry a varity of valus, this papr happns to us an xampl of what would happn to th wavlngth of a photon if it had th sam nrgy 3

4 as E 0. ombining quals from Formulas (II.3) and (II.4), w obtain: (II.5) Fundamntally ths two typs of nrgy hav diffrnt charactristics, but from a quantitativ prspctiv, it is possibl to combin thm as quals. Thus, a photon s frquncy ν is xprssd as follows. h ν = (II.6) Nxt, a photon s wavlngth λ bcoms: c λ = ν h = (II.7) Now, an lctron s ompton wavlngth λ is rprsntd by th following formula. λ = h (II.8) Th wavlngth of a photon with nrgy E 0 is th sam as th ompton s wavlngth λ of an lctron. Thus, (II.3) can b transformd as follows. E = m c 0 = λ ν (II.9) 4

5 In (II.9), λ is th wavlngth of a photon, not an lctron. Howvr, bcaus th right sids of (II.9) and (II.5) match, th following rlationship holds tru in th cas of a photon as wll. λ = h (II.10) III. Planck s onstant Drivd from th Various Enrgis of a Photon Th spcific nrgy hld by a photon was considrd in th prvious chaptr. This chaptr is a mor gnralizd discussion basd on a photon having various typs of nrgy. First, by gnralizing (II.5) w obtain th following: (III.1) Hr, m is not ncssarily th ntir mass of th lctron. Th mass of th lctron is bing gradually rducd du to th mission of photons, and m corrsponds to th rducd mass of that lctron. (whn 0 < m) In othr words, (III.1) is saying that th rduction in lctron nrgy is qual to th nrgy of th mittd photons. Th currnt rducd mass m is dfind as follows. m= a m (whn 0< a) (III.) Th momntum of a photon mittd from th lctron at this tim is xprssd as follows. p = = a (III.3) Also, sinc thr is an invrs proportional rlationship btwn a photon s momntum and wavlngth, th wavlngth of this photon is can b xprssd by th following formula. 5

6 λ a λ = (III.4) Thus, th product pλ of an mittd photon s momntum and wavlngth is: pλ = λ λ = (a )( ) (III.5) a = λ W can s that ultimatly, th product pλ of th momntum and wavlngth of any photon is th sam as th constant m cλ. That is, pλ = λ = h (III.6) onsidring (III.6), it is possibl to logically driv (II.) from (II.1). Thus, E = = λν (III.7) IV. Discussion W nxt substitut th following valus for physical quantitis in m cλ [4]. 31 m kg = (IV.1) c = m s m (IV.) λ = (IV.3) By doing so, m cλ bcoms as follows. 6

7 34 = J s λ (IV.4) Manwhil, Planck s constant has th following valu [4]. 34 h = J s (IV.5) m cλ and h ar a prfct match. Th currntly known valus for m or λ wr not dtrmind through xprimntation. m was dtrmind through prcis calculations from Rydbrg constant formulas, and λ was obtaind by substituting m in th formula λ =h/m c. Basd on masurd data from thortical formulas or xprimnts dsignd to rprsnt th fundamntal laws of physics, many fundamntal physical constants ar bing adjustd to avoid conflicts from arising btwn ths constants. Bcaus th formula to dtrmin an lctron s ompton wavlngth is λ =h/m c, naturally th modifid vrsion of this Formula (II.10) is tru. Logically, howvr, Planck s constant should thought of as a constant that only coms into xistnc onc Formula (II.1) is rwrittn into Formula (II.) to includ a photon s frquncy, and th subsqunt rcognition that th non-frquncy componnts λ form a constant which can b rplacd by h. In othr words, (II.10) can b intrprtd to man not m cλ and h ar idntical but instad to man m cλ is h. Thrfor, this papr dos not claim th discovry of any rlationship in (II.10). Rathr than naming this constant as Planck s constant h, w can simply rgard it as m cλ = pλ=const. Howvr, bcaus this constant has bn historically usd in othrs of Planck s rsarch, it has takn on th imag of bing a discovrd univrsal constant. V. onclusion According to xisting thory, Formulas (II.1) and (II.) hav bn thought to hav similar importanc. Howvr, according to our discussion, (II.1) is th mor fundamntal of th two. 7

8 Formula (II.) is Formula (II.1) rwrittn to also includ frquncy. Th right sid of (III.7), th product of th physical quantitis λ xcpt for frquncy, is a stady valu. Rgardlss of whthr m cλ is calld Planck s constant h, in this papr w conclud that Planck s constant h only cam into xistnc onc it was dfind. Howvr, not bing awar of what should hav bn dfind, this task was skippd, and thus Planck s constant was blivd to b a discovrd univrsal constant. Thus, it is valid to rgard Planck s constant not as a univrsal constant but as a physical constant on par with th fin structur constant α or th Rydbrg constant R. Acknowldgmnt I would lik to xprss my thanks to th staff at AN Translation Srvics for thir translation assistanc. Rfrncs [1] M. Plank, Phys. Gs., 37 (1900). [] A. Einstin, Ann. Phys. 18, 639 (1905). [3] A. Einstin, Ann. Phys. 17, 13 (1905). [4] P. J. Mohr and B. N. Taylor, ODATA rcommndd valus of th fundamntal physical constants: 00, Rv. Mod. Phys.77, No.1, 1 (005). 8