Planck constant estimation using constant period relativistic symmetric oscillator

Transcription

1 Plnk onstnt estition using onstnt period reltivisti syetri osilltor J. Br-Sgi * Applied Phys. Div. Soreq NRC, Yvne 818, Isrel. The eletrogneti wve quntu-energy depends only on its frequeny, not on the eitting syste s rdition power. The proportionlity onstnt between the frequeny nd the quntuenergy of the eletrogneti wve, the Plnk s onstnt is in the essene of quntu ehnis. This onstnt is known experientlly but till now there ws no lue for lulting its vlue on theoretil bsis. In the present work ethodology for lulting lower bound for Plnk s onstnt is presented, bsed on siple priniples. In order to get resonble good lower bound it is neessry to hve odel of reltivisti osilltor whose period is independent of its energy nd whih effiiently rdites eletrogneti energy. It is highly desired tht the thetis involved is siple enough to enble good insight into the results. Suh odel n lso be used for other investigtions, nd therefore, in this work potentil tht onserves the vibrtion period of syetri osilltors t reltivisti veloities is found nd nlyzed. The eletrilly hrged syste of onstnt period is used to lulte lower bound H of the Plnk s onstnt h. The vlue of H is sller thn h by ftor very lose to 3. The explntion of this ftor lso explins the vlue of Plnk s onstnt. Fro this vlue the fine struture onstnt vlue is lulted nd new interprettion of this onstnt obtined. PACS nuber(s): 14.7.Bh, 3.65.Sq, , 3.3.+p I. INTRODUCTION At the beginning of the th entury when the quntiztion of eletrogneti (EM) energy ws disovered, there were no theoretil tools to estite the vlue of Plnk s onstnt; onstnt tht reltes the energy of EM qunt to the frequeny of the eletrogneti wves. Lter it ws believed tht this is bsi onstnt so there ws no point to investigte its vlue theoretilly. There is need for n iproved odel to resonble estite Plnk s onstnt. This odel will reple the hroni osilltor odel. The siple hroni osilltor (SHO) is frequently used to explin different spets of physil phenoen. The siple nlytil solutions of the perti- * E-il ddress: j.br-sgi@fiuu.o

2 nent differentil equtions, both in the lssil nd quntu ehnis fres, ke this syste speil.the one diensionl (1D) SHO hs in ddition unique property nely, t non reltivisti veloities it hs onstnt period, independently of its energy. When 1D SHO is eletrilly hrged, its rdition frequeny is the se s the osilltions frequeny both ording to lssil eletrognetis nd quntu ehnil theories. A two diensionl (D) syetri SHO exhibits ore interesting properties, while retining the sipliity of single frequeny of osilltion nd rdition. It is well known tht t high energies, when reltivisti effets re iportnt the osilltion period is not onstnt [1]. It is energy dependent. The potentil for 1D osilltor tht ensures the onstnt period, independently of energy, t reltivisti veloities hs been lulted []. However the ext potentil ws not desribed by n nlyti expression. The first objetive of the present work is to find the potentil tht ensures onstnt period in the irulr ode of vibrtion (Syetri Constnt Period Potentil - SCPP) t reltivisti veloities. In this work, the potentil is expressed in losed-for. At reltivisti veloities, D eletrilly hrged syste is expeted to rdite its energy ore effiiently thn the 1D syste. For the generl ellipti odes of vibrtion there is need to fit potentil for eh vlue of eentriity. In this pper generl for for n pproxite potentil with one preter is presented. The vlue of this preter y be found nd n be heked for ury in the future. I hope tht the present work provides the thetil tools tht will help to enhne our understnding of reltivisti lssil ehnis nd reltivisti quntu ehnis. The otivtion for this study is the ttept to understnd the EM quntu-energy, i.e., the photon energy. This understnding strts with the siple observtion tht ording to lssil wve theory, In order tht EM rdition will hve defined frequeny within soe ury, it needs to be rditing t lest during one yle. Plnk s interprettion of blk body rdition [3] inludes the odel where the rdition is produed by eleentry systes, presubly eleentry hroni osilltors, with qunt of energy independent of the osilltors energy, despite the ft tht the rdition power inreses with the osilltors energy growth. The frequeny is one of the bsi properties of the photon; property tht deterines its energy. The obined result is tht t ny frequeny the EM quntu-energy ust be lrger thn the ount tht ny eleentry syste of onstnt period 1 n produe in one yle. At very high syste s energies, new hnnels, in ddition to the EM rdition, re opened for the relese of energy. Suh proesses of energy relese will liit the EM rdition of n osilltor. A high energy osilltor is nturlly in the reltivisti veloities doin. In order to get lower bound for the energy quntu we need n eleentry syste with onstnt period tht rdites energy effiiently. For this purpose syetri onstnt period osilltor (SCPO) is seleted. A odel of prtile hving ss nd eleentry eletri hrge of n eletron e is used s the oving oponent of the SCPO. A opetitive energy relese proess is pir retion. Above the required energy of this proess y be doinnt. Only under the energy of we n be ertin tht n isolted hrged SCPO will eit its vilble energy s EM wves. 1 For rditing syste of onstnt period, the independene of its rdited EM quntu-energy fro the syste s energy follows fro the ft tht the EM quntu-energy only depends upon its frequeny, whih is onstnt.

3 The rdition power of the hrged SCPO t energy equls will be lulted. Clulting the energy rdited in one period of osilltion t tht power gives lower bound for the EM quntu E H. Here ν is the rdition frequeny nd H is lulted ftor. Copring this result with the experientl vlue of Plnk s onstnt yields surprisingly siple reltion h 3H. The error in this reltion is.14%. Although there is no theoretil explntion for this ftor of 3, it y be explined in the future. Considering this reltion s n expression for h in ters of H llows the lultion of the fine struture onstnt (within the se reltive error). This onstnt is inversely proportionl to the squre of the referene norlized elertion derived fro the SCPO odel t energy of. In the rosopi world the EM quntu-energy is desribed s inute ount of energy. For n eleentry syste of oleulr size, this is tully huge ount of energy tht nnot be rdited during single period of osilltion. Atully, ording to lssil eletrodynis lws in usul ses there is need for thousnds of periods of osilltions in order to produe one quntu. The quntu ehnil equivlent is the verge witing tie (or dey tie) until photon is reted, the se tie tht this nuber of osilltions tkes. The present work gives n intuitive explntion for the gnitude of the EM quntu. II. THE RELATIVISTIC SYMMETRIC CONSTANT PERIOD POTENTIAL For irulr otion the potentil tht ensures onstnt period T should openste for the entrifugl potentil (relted to the entrifugl fore); tht is the se potentil with opposite sign. Suh reltivisti entrifugl potentil is lredy known [4]. The derivtion for this potentil is repeted here in order to hve interedite results. In unifor irulr otion the tngentil veloity v nd rdius of rottion r re relted to the period T nd ngulr veloity ording to T r. ( 1) v This reltion is used to express the veloity of prtile rotting with ngulr veloity s funtion of the rdius r v r ( r ). ( ) In the irulr otion the reltivisti proper (or intrinsi) entripetl elertion is v r r, ( 3) r r

4 where r is rdil unit vetor nd 1/ 1 s funtion of r. Inserting () r yields the proper elertion r/ r. ( 4) r 1 r / The fore tht genertes this proper elertion t ny r, for given period T is ording to Newton s seond lw: The potentil is found by integrtion r F r. ( 5) 1 r/ F r.5 ln 1 ( 6) V d q 1 ln q 1. 1 q Here diensionless vrible q r r T is used in order to siplify expressions. This onstnt-period potentil is generliztion of the SHO into the reltivisti regie in two or three diensions. There is no dvntge of working in 3 diensions, therefore in the present work we will restrit to the two diensionl frework. The onstny of the period for ny plitude (nd therefore for ny energy), is only gurnteed for irulr otion. A. Properties of the syetri potentil of onstnt period A grph of the potentil, in units of the rest ss energy, s funtion of norlized distne is shown in figure 1.

5 FIG. 1. A ut of the SCPP through the origin in units of rest ss energy. q r / T 1. Centrl region As r tends to zero, the potentil tends to behve s the usul syetri hroni osilltor V r x y r. Edge region 1 1. There is potentil singulrity t r T. It posts brrier tht prevents irultion of prtile on n orbit of iruferene longer thn T. B. Coprison with the one diensionl se The 1D osilltor of onstnt period ext potentil [] is not given nlytilly. Anlytil pproxitions re vilble; none of the re of logrithi for. The derived fore of one of the pproxitions is however siilr to the fore of SCPP: F liner x 4x 1 T 5 4. ( 7) C. Approxite reltivisti onstnt period potentil for ellipti otion This fore given by eq. (7) nd the fore derived in the present work n be written in ters of oon ellipse nottion using the eentriity preter nd the ellipse relted iruferene

6 of unit jor xis L( ) 4 ( ). The funtion is the oplete ellipti integrl of the seond kind [8]. It tkes the for Here xx yy F ellipse. ( 8) n( ) L( ) y 1 x T 1 xnd yre orthogonl unit vetors in the xy plne. For irulr orbit, Ln,,,1 nd for liner trjetory, Ln, 1,4,5 4. It is expeted tht for generl ellipse se the vlue of the exponent n n be expressed s funtion of the eentriity. At non reltivisti veloities, proper initil onditions re enough to ke the prtile enter into desired elliptil pth nd to osillte in the right period T. At the extree reltivisti se the prtile will stik to the elliptil singulrity line nd oplish round trip t period of L( ) / T. A proper hoie of the exponent n( ) will give good fit for the interedite veloities. III. OSCILLATOR PROPERTIES When prtile is osillting under the influene of the SCPP, it follows trjetory in the XY plne ording to its initil onditions. A. The nonreltivisti liit In the nonreltivisti liit ( E ) ll orbits of vibrtion will sty in the entrl region of the potentil nd will hve the se period of osilltions independent of energy or eentriity. B. The super-reltivisti liit In the super-reltivisti liit ( E ) the trjetory will be very lose to the line of singulrity. For irulr otion s initil onditionr, v : 1) r v, ) r v, the orbits re of onstnt period. In the next setions we will refer only to suh orbits. Devition of the Initil onditions fro these two rules leds to distorted trjetory tht is usully not periodi (not losed urve). Though for speil ses the trjetory is periodi. The period is thn energy dependent. See ppendix for lultion nd exples.

7 IV. ENERGY VERSUS VELOCITY AND RADIATION LIMIT A reltivisti prtile rotting in the SCPP possess totl vilble (not inluding rest ss) energy of 1 1 E 1 ln(1 ). 1 ( 9) Here we use the reltion found in eq. (), q. In the next setion the rdition of eletrilly hrged SCPO will be onsidered. Theoretilly, there is no obvious liit for the EM power it n rdite. The liit for the rdition ppers when opetitor energy relese proess exists. Suh proess is retion of prtiles pirs. Tht is when the energy of the SCPO rehes the level of. To be preise we ssue tht the prtile osillting under the influene of the SCPP is n eletron, nd the retion is of eletron-positron pir. Using eqution (9) the veloity of the SCPO t tht ritil energy of is deterined ording to ln(1 ). ( 1) The resulted vlue is A. Mxiu rdited energy t one yle The power rdited by elerting prtile, with n eleentry eletri hrge, e, is ording to Lror forul [5] e P, 3 6 ( 11) where is the prtile s elertion. The extension to reltivisti veloities [6] is de by substituting the proper elertion for [7]. The revolving prtile s proper elertion is given in eq. (3) where the rdius of revolution is onneted to the veloity ording to eq. (). It is useful to norlize the elertion of the prtile by the ngulr veloity nd the speed of light to get diensionless quntity. The definition nd expression for the present se ording to equtions (, 3) re def (1 ) (1 ). ( 1) Cobining equtions (11, 3, nd 1), the rdition power of the syste tkes the for

8 v e, 3 r ( 13) r e P 6 6 where 1 T is the osilltion frequeny. The energy E 1 rdited during one osilltion is just this power ultiplied by the period tie T. The result is (usingt 1) e E1 PT. ( 14) 3 In order to find the xiu rdited energy per period tht the hrged SCPO n eit one should substitute the liiting veloity vlue for pir retion to get norlized elertion. The xiu rdited energy is then E 1, e H. ( 15) 3 We introdued the onstnt H defined by the eqution bove. The vlue of this onstnt is H J s The energy E1, is lower bound to the EM quntu-energy: E h H. H is therefore lower bound for the Plnk onstnt h H. ( 16) The vlue of H is very lose to the vlue of Plnk s onstnt, this is n interesting issue to look t ore losely. This is done in setion V. B. Coprison with the one diensionl se Using the pproxite potentil, relted to the fore given by eq. (7), of the one diensionl osilltor of onstnt period (1DCPO) nd its equtions of otion [], it is stright forwrd to l- ulte its rdition when eletrilly hrged. When the osilltor s energy equls the rdited energy per yle is bout.6 ties less thn the energy rdited by the SCPO t se onditions. This result is not very intuitive. At non reltivisti veloities the effiienies (defined s the rdited energy per yle nd per syste s energy) of the two systes (1DCPO nd

9 SCPO) re identil. The rdition of reltivisti linerly elerting hrge gets ftor of 6 [7]. This provides the one diensionl se with n extr ftor opred to the irulr se [see eqution (13)]. The rdition is, therefore enhned by reltivisti effets. The ft is tht 1DCPO t high energies exhibits sw-tooth profile of position versus tie; so tht long ost of its trjetory there is lost no elertion. High elertion is obtined only t the turning points where the veloity (nd therefore ) is not high. Fro this oprison one n dedue tht diensionlity is iportnt onerning rdition effiieny, t reltivisti veloities. V. THE MATCHING FACTOR AND INTERPRETATIONS Nuerilly, to good pproxition, one n write h H. ( 17) 3 The ury of the reltion is.14%. Suh reltion is puzzling: On one hnd it n be identl, on the other hnd, s 3 is for exple, the digonl of unit ube; it y be sign tht there is hidden physil reltion. A. Vlues of fundentl onstnts Soe interesting onlusions ould be drwn for the fine struture onstnt α, if we onsidering H not only s the lower bound for h, but lso fix the reltion: h 3 H, nd leve it s n investigtion issue to explin it. Then, using eq. (15), we find: nd being e h, ( 18) 3 when inserting the expression for h one gets e, ( 19) h

10 3. ( ) 4 Here is the norlized elertion lulted ording to eqution (1) using the liiting veloity of the SCPO; with energy suffiient for pir retion [see vlue bellow eqution (1)]. It is usul to onsider the inverse vlue, for whih one gets: The vlue of is in greeent with the experientl vlue, within the se error, s in the vlue of h lulted ording to eq. (18), of.14%. The reltive errors re of the order of whih is eptble if good physil explntion to the ftor of B. Interprettion of lph 3 is vilble. The fine struture onstnt is thought by ny physiists to be even ore fundentl quntity thn the Plnk s onstnt, in quntu ehnis in generl nd espeilly in quntu eletrodynis. This diensionless quntity is first used in the lultion of reltivisti orretions to toi energy levels together with the spin-orbit oupling [9]. Fro the result of the present work it is obvious tht origin lies in the reltivity theory, i.e., the reltivisti onnetion between ss nd energy. Its tul vlue is deterined by the physis of rdition s result of elertion, ording to the lws of EM theory. To illustrte the ening of eq. () nd the norlized elertion tht ppers there let s ke use of it. The rte of photons prodution A by n eleentry osillting hrge ording to lssil eletrognetis is just the power eitted by the hrge divided by the EM quntu-energy of the se frequeny. Introduing norlized elertion ording to the definition in eqution (1), A beoes The ngle brkets denote tie verge. P e 1 4 A. ( 1) h h Inserting the vlue of α ording to eq. () into eq. (1) one n write the photon prodution rte s 1 A 3. ()

11 The quntity ppers in eqution () s referene vlue for the verge norlized is esure for how elertion of eleentry rditing systes. The rtio of to fst the syste eits photons on sle of to 1. ACKNOWLAGMENT The uthor grteful to Dr. Aos Sorgen (Universidd de Belgrno, Buenos Aires, Argentin) for his gret help in the revision of this pper nd to Asso. Prof. Ido Brslvsky (Ohio University, Athens, OH) for his enourgeent. APPENDIX The fore derived fro the SCPP tht ts on the prtile ording to Newton s lws in the inertil oordinte syste oving instntneously with the prtile is Bold letters designte vetors. ' F r. The elertion is divided into two oponents: one is prllel to the prtile veloity v, nd the other is perpendiulr to it )A1( ' ' v v v ' ' '. )A( The elertion trnsfortion to the oordinte syste tthed to the enter of rottion with fixed orienttion is ording to speil reltivity rules [1] 3 ' '. )A3( The veloity nd position re then lulted in Crtesin oordintes, using [, ], x y d vi i i x, y dt d ri v. i dt )A4(

12 We prefored nueril oputer integrtion of the set of equtions A1-A4 for different initil onditions. All oputtions were done for 1, nd T 1nd the results re nlyzed in the following setions. A. Super-reltivisti se Clultion ws de for initil veloity,.995, x y, with different initil lotions. When the initil lotion does not use irulr otion the prtile usully bounes between different points on the enlosing irulr brrier of potentil singulrity (See setion II.A.) long lines tht re lost stright, without losing loop. In ertin ses polygonl shpe is fored. Two exples, one for tringle nd seond for squre re shown in figure together with the irle resulting fro proper initil onditions. The periods of these 3 trjetories re different nd they re surized in tble I. FIG.. Clulted periodi trjetories in the super-reltivisti (.995 ) region on the xy plne. The third xis is the tie xis. Initil onditions re given in tble I. TABLE I. Periodi trjetories initil onditions nd resulted periods nd shpes. Initil onditions outoe No. x. y / x, Period shpe irle squre tringle Rounded tringle

13 B. Interedite to high energies At oderte reltivisti veloities, the non irulr trjetories re rounded. Fro bout.7 nd up, by fitting the vlue of the initil perpendiulr rdius, we find t lest one periodi trjetory in ddition to the irulr one. Suh periodi rounded tringle is shown in figure 3. The initil onditions nd the period for this se re surized in the lst row of tble I. FIG. 3. A lulted periodi trjetory in the interedite energy region (.75 ), on the xy plne. C. Low to interedite energies At very low nonreltivisti energies the usul ellipti trjetories re obtined, ll with the se period nd different shpes ording to their initil onditions. As the energy inreses the elliptil trjetories strt to show preession effets, typil to reltivisti orbits.. This effet inreses with inresing energy until it is no longer possible, in the generl se to hve defined period (unless initil onditions ensure irulr orbit). One exple of suh trjetory is shown in Figure 4 together with irle resulting fro the se vlue for x.

14 FIG. 4 Clulted trjetories on the xy plne for ses where.4,. 1) r Results in preessing ellipse. ) r,. /,.4 / Results in irle. D. Liner trjetories In ddition to the entioned trjetories, there re liner trjetories resulting fro the initil onditions r or r β. The period of suh trjetories hnges with energy, nd in the superreltivisti liit we get: T T 1. T Here is the low energy period of osilltion. The grph of the period s funtion of the initil veloity, for the se r is shown in figure 5. FIG. 5. Liner osilltions period versus initil veloity (initil position t the origin).

15 Referenes [1] H. Goldstein, Clssil Mehnis, nd edition (Edison-Wesley, Reding, MA, 1981), p. 34. [] Jung-Hoo Ki, Seung-Woo Lee, Hns Mssen nd Hi-Woong Lee, Phys. Rev. A, Vol. 53, No. 5, 991 (1996). [3] Mx Plnk, Annlen der Physik, vol. 1, no 4. April, 719. (19). [4] Moses Fynogold, Speil reltivity nd how it works, WILEY-VCH Verlg GBH & Co. KGA., Weinhei. Chp. 13, p. 68 (8). [5] J. Lror, Phils. Mg. 44, 94 (1897). [6] O. Heviside, Nture 67, 6 (19). [7] J. D. Jkson, Clssil eletrodynis, 3rd edition, John Wiley & sons, In. 111 River Street, Hoboken, NJ, p. 667 (1999). [8] L. M. Milne-Thoson, Hndbook of Mthetil Funtions with Foruls, Grphs, nd Mthetil Tbles, 1th edition, edited by Milton Abrowitz nd Irene A. Stegun, Dover, New York, p. 587 (197). [9] Dvid J. Griffiths, Introdution to quntu Mehnis, Prentie Hll, Upper Sddle River, New Jersey 7458 (1994), Chp 6.3. p. 35. [1] Ref. [7], p. 569 (1999).