THE PHOTON WAVE FUNCTION AND THE FRESNEL FORMULAS

Transcription

1 Romaa Reorts Physcs Vol 6 No P OPTICS THE PHOTON WAVE FUNCTION AND THE FRESNEL FORMULAS I-IOVITZ POPESCU P STERIAN M DOBRE Natoal Isttute for Lasers Plasmas ad Radato Physcs PO Box MG7 775 Magurele Romaa Physcs Deartmet Poltehca Uversty Bucharest Romaa E-mal: marceldobre@yahoocom (Receved October 9) Abstract Electromagetc heomea ca be ether classcally descrbed wth the hel of Maxwell's equatos or by the methods of quatum comutg ad Schrödger equato I the reset aer we wll descrbe the heomea of reflecto ad refracto usg oly quatum mechacal methods Fally we get Fresel s equatos wthout to call o the classcal electromagetc equatos The method volves the use of wave fuctos of the hoto the mometum reresetato comled wth a olarzato term ad kees the romse of a smle way of calculatg the behavor of hotoc devces as used otoelectrocs Key words: Schrödger tye equato Fresel formulas INTRODUCTION As t s well kow the quatum mechacal calculato methods are very advaced as comared to those of classcal hyscs oes For ths reaso descrbg a hyscal heomeo by the methods of quatum mechacs s usually much smler comarso to the same heomeo descrbed by the methods of classcal hyscs I the followg we wll get the Fresel's equatos usg oly the mathematcs of quatum mechacs Geerally the wave-artcle dualty aled to electromagetc waves mles that the lower the wavelegth that s the hgher the hoto eergy the more defed becomes the artcle behavor as s the case for X- ad Γ-rays The heomeo of hoto tuelg has bee secfed for the frst tme by Heseberg [] whe referrg to the assage of lght through a th fol of gold I ths semal study the roagato of the electromagetc wave through the th gold fol has bee assmlated to the hoto eetrato through a otetal barrer Ths cocet was later used also by other authors [-4] order to descrbe the hoto roagato through a delectrc usg the mathematcal devce of quatum mechacs

2 Photo wave fucto ad the Fresel formulas 36 THE LINK BETWEEN THE PHOTON WAVE FUNCTION AND THE ELECTROMAGNETIC FIELDS WAVE The method of the wave fucto of the hoto s reseted [5 7] wth the hel of the hexa-vector Q troduced by Plücker ad Kayley [8 9] Thus we wll defe QA = E + B QB = D+ H () wth D= εe B = µ H () The hoto wave fucto s a lear combato of these two vectors Ψ= KQ + KQ (3) A A B B where K A ad K B are comlex coeffcets For a homogeeous ad trasaret medum we have ε µ Ψ= C Q ( ) 8 A + Q B = C ε E + µ H (4) π + + 8π where C s the ormalzato factor By detfyg the costats eq (4) we obta the followg system of equatos KA +ε KB = C ε KB +µ KB = C 8π 8π µ (5) wth the solutos ε µ KA = C KB = C 8π + 8π + (6) I ths case the ormalzato codto reads + ED + HB ΨΨ ddd x yz= C ddd xyz 8π (7) The wave fucto from (4) ca also be rereseted the followg matrx form ε Ex + µ H x Ψ x Ψ C ε Ey + µ Hy Ψy 8π ε Ez + µ H Ψ z z or usg the algebra of s (8)

3 36 I-Iovtz Poescu P Stera M Dobre 3 δ= ι + ι + ι3 (9) δ δ= δ + δ =δ () the wave equato for hoto becomes [7] Ψ Ψ + Ψ Ψ = t + + ( ) ( v S ) () By addg ad subtractg the term v t Ψ we fally get Ψ Ψ + ( ) = c t c t I the case of roagato through a delectrc medum eq () ca be wrtte () K Ψ Ψ = c t ω K c Wth the otato troduced [] the otetal of delectrc medum has the form 3) U = ck = ω (4) Fally the eergy of a quatum artcle artcular that of a hoto ( m = ) whch moves a electromagetc feld characterzed by a otetal vector A ad a scalar oe Φ as well as our scalar otetal U assocated to the delectrc medum may be wrtte: ( ) ( ) E = qφ+ m c + U + c qa (5) 3 PHOTONS AND STEP POTENTIAL It should be emhaszed that the hoto wave fucto troduced above has ot the sgfcace of amltude of satal locato lke o-relatvstc quatum mechacs the oto of hoto coordate beg vod of ay hyscal cotet

4 4 Photo wave fucto ad the Fresel formulas 363 However the cocet roves to be beefcal ad s used by some authors who aly quatum mechacs methods to descrbe the electromagetc heomea wth the hel of hotos [] Let us further cosder the ste otetal from Fg assocated to the terface betwee two delectrc meda characterzed by the otetals U ad U U(x) U E U x Fg Oe-dmesoal otetal for alog U drecto eredcular to delectrc terface The wave fuctos of the hoto mometum reresetato solutos of Eq () are case of oe-dmesoal otetal from Fg the followg: xx xx x Φ = Ae + Be (6) xx x Φ = Ce (7) I Fg we suose that the cdet artcle our case the hoto moves from left to rght We have to ote that the mometa x ad x may be real or magary asmuch as the quatty ( E U ) may be ostve or egatve (see eq (5)) whch meas that we have a rogressve or a real decreasg exoetal wave We meto that there are meda wth the refractve dex < whch case the otetal U s magary ad the quatty ( E U ) ca become egatve I addto we wll also assume that the artcle - hoto ossesses a eergymometum quadrvector E ad a ut vector orthogoal to the movg drecto of the artcle the latter beg the hoto olarzato vector Cosequetly the wave fucto wll be comleted wth ths addtoal vector that s the the ew wave fucto s the roduct of the orgal wave fucto deedg o coordate Φ ( r ) ad the olarzato vector amely

5 364 I-Iovtz Poescu P Stera M Dobre 5 Ξ =Φ (8) ( r) The reflecto ad trasmsso coeffcets o the ste otetal from Fg are obtaed mmedately from the cotuty of the corresodg wave fuctos x Ξ = A + B xx xx e e (9) x Ξ = xx C e () 4 THE FRESNEL RELATIONS The cotuty codtos of (9) ad () at the otetal ste x = leads to B x x = () A x+ x C x = () A + x x Usg further the relatos () ad () ad the fact that the assage of artcles from medum characterzed by the otetal U to medum characterzed by otetal U coserves the artcle mometum arallel to the delectrc terface [] a smle calculato (see Aex) leads mmedately to B ( θ θ) s = (3) A s ad resectvely to C cosθ sθ = A s (4) We fd two artcular stuatos ay other case beg a lear combato of these two amely: a) the olarzato vector s eredcular to the cdece lae ; b) the olarzato vector s arallel to the cdece lae ;

6 6 Photo wave fucto ad the Fresel formulas 365 a) - b) Fg Idces ad refer to the cdet reflected ad trasmtted hoto: a) vectors ormal to the cdece lae ; b) vectors arallel to the cdece lae As ca be see from Fg a the eredcular olarzato vectors are arallel to each other ad so the drecto coses are = (5) = (6) Ths allows us to wrte for the amltude of the wave fuctos coformty to Eq (3) ad (4) the followg relatoshs

7 366 I-Iovtz Poescu P Stera M Dobre 7 B C ( θ θ) s = A s cosθsθ = A s that s Fresel relatos for the case of eredcular olarzato I the case of arallel olarzato for Fg b we get the olarzato vectors ratos: = cos ( θ θ ) (9) ad = cos π ( θ +θ) (3) Dvdg further the last two relatoshs t results cos ( θ θ) = (3) cos θ +θ ( ) Fally usg Eq (9) ad (3) Eq (3) ad (4) we get for hotos wth arallel olarzato vectors cotaed the cdet lae followg amltudes: s ( θ θ) cos( θ +θ) B = A (3) s θ +θ cos θ θ or more C ( ) ( ) cosθ sθ (7) (8) = A (33) s ( θ +θ) cos ( θ θ) C ( θ θ) ta B = A (34) ta cosθ sθ = A (35) s ( θ +θ) cos ( θ θ) The relatoshs (34) ad (35) are Fresel s relatos for the case of arallel olarzato We have obtaed ths way the Fresel relatos for eredcular as well as for arallel olarzato wthout ay aeal to Maxwell's equatos or to cotuty relatos for the electrc ad magetc felds the searato area betwee delectrcs

8 8 Photo wave fucto ad the Fresel formulas 367 REFERENCES W Heseberg Physkalsche Prze der Quatetheorer S Hrzel Verlag Stuttgard 93 4 footote & 4 L de Brogle The Reterretato of Wave Mechacs Foud Phys (97) 3 C Phlds C Dewdey BJ Hley Quatum Iterferece ad the Quatum Potetal Nuovo Cm 5 B (979) 4 N Marescu RE Nstor Quatum Descrto of Mcrowave Passve Crcuts Ca J Phys 68 (99) 5 R E Nstor I-Iovtz Poescu N Ioescu-Pallas A Schrödger tye exlaato of Fresel formulas Joural of Otoelectrocs ad Advaced Materals (7); htt://oeoero/joam/dexh?oto=magaze&o=vew&du=836&catd=6 6 I-Iovtz Poescu R E Nstor E Petrescu O the eergy trasfer a otcal couler Joural of Otoelectrocs ad Advaced Materals (7); htt://oeoero/joam/ dexh?oto= magaze&o=vew&du=837&catd=6 7 RE Nstor Quatum Asects of Photo Proagato Trasaret Ifte Meda Romaa Reorts Physcs (8); htt://wwwfmro/rr/8_6_3/ df 8 Ncholas Ioescu-Pallas Electrodyamcs of Movg Meda ad the Relatvty Theory Prert Isttute of Atomc Physcs Theoretcal Physcs Joh Davd Jackso Classcal Electrodyamcs Vol II Chater 7 Joh Wlley & Sos Ic New York 999 R E Nstor E Petrescu Wavegude Otogalvac Devce Czechoslovak Joural of Physcs (4); htt://adsabsharvardedu/abs/4czjph54483n Ncolae Marescu ad Rudolf Nstor Quatum Feature of Mcrowave Proagato a Rectagular Wavegude Z Naturforsch 45a (99) L Ladau E Lfchtz Mecaque Edtos Mr Moscou ANNEX To obta eq (3) ad (4) we cosder the followg: I order to obta the equatos (3) ad (4) let us cosder FgA Whe a artcle crosses the terface betwee two meda wth otetals U ad U resectvely the arallel mometum comoet to ths terface s coserved [] that s z P z P P x P z P x P x Fg A The hoto momets two delectrc meda wth <

9 368 I-Iovtz Poescu P Stera M Dobre 9 z z = (A) Takg further to accout that = wth the hoto mometum vacuum ad the refractve dex of the medum we ca wrte terms of the agles of Fg A the lght refracto law s θ = s θ (A) O the other had the ormal mometum comoet to the terface betwee the two meda s ot coserved ad s gve by ` x = cos θ (A3) or terms of the hoto mometum vacuum x cos θ cosθ x x cos θ cosθ = = = x + x cosθ + cosθ cosθ + cosθ ad usg (A) we obta eq (3) Smlarly s θ cos θ cosθ s θ s ( θ θ) = = + s θ cosθ + cosθ s x x x x s θ cosθ x cosθ = = x + x cosθ + cosθ cosθ + cosθ ad usg aga (A) we get eq (4) s θ θ s cos s = = + s θ cosθ + cosθ s θ +θ cos x θ θ θ ( ) x x s θ (A5) (A6) (A7) (A8)