Fundamenal Jounal of Mahemaical Phsics Vol 3 Issue 013 Pages 55-6 Published online a hp://wwwfdincom/ MATHEMATICAL FOUNDATIONS FOR APPROXIMATING PARTICLE BEHAVIOUR AT RADIUS OF THE PLANCK LENGTH Univesias Pendidikan Indonesia Jl DR Seabudhi No 9 Bandung Indonesia e-mail: andikaaiseawan@upiedu Absac In his heoeical sud we have appoximaed paicle behaviou a adius of he Planck Lengh l p such as posiion momenum and eneg has diffeen inepeaion wih he Heisenbeg unceain Heisenbeg unceain pinciple said ha we can no ge he fixed measuemen of boh posiion and momenum simulaneousl On he ohe hand his heoeical sud found ha we can measue posiion and momenum o posiion and ime simulaneousl even when = 0 Bu he poblem hen emeges fom he las equaion of his heo Thus we can no measue hem if = 0 because his measuemen is bounded b l p 0 1 Inoducion Fom man eas ago mico paicle like elecon is consideed o have wave popeies ([5-6]) The heoeical sud also ies o pove wh paicle is consideed o have wave popeies This is he las ilog of m wo pevious papes fo appoximaing paicle behaviou a adius of he Planck Lengh Kewods and phases: Planck lengh Heisenbeg unceain micoscopic ssem Received June 13 013 013 Fundamenal Reseach and Developmen Inenaional
56 Mahemaical Appoximaion Fom [1] and we se v0 = 0 = l p we can ewie one-dimensional equaion fo he paicle ha depends on ime as follows: cos( 90 θ) = 05 g sin ( 90 ) + θ (1) If we se θ = 0 fo all ime hen he coefficien of in (1) is an acceleaion of gavi ( g ) of fee fall on eah suface Bu i is no on eah suface we ae alking abou a micoscopic ssem Fom equaion (1) we ae ing o analse paicle behaviou a = l p B using he same equaion we can analse fo << e in which e is eah adius ([]) Since (1) is deived fom semicicula moion in [1] hen we ge he elaion beween and θ as follows: x cos θ = sin ( 90 ) cos x θ = θ = () In which x is a pojecion lengh of owad hoizonal axis Since x is a pojecion lengh of hus 0 < x < = l p Hee ou poblem is difficul o deemine he exac value of x because i is smalle han Planck lengh ( l p ) I is known ha l p is measuable minimum lengh (see [4]) Thus he ineval value of () is 0 < cos θ < 1 Now we will invesigae he cases aound minimum o maximum of cos θ 1 If we se cos θ ~ 1 (aound he maximum) hen equaion (1) become fee fall on eah suface because cos( 90 θ) = sin θ ~ 0 heefoe we can appoximae (1) wih = 05g Bu i is impossible because i is no on eah suface
MATHEMATICAL FOUNDATIONS FOR APPROXIMATING 57 b If we se cos θ ~ 0 (aound he minimum) hen () can be appoximaed sin ( 90 ) cos x θ = θ = ~ 0 (3) Fom (3) we ewie (1) = 05 cos( 90 θ) = 05 sin θ = A( ) sin θ (4) Equaion (4) means ha paicle equaion in (1) is a wave equaion a ve small adius B using igonome popeies we ge fom (3) Equaion (5) means o sin θ ~ 1 (5) sin θ ~ 1 (6) sin θ ~ 1 (7) Since (6) and (7) ae onl appoximaion values aound 1 and 1 so we can ewie (6) and (7) in he following ineval: Mulipling (8) wih A ( ) in (4) we have 1 sin θ 1 (8) 05 05 sin θ 05 min = 05 05 = max (9) Fom (9) we inepe ha he posiion of paicle is in his ineval Bu he mos
58 possible of finding paicle fom (9) is aound he maximum o minimum of Now we will deive fom (9) a = max bu we do no conside a = min because a his poin kineic eneg will be negaive = 05 (10) d d v = = (11) dν d = a = v (1) Fom (10) and (11) we have = 05v (13) 05mv = m (14) 05 p = m (15) Fom (15) Paial deivaive wih espec o momenum p = p (16) = (17) m v 1 = m( v ) (18) 1 = m( a ) (19) Subsiuing (1) ino (19) = (0) mv
MATHEMATICAL FOUNDATIONS FOR APPROXIMATING 59 p = m (1) = p m () We can appoximae () = p m (3) B using he same pocess we have paial deivaive fo posiion wih espec o ime = p m p = (4) m Fom (3) and (4) we educe (9) ino smalle ineval as follows min + < 0 < + (5) min 0 max max Fom (8) and (9) he pobabili of finding paicle in he ineval min is zeo Fom (11) we will calculae kineic eneg + max o v v = (6) 05mv mv = (7) mv Ek = (8) p Ek = (9) p Ek = (30) We knew ha he Planck lengh defined b hg =l p = (31) 3 πc
60 Hee h is Planck consan G is he consan of univesal gaviaion and c is he speed of ligh Subsiuing (31) ino (30) we have E k p hg = (3) 3 πc Subsiuing (3) o (3) we have p p hg = (33) Ek 4m 3 πc The bounda condiion is l p [4] hus p p E 4m k 1 (34) Now we will compae equaions (3) (4) (5) (3) (33) and (34) wih he Heisenbeg unceain Heisenbeg Unceain (see [6]) Fo Posiion and momenum h p π Fo Eneg and ime h E π Aiseawan Unceain fo Posiion and momenum = p m Unceain fo posiion and ime p = m Pobabili ineval of finding paicle 0 < max max + min min + < 0 Relaion beween eneg and ime Ek = p hg 3 πc Minimum condiion fo Wih: = p p E E 4m k p p hg k 4m 3 1 πc
MATHEMATICAL FOUNDATIONS FOR APPROXIMATING 61 3 Resuls and Discussion If he posiion of paicle can be measued exacl = 0 Fom Heisenbeg unceain he momenum will be unknown exacl p ~ I means we can no measue posiion and momenum simulaneousl and vice vesa On he ohe hand fom (5) if = 0 max max i means he posiion is ceain = max B using (3) o (4) if = 0 i means he momenum of paicle o ime ae ceain ( p = 0) o ( = 0) Unfounael he condiion whee momenum o posiion can be known exacl in he pevious discussion is limied b equaion (33) and (34) so ha minimum should be geae o equals han l p ( l p 0) I causes he posiion ime and momenum of paicle become a pobabili Equaion (33) means if we wan o make smalle han l p o equals o zeo we need ve big (o infinie) kineic eneg Equaion (33) also means if we wan o make smalle han (o infinie) oo l p o equals o zeo he mass of paicle should be ve big Refeences [1] A Aiseawan Esimaion models using mahemaical conceps and Newon s laws fo conic secion ajecoies on eah s suface Fundamenal J Mahemaical Phsics 3(1) (013) 33-44 [] A Aiseawan How o eliminae gavi effec fom moving bod nea he eah suface Fundamenal J Mahemaical Phsics 3() (013) 51-54 [3] W Beozzi Speed and kineic eneg of elaivisic elecon Ame J Phs 3 (1964) 551 [4] Luis J Gaa Quanum gavi and minimum lengh In J Mod Phs A (10) 1995 [5] D Hallida and R Resnick Fisika:jilid Tanslaed b: Panu Silaban and Ewin Sucipo 1984 Elangga Jakaa 1978 [6] Kenneh S Kane Fisika Moden (Moden Phsics) Tanslaed b: Hans J
6 Wospakik and Sofia Niksolihin UI-Pess Jakaa 199 [7] Edwin J Pucell and D Vabeg Kalkulus dan Geomei Analiik: jilid 1 Tanslaed b: I Noman Susila Bana Kaasasmia and Rawuh Elangga Jakaa 003 [8] Edwin J Pucell and D Vabeg Kalkulus dan Geomei Analiik: jilid Tanslaed b: I Noman Susila Bana Kaasasmia and Rawuh Elangga Jakaa 003