NUMERICAL CALCULATION OF THE WIGNER ROTATION TAREK HOSSAIN & SHAH ALAM

Transcription

1 Intenatinal Junal f Physis and Reseah IJPR ISSNP: 5-3; ISSNE: l 7 Issue Jun 7-6 TJPRC Pt Ltd NMERICAL CALCLATION OF THE IGNER ROTATION ABSTRACT TAREK HOSSAIN & SHAH ALAM Depatment f Physis Shahjalal niesity f Siene and Tehnlgy Sylhet Bangladesh e hae deied the fmula f igne tatin in tw ways Initially we hae alulated igne tatin f π µ and e deay hange Late we hae alulated hange f igne tatin f Bb with espet t missin ntl e epesented speial and mst geneal Lentz tansfmatins The elity additin fmula f speial and mst geneal Lentz tansfmatins ae lealy explained e hae alulated the igne tatin with espet t diffeent Bb elity The gaph f the igne tatin with espet t diffeent Bb elity has pltted e hae used simulated data f applying the igne tatin fmula in pin deay hain and nluded the esult KEYORDS: Speial Lentz Tansfmatin; Mst Geneal Lentz Tansfmatin; Lentz Tansfmatin; elity Additin Fmula Mst Geneal Lentz Tansfmatin; elity Additin Fmula Speial Lentz Tansfmatin; igne Rtatin PACS: 33 P Reeied: Feb 8 7; Aepted: May 3 7; Published: Jun 3 7; Pape Id: IJPRJN7 INTRODCTION igne tatin esults fm the fmatin f tw nn-llinea Lentz bsts that is nt a pue bst but is the mpsitin f a bst and a tatin This tatin is knwn as Thmas tatin Thmas igne tatin igne tatin The tatin was diseed by Thmas in 96 [] and deied by igne in 939 [] hen a sequene f nn-llinea bsts etuns the spatial igins f a sequene f inetial fame t the stating pint then the sequene f igne tatins mbine t pdue a net tatin alled the Thmas peessin [3] Cnside fu inetial fames f efeenes S π µ and e whee the fame S is the lab fame whih Oiginal Atile is at est and the fame π is ming with unifm elity with espet t S µ is ming with unifm elity with espet t π e is ming with unifm elity espet t µ e want t find the elity f the eletn with espet t lab fame S Thee ae tw ways t get the elity f eletn with espet t lab fame Fistly we an take the Lentz sum f and ie whee dentes the Lentz sum and then we an take the Lentz sum f and ie Sendly we an take the Lentz sum f and ie and then we an take the Lentz sum f and ie The angle between these tw elities is igne tatin and wwwtjpg edit@tjpg

2 Taek Hssain & Shah Alam Figue : igne Rtatin f π µ and e Deay Change e an als explain the igne tatin by the fllwing ways Let us nside Alie is ming with elity with espet t missin ntl and Bb is ming with elity espet t Alie as shwn in Figue nftunately missin ntl annt dietly bsee the elity f Bb The elatiisti mbinatin f elities is hw we dedue the elity f Bb as seen by missin ntl ug the elities and e must be lea that is measued in Alie's est fame whilst and ae measued in missin ntl's est fame e deie a simple fmula f this elity and it is quantitatie esults the elatiisti mbinatin f elities and [4] Missin ntl bsees Bb as the spaeaft labeled B and t be ming at elity but pinting in a dietin tated by the igne tatin angle Ω Figue : The igne Rtatin f Bb with Respet t Missin Cntl LORENTZ TRANSFORMATION The tansfmatin whih elates the bseatins f psitin and time made by the tw bsees in tw diffeent inetial fames is knwn as Lentz tansfmatin Impat Fat JCC: NAAS Rating: 4

3 Numeial Calulatin f the igne Rtatin 3 Speial Lentz Tansfmatin Figue 3: The Fame S is at Rest and the Fame S is Ming with Respet t S with nifm elity Alng x-axis Let us nside tw inetial fame f efeenes S and S whee the fame S is at est and the fame S is ming alng x-axis with elity with espet t S fame The spae and time dinates f S and S ae x y z t and x y z t espetiely The elatin between the dinates f S and S whih is alled speial Lentz Tansfmatin an be witten as [5-8] x x t y y z z t x t And x x t y y z z t x t Speial Lentz Tansfmatin is ne dimensinal The elities f ming fames ae alng x-axis S thee is n igne tatin Speial Lentz Tansfmatin Mst Geneal Lentz Tansfmatin Figue 4: The Fame S is at Rest and the Fame S is Ming with Respet t S with nifm elity A Lng any Abitay Dietin wwwtjpg edit@tjpg

4 4 Taek Hssain & Shah Alam hen the elity f S with espet t the S is nt alng x-axis ie the elity has thee mpnents x y and z then the elatin between the dinates f S and S whih is alled mst geneal Lentz tansfmatin an be witten as [5 8-] X X [{ X / }{ / / } t / / ] 3 t / / t X / and X X X t t X t 4 hee and Hee X and X is the spae pat f the S and S fame espetiely 3 RELATIISTIC ELOCITY ADDITION FORMLA OF LORENTZ TRANSFORMATION 3 Relatiisti elity Additin Fmula f speial Lentz Tansfmatin Cnside thee inetial fames f efeene S S and S whee the fame S is at est and the fame S is ming alng X-axis with elity with espet t S fame and the fame S is ming alng X-axis with elity u with espet t S Then Fm Equatin we hae x t x t x t Figue 5: elity Additin f Speial Lentz Tansfmatin Diiding numeat and denminat byt we get x t w t t t x t t Impat Fat JCC: NAAS Rating: 4

5 Numeial Calulatin f the igne Rtatin 5 wwwtjpg edit@tjpg Putting t x u we hae u u w 5 Equatin 5 is the elatiisti Einstein elity additin theem f speial Lentz tansfmatin 3 Relatiisti elity Additin Fmula f Mst Geneal Lentz Tansfmatin Figue 6: elity Additin f Mst Geneal Lentz Tansfmatin Cnside thee inetial fames f efeene S S and S whee the fame S is at est and the fame Sis ming with elity with espet t S fame and the fame S is ming with elity u with espet t S fame Fm the equatin 4 we hae X t t X X t X 6 Diiding numeat and denminat f equatin 6 bytwe get u u u w 7 hee t X w t X u Equatin 7 is the elatiisti elity additin theem f mst geneal Lentz tansfmatin

6 6 Taek Hssain & Shah Alam Impat Fat JCC: NAAS Rating: 4 33 Relatiisti elity Additin Fmula f Mst Geneal Lentz Tansfmatin in the Case f Pependiula elities Figue 7: Pependiula Relatiisti Cmbinatin f elities between Bb and Alie Let us nside the speial ases f pependiula elities between Alie and Bb Alie is ming with elity with espet t missin ntl and Bb is ming with elity with espet t Alie as shwn in Figue 7 The elity f Bb is suppsed t ead by missin ntl ug equatin 7 it an be witten as { } s 9 s 9 hee and 8 Similaly ug equatin 7 the elatiisti mbinatin f elities an be witten as { } s 9 s 9 hee 9 These fmulas ae extemely useful t intdue the nept f elatiisti ally mbining elities

7 Numeial Calulatin f the igne Rtatin 7 wwwtjpg edit@tjpg 34 Relatiisti elity Additin Fmula f Mst Geneal Lentz Tansfmatin in the Case f Paallel elities Figue 8: Paallel Relatiisti Cmbinatin f elities between Bb and Alie The speial ases f paallel elities between Alie elity and Bb elity ae shwn in Figue 8 The elatiisti mbinatin f paallel elities fmula is usually gien in textbks [ 6-9] g equatin 7 the elities an be witten as { } s s hee and g equatin 7 the elatiisti mbinatin f elities an be witten as { } s s hee S we an wite Hee The magnitudes f the tw mbined elities and ae the same and hene thee will be n esulting igne tatin Thmas peessin

8 8 Taek Hssain & Shah Alam 4 DERIATION OF THE FORMLA OF IGNER ROTATION e hae deied the fmula f igne tatin in tw ways Initially we hae alulated hange f igne tatin f Bb with espet t missin ntl Late we hae alulated igne tatin f π µ and e 4 Methd One Calulatin f igne Rtatin Figue 9: A Me Cet Intepetatin f the Relatiisti Cmbinatin f elities In figue 9 Alie is ming with elity with espet t missin ntl while Bb is ming with unifm elity with espet t Alie whih is shwn by the slid line hen missin ntl bsees Bb elity it is suppsed t ead but it bsees a tatin f angle Ω whih is the igne tatin angle The dashed lines indiate Alie has elity as measued by missin ntl and Bb has elity as measued by Alie as shwn hen missin ntl bsees Bb elity it is suppsed t ead But it als bsees a tatin f angle Ω This angle is als knwn as igne tatin Let us nside with the ase f and being pependiula and hene with the elatiisti mbined elities and as defined by 8 and 9 espetiely e an wite But Figue : Paallel and Pependiula Dempse the elities is Illustated in Subfigues a and b Respetiely Impat Fat JCC: NAAS Rating: 4

9 Numeial Calulatin f the igne Rtatin 9 In subfigue a we dempse the elity f Bb as measued by Alie int the mpnents and paallel and pependiula t espetiely The elatiisti ally mbined elity is the elity f Bb as seen by missin ntl In subfigue b we see the S fame whih is bseed t hae elity ntl whih epesents the elatiisti mbinatin f elities and In the elity pependiula t As and ae llinea by the elity f S as measued by missin ntl is by missin S fame Bb is measued t hae Thee is n igne tatin f the illustated in Figue b whee we hae elity as measued in the S fame elatie t missin ntl Theefe we an think f a new situatin as S fame Sine S ming at elity and elatie t missin ntl and Bb ming at sme ae pependiula then we an wite 3 hee Thus as missin ntl sees S ming at elity and the bsee S sees Bb t be ming at the pependiula elity we see that the elity f Bb with espet t missin ntl is gien by 4 Equatin an be witten as 5 And hene 4 bemes wwwtjpg edit@tjpg

10 Taek Hssain & Shah Alam Impat Fat JCC: NAAS Rating: 4 6 Similaly we find 7 These ae the mst elementay fmulae f the mpsitin f geneal elities e use a simila pedue t nside the igne tatin whee we must hae 8 As peiusly desibed f the pependiula ase the igne tatin angle Ω is exatly the angle between and as measued by missin ntl The igne tatin Ω angle then fllws fm the ss-pdut f the ets and g 6 7 and 8 we an wite Ω 9 Ω Ω Ω This an be simplified t

11 Numeial Calulatin f the igne Rtatin wwwtjpg edit@tjpg Ω θ hee θ is the angle between and as measued by Alie Hwee eaanged t gie s θ Hene an be witten as θ Ω θ Ω Ω θ hih sme may egnize as Stapp's elegant fmula [8] Similaly ug the definitin f the dt-pdut t find the igne tatin angle Ω ne finds s Ω s Ω s Ω Ω s Equatin is the igne tatin angle Ω between the Bb and Alie

12 Taek Hssain & Shah Alam 4 Numeial Calulatin Let us nside the Alie elity 4C whih is nstant and the diffeent elity f Bb and espnding igne tatin alues alulated ug equatin in the fllwing table Table Bb elity igne Rtatin The Alie elity is nstant 4C and the hange f Bb elity with espet t hange f igne tatin as shwn in the figue 6 4 igne tatin in 8 6 Figue : The Change f igne Rtatin with Respet t Diffeent Bb elity 43 Methd Tw Calulatin f igne Rtatin The igne tatin an als be alulated thugh the fllwing pess In figue be the elity f mun with espet t lab fame then ading t the elity additin fmula f the mst geneal Lentz tansfmatin [8- ] we an wite Nw if mun mes with elity with espet t lab fame and eletn mes with elity espet t mun then ading t the elity additin fmula f the mst geneal Lentz tansfmatin [3] we an wite 6 7 Bb elity in unit f C Impat Fat JCC: NAAS Rating: 4

13 Numeial Calulatin f the igne Rtatin 3 wwwtjpg edit@tjpg 4 Again fm mun and eletn ading t the elity additin fmula f mst geneal Lentz tansfmatin [9-3] the esultant elity f and an be witten as 5 Finally if eletn mes with elity with espet t pin then the esultant elity [9] eletn with espet t lab fame an be witten as 6 Speifially t illustate igne tatin f elity ets in unit f ae defined as 5 y u x u elity f pin elatie t lab fame; 5 3 y x elity f mun elatie t pin; 4 y w x w elity f eletn elatie t mun The espnding fats ae as fllws u Fm equatin 3 and 4 we hae A say

14 4 Taek Hssain & Shah Alam Fm equatin 5 and 6 we get B e knw that A B AB s Ω A B Ω s AB say hee 737i 9675 j k A A And 6658i j k B B A B 979 Ω s Hene Again let u x u y 5 x y 4 wx wy 5 3 u u w w 6 3 x y x y Be tw sets f elity ets f pin deay hainπ as figue then ug equatins 345 ae µ e and 6 we hae the elity ets f eletn elatie t lab fame and x y & and espetiely g simila pess as peius we hae the igne tatins in this ase App and Ω s Ω s App espetiely 5 CONCLSIONS e hae deied the fmula f igne tatin in tw diffeent ways e hae bseed that thee is n igne tatin f tw linea Lentz bsts e hae alulated the numeial alues f igne tatin f diffeent ases e hae bseed that the igne tatin ineases due t the inease f elity g equatin we an easily alulate the igne tatin with espet t diffeent Bb elity Simulated data in equatin 6 is used f applying the igne tatin fmula in pin deay hain Impat Fat JCC: NAAS Rating: 4

15 Numeial Calulatin f the igne Rtatin 5 REFERENCES LH Thmas Thmas Natue EP igne Ann Math J A Rhdesa M D Semn Ameian Junal f Physis K O Dnnell M isse Eupean Junal f Physis 3 p C Mlle Oxfd niesity Pess Lndn 95 6 A R Baizid and M S Alam Ameian Junal f Mathematis and Statistis A R Baizid and M S Alam Ameian Junal f Mathematis and Statistis A R Baizid and M S Alam Intenatinal Junal f Reipal Symmety and Theetial Physis lume N 4 9 MS Alam P Pakistan Aad f Si MS Alam S Bauk Ameian Institute f Physis Canada R Resnik Intdutin t speial elatiity iley Easten limited 994 J D Jaksn iley New Yk 3 d ed pp R D Sad Benjamin New Yk Chap H Azelie`s Relatiisti Kinematis New Yk pp H Gldstein C Ple and J Safk Classial Mehanis Addisn esley New Yk 3d ed pp G P Fishe Am J Phys J D Jaksn Classial Eletdynamis iley New Yk 3 d ed pp R D Sad Relatiisti Mehanis Benjamin New Yk Chap H Azelie`s Relatiisti Kinematis New Yk pp J D Jaksn Classial Eletdynamis Chiheste: iley pp K Rebilas Am J Phys K Rebilas Phys F Rhlih Ann Phys C Misne K S Thne and J A heele Gaitatin San Fanis CA: Feeman pp K Rebilas Eu J Phys 34 L55 L 3 6 G P Fishe Ameian Junal f Physis I Ritus Physis-spekhi H P Stapp Physial Reiew E P igne Ann Math G P Fishe Am J Phys wwwtjpg edit@tjpg

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