Transformation of Orbital Angular Momentum and Spin Angular Momentum

Transcription

1 Aerian Jornal of Matheatis and Statistis 6, 65: 3-6 DOI: 593/jajs6653 Transforation of Orbital Anglar Moent and Spin Anglar Moent Md Tarek Hossain *, Md Shah Ala Departent of Physis, Shahjalal Uniersity of Siene and Tehnology, Sylhet, Bangladesh Abstrat The Lorentz transforation an be eplained in to ays ie the speial Lorentz transforation and ost general Lorentz transforation The length ontration, tie dilation and eloity addition forlas for the speial and ost general Lorentz transforations are learly eplained The ariation of ass de to the relatiisti eloity has also eplained Using aboe entioned paraeters e hae deried the forlas of the transforation of orbital anglar oent and spin anglar oent for speial Lorentz transforation We hae allated the ales of orbital anglar oent for rest and oing obserers The ales of orbital anglar oent hae plotted for the different eloities of the oing obserers Keyords Speial Lorentz transforation, Most general Lorentz transforation, Anglar oent, Length ontration, Veloity addition forla Introdtion Lorentz Ttransforation The transforation hih relates the obserations of position and tie ade by the to obserers in to different inertial fraes is knon as Lorentz transforation Speial Lorentz Transforation respetiely The relation beteen the oordinates of S and S, hih is alled speial Lorentz Transforation, an be ritten as [, 3-5 ' t / yy z z t ' t / / 3 Most General Lorentz Transforation Figre The frae S is at rest and the frae S is oing ith respet to S ith nifor eloity along -ais Let s onsider to inertial frae of referenes S and S here the frae S is at rest and the frae S is oing along -ais ith eloity ith respet to S frae The spae and tie oordinates of S and S are,y,z,t and,y,z,t * Corresponding athor: tarekphysst@gailo Md Tarek Hossain Pblished online at Copyright 6 Sientifi & Aadei Pblishing All Rights Resered Figre The frae S is at rest and the frae S is oing ith respet to S ith nifor eloity along any arbitrary diretion When the eloity of S ith respet to the S is not along -ais ie the eloity has three oponents, y and z then the relation beteen the oordinates of S and S,

2 Md Tarek Hossain et al: Transforation of Orbital Anglar Moent and Spin Anglar Moent hih is alled ost general Lorentz transforation, an be ritten as [-5 X X [ X / / / } t / / t / / t X / Here X and X is the spae part of the S and S frae respetiely Conseqene of Lorentz Transforation The length of any objet in a oing frae ill appear shortened or ontrated in the diretion of otion The aont of ontration an be allated fro the Lorentz transforation The length is ai in the frae in hih the objet is at rest If the length L is easred in the oing referene frae and the length L is easred in the rest referene then sing speial Lorentz Transforation The relation beteen L and L an be ritten as [ L L 3 This is the forla of length ontration for Speial Lorentz transforation Length Contration of Speial Lorentz Transforation Figre 3 A frae is at rest and another frae is oing along X-ais ith respet to the rest frae A rod of length L is at rest ith respet to the oing frae Figre A frae is at rest and another frae is oing along arbitrary diretion ith respet to the rest frae A rod of length L is at rest ith respet to the oing frae

3 Aerian Jornal of Matheatis and Statistis 6, 65: Length Contration of Most General Lorentz Transforation Let, L Using Eq, e an rite [ t [ t Where, and Therefore, osθ osθ} L L [ here, L This is the forla of length ontration for the ost general Lorentz transforation If is along the -ais, then θ We get fro Eq L [ / or, L L This is the sae as Eq 3 3 Tie Dilation of Speial Lorentz Transforation A lok in oing frae ill be seen to be rnning slo or dilated aording to Lorentz transforation The tie ill alays be shortest as easred in the rest frae The tie easred in hih the lok is at rest alled the proper tie If the tie interal T t t is easred in the oing referene frae, than T t t an be allated sing the speial Lorentz transforation Eq T t t [t / t / The tie easreents ade in the oing frae at the sae loation, so the epression redes to T t t Or, T T This is the forla of tie dilation for the Speial Lorentz transforation Tie Dilation of Most General Lorentz Transforation T t t an be allated sing the ost general Lorentz transforation T t t t t or, T T 5 This is the forla of tie dilation for the ost general Lorentz transforation hih is the sae as that of the Speial Lorentz transforation 5 Relatiisti Veloity Addition Forla for Speial Lorentz Transforation Consider to systes S and S, S be the grond frae and S be the frae of the train, hose speed relatie to the grond is The passenger s speed in the S is ' The passenger s speed relatie to the grond an be ritten as [ 6 In nit of, eqation 6 an be ritten as 7 Eqation 7 is the relatiisti or Einstein eloity addition theore for speial Lorentz transforation Figre 5 A frae is at rest and another frae is oing along X-ais ith respet to the rest frae A lok is at rest ith respet to the oing frae

4 6 Md Tarek Hossain et al: Transforation of Orbital Anglar Moent and Spin Anglar Moent Figre 6 A frae is at rest and another frae is oing along arbitrary diretion ith respet to the rest frae A lok is at rest ith respet to the oing frae 6 Relatiisti Veloity Addition Forla for Most General Lorentz Transforation In this ase the relatiisti eloity addition theore an be ritten as [6 [ } 8 Where t Ptting, eqation 8 takes the for [ } 9 Eqation 9 is the relatiisti eloity addition theore for ost general Lorentz transforation 3 Variation of Mass 3 Variation of Mass of Speial Lorentz Transforation Let there be to systes S and S, later is oing ith eloity relatie to forer Let there be to partiles in syste S traeling ith eqal and opposite eloities, ie and - along e ais Let these to partiles ipinge and after ipat oalese into one partile Aording to the priniple of onseration of oent, this oalesed partile is at rest ith respet to the oing frae Let & are the asses of the to partiles and & are the eloities of the partiles as seen fro S frae No for the partile traeling ith eloity, e hae, And for the partile traeling ith eloity -, Sine after ipat the oalesed partile in S traeling ith eloity relatie to the syste S, as the partile is at rest in syste S Aording to the priniple of onseration of oent as ieed fro S, Or,

5 Aerian Jornal of Matheatis and Statistis 6, 65: No ptting the ales of and in the aboe eqation by onsidering, 3 No sqaring & e get, And Sbtrating these, e get, Or, } } Or, Or, } ± } [As <, taking - e sign Ths Sbstitting this ale in eqation 3 e get,

6 8 Md Tarek Hossain et al: Transforation of Orbital Anglar Moent and Spin Anglar Moent No if the partile of ass is at rest, then orresponding eloity We an rite ith the sal notations and, This is the forla for the ariation of ass ith eloity in speial Lorentz Transforation Ths fro the aboe forla it is lear that the ass of a body inreases ith inrease of eloity 3 Variation of Mass of Most General Lorentz Transforation No e attept to find ot the ariation of ass ith eloity in the ase of ost general Lorentz transforation Consider to inertial fraes S and S The later is oing ith eloity in the arbitrary diretion relatie to the forer Here has three oponents Let s no onsider to partiles of asses and obsered fro the frae S The partiles oing ith eqal and opposite eloity, ie and obsered fro the frae S and they are attepted to ipinge The asses of the partiles are eqal easred fro the frae S After ipat they oalesed into one partile and beoe rest relatie to the frae S Let and be the eloity of the partiles ieed fro the frae S It shold be noted that after ipat the oalesed partile oing ith the frae eloity S ith respet to the rest frae S Let s onsider ; Aording to the eloity addition rle of ost general Lorentz transforation eqation 3, the eloity of the partile of ass obsered fro the frae S is, [ In fat is the eloity of the S frae relatie to the S frae In ters of eqation an be ritten as,

7 Aerian Jornal of Matheatis and Statistis 6, 65: [ 5 No the eloity of the partile of obsered fro the frae S is, [ 6 Let and be the asses of the partiles oing ith eloity and No aording to the priniple of onseration of oent as ieed fro S is, Or, 7 No ptting the ale of and fro eqation 5 & 6 in eqation 7 e get, [Sine AB BA 8 Sqaring 5 e hae, } } 9

8 Md Tarek Hossain et al: Transforation of Orbital Anglar Moent and Spin Anglar Moent No Sqaring 6 e get, } } Sbtrating fro 9 e get, [ Or, [ [ Or, ± [Sine <, e sign ignored Ptting the ale of fro, eqation 8 beoes,

9 Aerian Jornal of Matheatis and Statistis 6, 65: 3-6 If the partile of ass is at rest, and sing the sal notations and, the aboe eqation beoes, This is the forla for the ariation of ass ith eloity in ost general Lorentz Transforation Anglar Moent Let s onsider a partile of ass, hih is rotating abot a irlar path of radis r along antilokise diretion as shon in the figre 7 At a ertain tie t, the linear oent of the partile is, p The obserer O is at rest ith respet to the laboratory and the obserer O is oing ith nifor eloity along -ais ith respet to laboratory [5-6 Aording to the definition, the anglar oent of the oing partile obsered by the obserer O is L r P We ant to find the anglar oent of the oing partile obsered by the obserer O Figre 7 Rotating partile obsering by to obserers Anglar Moent Obsered by a Rest Obserer For the figre-7, the anglar oent an be ritten as, L r p r p sin φ Or, here φ is the angle beteen r and p L r p sin 9 At the point A, φ 9 Again, p Ptting the ale of p in eqation, than e an rite L r L r Where, This is the anglar oent forla obsered by a rest obserer Anglar Moent Obsered by a Moing Obserer For figre-7 the eloity of the oing partile ith respet to the oing obserer the relatiisti eloity addition theore for ost general Lorentz transforation eqation 9 an be ritten as [ } Where, [ }

10 Md Tarek Hossain et al: Transforation of Orbital Anglar Moent and Spin Anglar Moent Again, p } [ Here, } [ } [ Let A, B, And C A A A A A } } The anglar oent of the oing partile obsered by the oing obserer an be ritten as, p r L φ sin p r L r } } sin φ r o o } } sin φ Where,

11 Aerian Jornal of Matheatis and Statistis 6, 65: } r o o } sin φ 3 Let the obserer O is oing ith nifor eloity along -ais ith respet to laboratory The oing obserer O obseres the angle in rotating partile be φ sinφ tan φ osφ o sin 9 tan φ os9 tan φ tan φ o tan φ φ tan No ptting the ale of φ in eqation 3, L r o o }, } sin tan This is the anglar oent eqation obsered by a oing obserer 3 Nerial Callation Let s onsider the partile is ball So rest ass of the ball 79 Kg and radis a59 - We hae allated the anglar oent of the ball for different eloities and presented the ales in the folloing table Vale of in nit of L in kg s The obserer O is at rest ith respet to the laboratory and the obserer O is oing ith nifor eloity along -ais ith respet to laboratory The hange of anglar oent obsered by a oing obserer as shon in the figre-8 Using eqation the hange of the anglar oent obsered by a oing obserer and rest obserer data is shon belo L in kg s - L in kg s The partile of ass, hih is rotating abot a irlar path of radis r along antilokise diretion as shon in the figre 7 The partile eloity is The hange of the anglar oent obsered by a oing obserer and rest obserer as shon in the figre-9

12 Md Tarek Hossain et al: Transforation of Orbital Anglar Moent and Spin Anglar Moent L ' in kg s in nit of Figre 8 The hange of anglar oent obsered by a oing obserer 5 L ' in kg s L in kg s - Figre 9 The hange of the anglar oent obsered by a oing obserer and rest obserer 5 Spin Anglar Moent A solid spherial ball of ass and radis a is oing abot a irlar path along antilokise diretion and it also rotating abot its on ais At a ertain tie t, the linear oent of the ball is, P The obserer O is at rest ith respet to the laboratory and the obserer O is oing ith nifor eloity along -ais ith respet to laboratory The orbital anglar oent of the oing ball obsered by the obserer O an be ritten as L r P We ant to find the spin anglar oent of the oing ball obsered by both the obserers 5 Spin Anglar Moent Obsered by a Rest Obserer Spin is the anglar oent hih is assoiated ith a rotating objet like a spinning golf ball The spin anglar oent of sh a rigid body an be allated by integrating oer the ontribtions to the spin anglar oent de to the otion of eah of the infinitesial asses aking p the body The ell knon reslt is that the

13 Aerian Jornal of Matheatis and Statistis 6, 65: spin anglar oent S an be ritten as [9 S Iω 5 Where I is the oent of inertia and ω is its anglar eloity of the ball The oent of inertia is deterined by the distribtion of ass in the rotating body relatie to the ais of rotation The oent of inertia of the ball an be ritten as [9 I a 6 5 The anglar eloity of the spinning otion of the ball an be ritten as, ω π 7 T Where T is the tie needed for one reoltion abot on ais of the ball obsered by rest obserer Ptting the ale of I and ω fro eqation 6 and 7 into eqation 5 e get, π S a 5 T πa S 5T 8 This is the forla of Spin anglar oent of the ball obsered by the rest obserer O 5 Spin Anglar Moent Obsered by a Moing Obserer The Spin anglar oent obsered by the oing obserer an be ritten as, S I 9 Where I is the oent of inertia and ω is its anglar eloity of the ball obsered by the oing obserer The oent of inertia of the ball obsered by the oing obserer an be ritten as, I a 3 5 The anglar eloity of the spinning otion of the ball obsered by the oing obserer an be ritten as π ω 3 T Where T' is the tie needed for one reoltion abot on ais obsered by oing obserer Ptting the ale of I and ω fro eqation 3 and 3 into eqation 9 e get, S π a 5 T π a S 3 5T ω Using eqation 3 e an rite, a a Where, Using eqation 5 e an rite, T T Where, Using eqation 8 e an rite, Where, So, eqation 3 an be ritten as, a π S 5T a π S 33 5T This is the forla of Spin anglar oent of the oing ball obsered by the oing obserer O 6 Conlsions We hae deried the forla for the transforation of orbital anglar oent and spin anglar oent for speial Lorentz transforation We hae allated the ale of orbital anglar oent obsered fro the rest obserer and oing obserer The graph of the orbital anglar oent erss eloity of the oing obserer has plotted We hae obsered that the orbital anglar oent inreases de to the inrease of the eloity of the obserer The orbital anglar oent obsered fro the oing obserer and rest obserer has also plotted We hae also obsered that the orbital anglar oent obsered fro the oing obserer is inreased de to the inrease of anglar oent obsered fro the rest obserer We hae deried the forla of spin anglar oent obsered fro the rest obserer and oing obserer The eqation 33 learly shos that the spin anglar oent dereases de to the inrease of the eloity of the obserer REFERENCES [ R Resnik, Introdtion to speial relatiity, Wiley Eastern liited, 99 [ C Moller, The Theory of Relatiity, Oford Uniersity press,

14 6 Md Tarek Hossain et al: Transforation of Orbital Anglar Moent and Spin Anglar Moent London, 97 [3 A R Baizid and M S Ala, Appliations of different Types of Lorenz Transforations Aerian Jornal of Matheatis and Statistis, 5: 53-63, [ A R Baizid and M S Ala, Properties of different Types of Lorenz Transforations Aerian Jornal of Matheatis and Statistis, 33: 5-3, 3 [5 A R Baizid and M S Ala, Reiproal Property of different Types of Lorenz Transforations International Jornal of Reiproal Syetry and Theoretial Physis, Vole, No, [6 MS Ala, Stdy of Mied Nber, Pro Pakistan Aad of Si, 37: 9-, [7 TA Nieinen, AB Stilgoe, NR Hekenberg and H Rbinsztein-Dnlop Anglar oent of a strongly fosed, H 8 [8 Barnett, SM and Allen L Orbital anglar oent and no paraial light beas, Opt Co,, , 99 [9 S Olszeski, Eletron Spin and Proton Spin in the Hydrogen and Hydrogen-Like Atoi Systes, Jornal of Modern Physis, 5, 3-,