The Hooke-Newton transmutation system of magnetic force

Transcription

1 Jounal of Physs Communatons PAPER OPEN ACCESS The HookeNewton tansmutaton system of magnet foe To te ths atle: DeHone Ln 08 J. Phys. Commun Related ontent Quantum Mehans: Spheally symmet potentals M Saleem A massndependent onfomal quantum loak DeHone Ln Beyond the unfed model S Fauendof Vew the atle onlne fo updates and enhanements. Ths ontent was downloaded fom IP addess on 7/09/08 at 3:00

2 J. Phys. Commun. (08) OPEN ACCESS RECEIVED 5 Januay 08 REVISED 5 May 08 ACCEPTED FOR PUBLICATION 0 June 08 PAPER The HookeNewton tansmutaton system of magnet foe DeHone Ln Depatment of Physs, Natonal Sun Yatsen Unvesty, Kaohsung, Tawan Emal: dhln@mal.nsysu.edu.tw Keywods: onfomal tansfomaton mehans, dualty of foe, magnet HookeNewton tansmutaton PUBLISHED 9 June 08 Ognal ontent fom ths wok may be used unde the tems of the Ceatve Commons Attbuton 3.0 lene. Any futhe dstbuton of ths wok must mantan attbuton to the autho(s) and the ttle of the wok, ounal taton and DOI. Abstat Reently, t was epoted that the HookeNewton tansmutaton of magnet foe an be geneated by a onfomal mappng n a unfom magnet feld. We pefom the lassal analyss of the tansmutaton system n ths pape. Fst, the aton vaables ae alulated and the enegy s expessed n tems of the vaables. The quantum spetum of the system s manfested by the ondton of the angula vaables fo the losed taetoes. Seond, the taetoy equatons ae pesented fo the hage n the tansmuted Coulomb feld whose haatests, attatve o epulsve, ae detemned by the sgnatues of both the angula momentum and the hage. It s shown that, gven the eleased enegy E and angula momentum p of the haged patle, the types of taetoes an be lassfed by the tal enegy assoated wth the magnet feld stength E = w p C 4. The featue makes the system only tap the patles wth enegy E<E C and the defnte sgnatues of angula momentum and hage.. Intoduton It was ponted out by Issa Newton that thee exsts dualty between Hooke s lnea foe law and the nvese squae foe law (Co. III, Pop, VII, []). Newton evealed ths dualty by showng a tansmutaton between these two knds of foes when gven a planet obt, and, amazngly, poved t only wth the sklls of elementay geomety. In moden temnology, the tansmutaton s establshed by a quad onfomal mappng (see, e.g., appendx I n [, 3], and seton 6 n [4]). Reently, t was shown that thee exsts a magnet foe veson of the HookeNewton (HN) tansmutaton fo a hagedquantumpatle n a unfom magnetfeld [5]. Ths s moe o less unexpeted sne the magnet foe s qute dffeent fom the ental foe feld, e.g., dong no wok, and unlke the felds of the Newtonan gavtatonal attaton and of eletostat nteaton geneated by a massve obet and a hage, whh ae the soues of the nvese squae foe felds, thee s no magnet monopole dsoveed so fa. Although the magnet veson of the HN tansmutaton s due to a onfomal mage, t s possble to ealze the system by entepetng the onfomal stutue as a physal effet by tansfomaton desgn means [6 8]. In efeene [5], the quantum mehans of the magnet tansmutaton system have been dsussed. The pupose of ths atle s to ay out the lassal analyss fo the system. The pape s aanged as follows: n seton, the physal ealzaton of the mehanal onfomal mage s fomulated. In seton 3, the HN tansmutaton of a hage n the unfom magnet feld s nvestgated by way of the aton and angula vaables. The enegy of the losed taetoy s expessed n tems of the aton vaables. The quantum spetum of the system s then dentfed afte the Boh s quantzaton ules have been adopted fo the aton vaables. It follows fom the aton epesentaton of enegy that the angula vaables ae alulated, fom whh the elaton between the allowed quantum bound states and the ondtons of the angula vaables fo the losed taetoes ae establshed. Seton 4 s used to obtan the taetoy equatons of a hage n the tansmutaton system. It s shown that the types of taetoes an be ontolled by the stength of the magnet feld. Fnally, the onluson s povded n seton The Autho(s). Publshed by IOP Publshng Ltd

3 J. Phys. Commun. (08) DH Ln. Fomnvaant HamltonJaob equaton and mehanal onfomal mappng The pupose of ths seton s to show that a fomnvaant HamltonJaob (HJ) equaton an be geneated by a knd of onfomal mappng, and one an endow the onfomal fato n the equaton wth the meanng of a potental feld suh that t s possble to ay out the tansfomed system by physal means. The evoluton of a hage movng n a D spae s govened by the soluton of the HJ equaton å = () M g S S E, q q whee S=S(q) s the edued aton (p. 49, [9]). It may be egaded as a funton of the oodnates. The patal devatves of the aton wth espet to the oodnates ae equal to the oespondng genealzed momenta: S = p. () q Consde the fomnvaant HJ equaton due to a tansfomaton å = () M g S S E, 3 q q whee the new oodnate vaables q = q ( q) ae only the funton of q, and the met oeffents ḡ ae defned by g = g( q) obtaned by eplang the vaable q wth q fo the nvese of the ovaant met omponents g (q) alulated by the standad defnton Equaton (3) an be expessed as Sne q s only the funton of q, Equaton (5) edues to = x x + y y g() q () q q q q. 4 å g q M q k,, q q å M g dq dq S q q q Hee, let us ntodue the met oeffents fo the new oodnate system { q } k S = E. () 5 qk dq = d, () 6 dq S S = E. () 7 q q = x x + y y G( q) q q q q to fgue out the tansfomaton leadng to the nvaant HJ equaton. It s easy to fnd the elaton between the met oeffents g (q) fo the systems { q } and G( q) fo { q } Ths makes (7) tun nto M å G () 8 = dq g. () 9 dq ( Gg ) g S S = E ( 0) q q fo g =/g. Fom ths, one gets to the ondtons of onfomal mappng that geneate the nvaant equaton (3) G g G G =, and G g =, g g whee the detemnants G = G and g = g. The smultaneous equatons detemne the defnte fom of the tansfomaton. Now equaton (0) beomes M G å g S S = E. g q q

4 J. Phys. Commun. (08) DH Ln The nvaant squaed dstane between two neghbong ponts of the tansfomed spae s depted by dl g = G å g dqdq. ( 3) One nteestng outome whh ould emege fom the HJ equaton s that the moton of the patle n the onfomal spae an be obtaned wth a physal effet by movng the fato ḡ G to the ght hand sde of the equaton, whh yelds å = ( ) M g S S E U q, 4 q q wth the effetve potental U() q = g G E, 5 whee E s the eleased enegy of the patle. Ths entepetaton tuns the geomet effet of the tansfomaton nto a eal mehanal effet. It allows us to eveal and onstut novel systems though onfomal mappng. Equaton (4) s effetve fo hghe dmensonal spae as long as the onfomal ondtons G g = G g ae solvable. In the omng setons, the fomulaton s appled to nvestgate the HN tansmutaton n the unfom magnet feld. 3. Aton vaables and the enegy of the HookeNewton tansmutaton system n the unfom magnet feld The aton vaables of the tansmutaton system ae alulated n ths seton so as to obtan the system enegy. To geneate the tansmutaton, let us evaluate the tansfomaton funtons wth espet to the ylndal oodnate system (ρ, ). Aodng to the defntons of ḡ and G n (9), we have ( g ) = dag(, ), and ( G) = dag ( ),, 6 ( ) whee = d d, and = d d. Solvng the equatons n () gves the tansfomaton funtons n =, and = n, ( 7) ln whee the paamete ν s a eal numbe, and λ has the unts of length used to balane the unts. Fo ou onsdeaton of the HookeNewton tansmutaton, the paamete ν=, we have =, and =. ( 8) l Substtutng the tansfomaton nto the omponents n (6) and a shot alulaton gve the onfomal fato So we obtan the nvaant dstane of tansfomed spae g G = = l. ( 9) dl = + l ( d d ) ( 0 ) fom (3), and the tansfomed HJ equaton wth + = M p p E U, 4 U = E, l fom (4). It tuns out that the geomet effet s now aheved physally by the potental U. It s not dffult to geneate the potental feld sne t oesponds to a epulsve lnea foe whh an be podued by the elet feld E=F/q= U/q. Now let us tun ou attenton to onsde the nfluene of a unfom magnet feld on the onfomal mage of a hage s moton. Fo the magnet feld deted along the Zaxs, B = Beˆ z, the HJ equaton s obtaned by eplang p /ρ wth the mnmal ouplng ( p qa) sne the magnet feld s alulated though B = A, and one an hoose the veto potental A ( B )ˆ e. The eplaement 3 =

5 J. Phys. Commun. (08) DH Ln guaantees the onsevaton of the total momentum of the hage and feld. The moton of the hage q s now govened by the HJ equaton + = M p p qa E U, 3 wth A =Bρ /. The angula vaable s obvously a yl oodnate. So the genealzed momentum = S p = onstant. ( 4) Ths s the angula momentum of the system. The oespondng aton vaable to the momentum s It follows fom (3) that the momentum p ρ an be evaluated by I = pd = p. ( 5) p p = M( E U) ( p qa) w = + w p M p M 8ME l 4, ( 6) whee the yloton fequeny ω =qb/m has been ntodued. Usng the new oodnate vaable = l, the adal aton vaable s alulated fom the ntegaton I = p d p ( p ) M( l ) w ( p ) M ( l ) ( w ) = + + ME d. ( 7) p We note that ths s an aton vaable fo an nvese squae foe geneated by the Coulomb feld ( lw p 8). Theefoe, the mappng n (8) does ndeed geneate the HN tansmutaton of the magnet foe n the unfom magnet feld. It s emakable to note that the foe whh s attatve o epulsve s detemned by both the sgnatues of hage q n ω and the angula momentum p. Gven a knd of haged patle n the Coulomb feld, a ght o left tun wll detemne the patles wth losed o unlosed taetoes. To pefom the ntegaton of I ρ, let us put a = ( p l) Mw p ( l) ; b = ; ( 8) w 8ME Mw 8ME l l M then l w = M 8ME max I p l ò a + b d, ( 9) 4 mn whee = b + b a; = b b a. ( 30) max Sne 0, t mples b0. So the sgnatue of q has to be the same as that of the angula momentum p, and enegy satsfes mn M E < ( lw ). ( 3) 3 The ondtons of the sgnatue and the nequalty atually stand fo the attatve foe and losed taetoy ondtons as we shall see late. Wth the ntegal fomula (p. 0, [0]), ò a + bx xdx x + x x = + x x a b b ( a b ) a sn b sn. ( 3) x b a b a 4

6 J. Phys. Commun. (08) DH Ln The adal aton vaable s found to be I = I Mw Mw 8ME l. ( 33) Solve the equaton fo the enegy. It s expessed n tems of aton vaables by M ( w ) ( l ) ( I ) E =. ( 34) ( I + I ) Aton vaables ae adabat nvaant. Let us hoose the Boh s quantzaton ondtons fo the atons I = ( n + ), n = 0,,,, ( 35) and I = m, m =,,. ( 36) Hee we emembe that I = p ¹ 0. Othewse, the attatve foe no longe exsts. The enegy tuns nto the dsete spetum M( w ) ( l ) ( m ) E =. ( 37) ( n + m + ) Ths s exatly the quantum spetum of the tansmutaton system (See equaton (46) n [5]). Equatons (34) and (37) show that the exstene of ω, and thus the magnet feld, s the ual pont of the losed taetoes and bound states. Wth the epesentaton n (34 ), the angula vaables an be alulated, yeldng whee C s the pefato n fomula (34), and w = E = ( I ) C I ( I + I ) 3, ( 38) w = E I II =C ( I + I ) 3. ( 39) The ato s w w = I I. 40 When the quantum ules n (35) and (36) ae nseted, the ato beomes w w = m m = n + ( n + ). 4 It s a atonal numbe, and the ondton of the losed taetoes. Ths exhbts the oespondng elaton between the bound states n quantum mehans and the losed taetoes n lassal mehans. Befoe fnshng the dsusson of the seton, let us detemne the onstant λ. The best hoe s l = 8, ( 4) M w p the haatest length of the system. The expesson s patulaly useful when we would lke to ompae the nteaton stengths of the magnet HN system and the elet Coulomb system, as we shall see n the notes of the onluson. Futhemoe, the hoe also makes the wave funton nomalzed to unty [5]. To obtan the expesson, t s noted that the haatest length of the hydogen atom system wth the elet Coulomb nteaton q e s the Boh adus ab = Mq e wth q e beng the hage of an eleton. We now have the nteaton of the tansmutaton Coulomb feld ( lw p 8), and the haatest length of the system In [5], the quantum numbe m n λ was fatoed out and put asde evey a n the wave funton, see equaton (49) theen. A bette hoe s absobng the fato m nto the dmensonless vaable a = a beomng l = l wth l = 8 M w ( m ) as pesented n ths pape sne the Coulomb nteaton of the tansmutaton system depends on the quantum numbe m. 5

7 J. Phys. Commun. (08) DH Ln a T = M( lw p 8 ). 43 Hee ÿs ust a onstant quantty, wth no elevane to the quantzaton. The onstant s then found by the ato l = = a 8 ( lm w p ) T 8 ( M w p ), ( 44) 8 ( M w p ) whh ndates that l = a = 8 T ( M wp ). Ths ompletes the poof of (4). Let us estmate the ode of λ when the magnet feld B= Tesla. Aodng to quantum mehans, the angula momentum s always quantzed by p = m, m =,,, so that l = 8. m M w ( 45 ) The substtuton of the yloton fequeny w = qb M».6 0 Hz fo an eleton unde the feld, and ÿ= Mev se gves A weake magnet feld wll nease the length. l» 8. 0 mm. ( 46) m 4. Taetoy equatons fo the HookeNewton tansmutaton system of magnet foe The taetoy s defned by the equaton n the HJ fomulaton of mehans (p. 49, [9]) + S = onstant. ( 47) p Equaton (3) shows that the dffeental fom of the HJ equaton s S p + qa = ( E U). ( 48) M Sne S=S(ρ ) s a funton of ρ, and ds d = ( l) ds d, solvng the above equaton fo the fato ds d gves l w ds M ( l ) Mw ( p ) ( p ) = ME +. ( 49) d 4 The edued aton s then gven by l w M ( l ) Mw ( p ) ( p ) S = ò ME + d. ( 50) 4 Usng the epesentaton, we get S ( p ) ( lmw 4) = ò d + ò d ( 5) p Y() Y() by ntodung the abbevaton lmw ( l ) Mw ( p ) ( p ) Y () = ME + 4 fo shot. To go futhe, let us defne ( 5) lw M w º = E E E, ( 53) 4 4 p whee we have used (4) to get to the seond equalty. The aton (50) may be fo the attatve o epulsve foe dependng on the ouplng onstant (λω p )/8>0o<0. We shall fst onsde the attatve ase. 4.. Attatve foe Equaton (50) shows an nvese squae foe geneated by the Coulomb feld ( lw p 8). Attaton s aused by the ouplng (λω p )/8>0. The alulaton of S p would be dvded nto thee egons:. Ē < 0,. 6

8 J. Phys. Commun. (08) DH Ln Ē = 0, and 3. Ē > 0, aodng to the stength of the magnet feld when the eleased enegy and angula momentum of the hage ae gven Taetoy equaton fo Ē < 0. In ths ase the stength of the magnet feld s stong enough suh that 4 w p > E. ( 54) The fst ntegal n (5) an be manpulated to yeld ò ( p ) d Y() = ò p d, ME + ( Ma) p = ò p d, ME + ( Ma p ) ( p Ma p ) [ ME + ( Ma p )] ( 55) whee the abbevatons, have been used. Put ò p x = p ( lw p) = a =, and, 56 8 ( p Ma p ) M E + ( Ma p ). ( 57) One gets the analyt esult of the ntegal ò dx = os x. ( 58) x The seond ntegal n (5) ò d = ò d Y() ME + ( Ma) p = ME [ a ( )] E p ( M ) E d. ( 59) [ a ( )] E {[ a ( E )] p ( M )} E Put y = a ( ) E. ( 60) [ a ( )] E p ( M ) E The ntegal beomes dy ò ME y = os y. ( 6) ME The taetoy equaton fo Ē < 0 s then gven by l w M x y = os os onstant. 6 4 ME It s onvenent to ntodue the paametes The ange of e an only be Ep p e =, and L =. ( 63) Ma Ma 0 e < ( 64) fo the ase Ē < 0. It may be efeed to as the quaseentty. Wthout loss of genealty, the onstant n (6) s hosen as zeo, and the taetoes must satsfy the equaton 7

9 J. Phys. Commun. (08) DH Ln Fgue. Taetoes of the quaseentty e = 3 fo a hage n the attatve foe feld of the magnet HookeNewton tansmutaton system. The paamete L= was hosen n all ou smulatons fo the atle. = L ( e) L os os, ( 65 ) e e e whee the mnus (plus) sgn s fo the negatve (postve) haged patle. To eflet the yl haatest of fo a geneal value of e, let us take the osne opeaton wth espet to both sdes of (65 ), yeldng os = x os os x x sn os x, ( 66) e e whee L ( e) L x=, x =. ( 67 ) e e The top sgn s fo the postve hage. It s possble to obtan an altenatve epesentaton of the taetoy equaton wthout esotng to the nvese funton. In ode to pove the statement, we need the known equaltes (p. 58, [0]) os ( b os ( x)) = [( x + x ) b + ( x x ) b], ( 68) os ( xy x y ), fo x + y 0, osx + osy = ( 69) p os ( xy x y ), fo x + y < 0, and = os ( xy + x y ), fo x y, osx osy ( 70) os ( xy + x y ), fo x < y. It s easy to vefy that the taetoy equaton an be expessed n tems of os = [ xz x z ], ( 7) whee wth b = z = [( x + ) b + ( ) b x x x ], ( 7) e. Fo the speal ase β=, the equaton edues to os = [ x ( x ) x ( x ) ]. ( 73) The top sgn s fo the postve hage. Fgue shows the pattens of the taetoes wth the value β=,.e. e = 3, whee L= was hosen. The left patten s fo the negatve hage oespondng to the mnus sgn n (66) o the plus sgn n (7). The pattens an be obtaned ethe though (66) o by (7). Fgue shows the taetoes fo seveal dffeent values of e. They exhbt smple desgns when the hage s movng n the stonge magnet felds, oespondng to the smalle values of e. Eah of them only spans a smalle egon of the taetoy plane. When the hage s movng n the weake magnet felds, the lage values of e, the taetoes 8

10 J. Phys. Commun. (08) DH Ln Fgue. Taetoes fo seveal dffeent values of e<. A smalle value of e oesponds to a stonge magnet feld. The pattens show that the taetoes of the hage n a weake feld ae moe omplated than the taetoes n a stonge magnet feld. Two banhes of taetoes ae allowed n the stonge feld. get omplated. Ths s easonable sne the tappng foe of the unfom magnet feld F ω ρ,s popotonal to the stength of the magnet feld n ω. The quanttatve poof of the statement that the stength of the magnet feld ontols the value of e fo a hage wth gven eleased enegy and angula momentum an be poved as follows: fom (56) and (4). So the fato n e a lw p = = w 8 4M p 74 Ep a = 8 E p w = E w = E M 4p w 4p ( 75) fom (53) and (54), whee p = p. The substtuton of the fato fo e n (63) then gves = E e w <. ( 76) 4p Fgue 3 shows the taetoes fo e=0.99 and The taetoes ae always fnte, and losed. Some ponts n fgue 3 whee the taetoy s not smooth should be due to the fat that the seond tem wth os ( x) e n (66) osllates vey apdly when e. The losed haatest of the taetoy fo e an also be nspeted by detly takng the lmt n equaton (66). Unde the opeaton, the seond nvese osne funton n (66) has the multvalued lmt p ( e) L os n. ( 77) e Ths makes the asymptot epesentaton of the taetoy equaton beome os np e L =. ( 78 ) e It s easy to see that the taetoes of the equaton ae ellpt wth two dffeent tltng angles fo a gven n. 9

11 J. Phys. Commun. (08) DH Ln Fgue 3. Taetoes fo e=0.99, and e= The pattens show that, gven the stength of the magnet feld, the taetoes ae always losed as long as the enegy E and angula momentum p of the haged patle satsfes E w p 4. < 4... Taetoy equaton fo Ē = 0. In ths ase the stength of the magnet feld satsfes the ondton 4 w p = E. ( 79) The ondton s equvalent to e=. As alulated above, the fst ntegal n (5) an be pefomed, and gves The seond ntegal fo Ē = p ò d a ( M ) p p Ma p = = L os a os. ( 80) M p 0 beomes lmw 4 ò Ma p d L =. ( 8) L 0

12 J. Phys. Commun. (08) DH Ln Fgue 4. Taetoes of the tal value e= fo the magnet tansmutaton system. The fst (seond) ow s the taetoy fo a negatve (postve) haged patle. It s a unfom spal. The enegy and angula momentum of the hage n ths ase satsfes the tal ondton E w p 4. = Aodngly, the equaton of the taetoy s = L L os. ( 8) L The plus s fo the negatve haged patle. The pattens n the fst ow of fgue 4 exhbt the taetoy, whee L= was hosen. It s a knd of unfom spal. The pattens n the seond ow ae fo the postve haged patle Taetoy equaton fo Ē > 0. The stength of the magnet feld satsfes the ondton The equvalent ondton to ths s e>. The fst ntegal n (5) n ths ase s ò 4 w p < E. ( 83) ( p ) d Y() p = ò d, ( M E + a ) p = ò dx = os x, ( 84) x

13 J. Phys. Commun. (08) DH Ln whee We have the seond ntegal whee ( p Ma p ) x =. ( 85) ME + ( Ma p ) lmw ò d 4 Y() lmw = ò d 4 ( ME) + ( Ma) p lmw = 4 y = ME osh y, 86 + a ( E). ( 87) [ a ( )] E p ( ME) + The soluton of the nvese hypebol osne has taken the banh osh y > 0 fo y>0. Intodung the paametes the taetoy equaton s found to be = os L Ep p e = +, and L =, ( 88) Ma Ma ( e ) L + e e osh. 89 e As above, the plus s fo the negatve hage. The pattens n the fst (seond) ow of fgue 5 show two taetoes of the negatve (postve) haged patle podued fom the equaton. They ae a nonunfom spal. The dvestes of the taetoy ae dstngushed by the tal enegy E = w p C 4. Let us estmate ts ode fo an eleton. Consde the angula momentum p = m, m =,,. E C w w = =. 4 p 4 m Fo the magnet feld B= Tesla, ÿω 0 4 ev. We have E 0 4 m» 4 C ev. ( 90) Wth l» 8. 0 m μm fom (46 ), the elet feld whh geneates the oespondng effetve potental s gven by 8EC E = U qe = eˆ. lqe 8 ( 04 4 m ) ev m = e» m ( m ) ( ) ˆ 3 0 0e 8. 0 m.6 0 C ˆ ( N C ), ( 9 ) 4 9 whee the vaable s smply hosen as ρ= mete (m). A ealzed setup may be muh smalle. The stength of the elet feld s about /0 of an eleton n a hydogen atom. Fo the magnet feld B=00 Gauss, w» Hz, and l» 8. 0 m μm, the tal enegy would deease to E 0 4 m» 6 C The oespondng elet feld fo the potental U s ev. ( 9) E» 3 06eˆ ( N C ). ( 93) The stength s about 0 tmes the elet feld n a photoope and s easy to aheve. Equatons (90) and (9) show that the elet and magnet felds of the magnet tansmutaton system an selet and tap eletons wth extemely low enegy and a defnte sgnatue of angula momentum. Ths funton s obvously effetve fo abtay haged patles. It thus offes a novel means fo the hoosng and tappng of the low enegy haged patles n the unfom magnet feld.

14 J. Phys. Commun. (08) DH Ln Fgue 5. Taetoes of the value e=.0 and e=. n the attatve foe feld. The pattens n the fst ow ae fo the negatve haged patle. The taetoes n the ase e> devate fom the unfom spal patten, and beome nonunfom spals. The stength of the magnet feld n ths ase satsfes E w p 4. > 4.. Repulsve foe In aodane wth equaton (50), the foe geneated by the Coulomb feld of the tansmutaton system would be epulsve when the hage q and momentum p have dffeent sgnatues,.e., a = ( lw p ) 8 < 0n ths ase. Wth the defntons Ep p e = +, and L =, ( 94) Ma M a t an be poved that only the ase of Ē > 0, namely e>, has a physal soluton. The othe two ases, Ē = 0 and Ē < 0, only have solutons n the omplex plane. A devaton smla to the pevous subseton shows that the taetoy equaton s gven by = os L + ( e ) L e e osh. 95 e The top (bottom) patten n fgue 6 s a taetoy of the negatve (postve) haged patle geneated fom the equaton wth the plus (mnus) sgn. It s a nonunfom spal lke the pevous ases of Ē > 0 and α>0. Howeve, t esapes moe qukly fom the egon of the eletomagnet feld. 5. Conluson Ths atle pefoms the lassal analyss fo the HookeNewton tansmutaton n the unfom magnet feld. The fst pat s devoted to dsussng the physal ealzaton of mehanal onfomal mappng of the othogonal systems based on the fomnvaant HamltonJaob equaton. Then, t s appled to alulate the aton and angula vaables of the tansmutaton system sne they ae the easest way to manfest the oheene of enegy between lassal and quantum mehans. The seond pat of the atle s used to evaluate the dffeent knds of taetoy equatons fo the moton of a hage n the system. It s shown that the types of taetoes fo the attatve feld ae lassfed by the magnet feld paamete w p 4. Although the tansmutaton system s eated by the quadat onfomal mappng, t s possble to atualze the system n physs by entepetng the onfomal stutue as an aton of the foe feld. Seveal notes ae woth makng as follows: () The nteaton stength of the tansmuted Coulomb feld s weak, but adustable. To appeate 3

15 J. Phys. Commun. (08) DH Ln Fgue 6. Taetoes of haged patles n the epulsve Coulomb feld of the tansmutaton system fo the value e=.0. The taetoy above s fo the negatve hage. It s a nonunfom spal. It devates fom the eletomagnet feld moe qukly than the taetoy n the attatve feld as shown n fgue 5. ths quanttatvely, we assume that the angula momentum s to take the quantzaton values p = m, m =,,,aodng to quantum mehans. Fo an eleton movng n the magnet feld of B= Tesla, l» 8. 0 m μm fom (46), and w» 0 4 ev. The tansmuted fnestutue onstant s gven by a lw p M m lw = = = m. ( 96) 8 8 Hee we have used a suffx M to denote that the ouplng s due to the magnet feld. The ato between the tansmuted and elet fnestutue onstants s a a M e» m = m. ( 97) Theefoe, the nteaton stength s only the ode of /000 elatve to the ommon elet Coulomb nteaton when B= Tesla and m s a small ntege. Howeve, t s not mpossble to evese the outome as long as the hage s n the stonge magnet feld and has a lage angula momentum m. () The sgnatue of the angula momentum p fo a patle wth negatve (postve) hage n the magnet feld B = Beˆ z an only be pesbed as negatve (postve) n the pesented analyss. Othewse, t would volate the physal fat that a haged patle s always tapped n a stongenough unfom magnet feld. Nevetheless, quantum mehans allows an altenatve hoe of the sgnatue sne t s wellknown that the angula momentum of a patle s always quantzed by p = m n the D ental foe feld poblem. Consequently, t s plausble to postulate that the pesented tansmutaton system n quantum mehans ould be appled to selet and tap haged patles wth the oespondng sgnatues unde the tal enegy E = w p C 4 sne E C s also the uppe bound enegy fo a bound state aodng to equatons (37) and (53). () In geneal, thee ae many knds of tansmutatons that an be geneated fom the magnet foe of the unfom magnet feld by onfomal mappng. Ths s poved by takng the geneal onfomal tansfomaton fom (7) Substtutng them nto equaton (3) gves se to n =, and = n. ( 98) ln nn dl = ( d + d ). ( 99) ln The onfomal fato thus esults n a dffeent knd of tansmutaton of foe when a dffeent value of ν s hosen. Let us llustate t wth an example by hoosng ν=. The adal aton vaable beomes n ths ase 4

16 J. Phys. Commun. (08) DH Ln p w M l8 Mw pl4 I = + ME d. ( 00) p 6 4 Ths s obvously a ental foe feld poblem. The seond tem n the squae oot s ust the entfugal enegy fo a moton n the feld. The thd tem stands fo an attatve foe popotonal to 7, whle the fouth tem s a epulsve (attatve) foe when ω, and p have the same (dffeent) sgnatues. Hee we see agan that the topologal sgnatue of angula momentum s also sgnfant, not ust the sgnatue of the hage. Physally, the system an be eated by applyng the effetve potental U = ( l4 4) E and the unfom magnet feld to the envonment of the hage. All n all, the pesented system has hybd popetes of the Coulomb and smple hamon osllato (SHO) systems. Fst, t has an unlosed taetoy lke the Coulomb system, whle the SHO only allows a losed obt. Seond, the system has a losed taetoy (oespondng to the dsete bound state at the quantum level) whle the enegy s estted by the potental of the SHO type M w l < ( ) E ( ). ( 0) Fnally, unlke the Coulomb system, the tappng foe s due to the lnea foe of the onstant magnet feld. Ths makes t easy to aheve n a laboatoy. Tansmutaton o dualty between two dffeent systems establshed by a tansfomaton has long been of nteest. It naates the ntenal onneton of two systems, and the unty of the dffeent banhes. It s ou hope that ths pesentaton wll be helpful n futhe evealng nteestng systems and applatons. Aknowledgments The autho thanks C R Hangton fo he tal eadng of the manuspt, and Pofesso P G Luan fo the dsusson. My wok has been suppoted by the Mnsty of Sene and Tehnology of Tawan unde Contat No. MOST 05M0007MY3. ORCID Ds DeHone Ln https: /od.og/ Refeenes [] Newton I 687 Phlosophae Natuals Pnpa Mathemata (Cambdge: Cambdge Unvesty Pess) [] Anol d V I 990 Huygens & Baow, Newton & Hooke (Basel: Bkhäuse) [3] Needham T 993 Am. Math. Mon [4] Leonhadt U and Phlbn T 00 Geomety and Lght: The Sene of Invsblty (New Yok: Dove) [5] Ln D H 06 Eu. Phys. J. D 70 4 [6] Leonhadt U 006 Sene [7] Pendy J B, Shug D and Smth D R 006 Sene [8] Zhang S, Genov D A, Sun C and Zhang X 008 Phys. Rev. Lett [9] Landau L D and Lfshtz E M 998 Mehans 3d edn (Oxfod: ButtewothHenemann) [0] Gadshteyn I S and Ryzhk I M 994 Table of Integals, Sees, and Poduts 5th edn (New Yok: Aadem) 5