Brazilian Journal of Physics, vol. 29, no. 3, September, Classical and Quantum Mechanics of a Charged Particle

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1 Brazilian Journal of Physis, vol. 9, no. 3, September, Classial and Quantum Mehanis of a Charged Partile in Osillating Eletri and Magneti Fields V.L.B. de Jesus, A.P. Guimar~aes, and I.S. Oliveira Centro Brasileiro de Pesquisas Fsias Rua Dr. Xavier Sigaud 150, Rio de Janeiro , Brazil Reeived 7 May, 1999 The motion of a partile with harge q and mass m in a magneti eld given by B = kb 0 + B 1[ios(!t)+jsin(!t)] and an eletri eld whih obeys re =,@B=@t is analyzed lassially and quantum-mehanially. The use of a rotating oordinate system allows the analytial derivation of the partile lassial trajetory and its laboratory wavefuntion. The motion exhibits two resonanes, one at! =! =,qb 0=m, the ylotron frequeny, and the other at! =! L =,qb 0=m, the Larmor frequeny. For! at the rst resonane frequeny, the partile aquires a simple losed trajetory, and the eetive hamiltonian an be interpreted as that of a partile in a stati magneti eld. In the seond ase a term orresponding to an eetive stati eletri eld remains, and the partile orbit is an open line. The partile wave funtion and eigenenergies are alulated. PACS: 1.75.A, 5.50.G, D, 1.0 Keywords: magneti resonane, ion trapping, isotope separation, magneti onnement I Introdution The dynamis of harged partiles in eletri and magneti elds is of both aademi and pratial interest. The areas where this problem nds appliations inlude the development of ylotron aelerators [1], free eletron lasers [], plasma physis [] and so on. In this paper one onsiders the lassial and quantum dynamis of a partile with harge q and mass m ated by a magneti eld given by B = kb 0 + B 1 [ios(!t)+jsin(!t)] (1) and an eletri eld whih relates to B through the Faraday law: () Both elds an be derived from the vetor potential: A =, 1 R B (3) When B 1 is given by two pairs of rossed Helmholtz oils, the approximation of homogeneity impliit in equation (1) is valid in a region of about 0% of the volume enlosed by the oils. Similar assumption have been taken by other authors [, 3]. The lassial equations of motion an be obtained from the lagrangian: L = m ( _ X + _ Y + _Z )+ q _ X[ZB 1 sin(!t), YB o ]+ + q _ Y [XB o, ZB 1 os(!t)] + q _ Z[Y B 1 os(!t), XB 1 sin(!t)] () whereas to study the quantum dynamis we need the hamiltonian: H = 1 m (P x + P y + P z ), q m [B 1os(!t)L x + B 1 sin(!t)l y + B 0 L z ]+ + q m fb 0 (X + Y )+B 1 Z, B 0 B 1 Z[Xos(!t)+Y sin(!t)]+ + B 1[Y os(!t), Xsin(!t)] g (5)

2 5 V. L. B. de Jesus et al. II The lassial motion The lassial equations of motion an be easily obtained from (). In order to eliminate the time dependene of the lagrangian, we perform the following transformation of oordinates: i = i 0 os(!t), j 0 sin(!t) j = i 0 sin(!t)+j 0 os(!t) (6) k = k 0 d In the nulear magneti resonane (NMR) literature [5] these transformations are interpreted as leading to a system of referene whih rotates with angular frequeny! in respet to the laboratory oordinate system. From (6), using lower ase to indiate the variables in the rotating system, the eetive lagrangian beomes: L ef f = m (_x +_y +_z ), q _xy(b o +! )+ q _y[x(b o +! ), zb 1]+ + q _zyb 1 + q! (B o +! )(x + y ), q!b 1xz (7) This lagrangian an be written in the usual ompat form: L ef f = T + q _r A ef f, q ef f () the ee- where T is the partile kineti energy, A ef f tive vetor potential given by: A ef f (r) =, 1 r B ef f ; (9) with B ef f being the eetive magneti eld (dened below), and the eetive salar potential: ef f (r) = 1!B 1xz,! (B o +! )(x + y ) (10) where = q=m is the partile harge-to-mass ratio. Thus, one has for the partile in the rotating frame the following equations of motion: x = _y(b o +! )+x!(b o +! ), z!b 1 y = _zb 1, _x(b o +! )+y!(b o +! ) (11) d z =, _yb 1 + x!b 1 It is useful to dene an eetive eletri eld E ef f. The expressions for the eetive elds are: E ef f = x!(b o +! ), z! B 1 i y!(b o +! ) j 0, x!b 1 k 0 (1) B ef f = B 1 i 0 +(B o +! )k0 (13) Therefore, for a xed value of!, eah partile with a given harge-to-mass ratio,, will feel dierent effetive elds. Note that B ef f diers from that in the NMR ase by a fator `' in the \apparent eld"!= [5]. With these denitions, the form of the Lorentz fore is preserved in the rotating system: F ef f = qe ef f + qv B ef f (1) One an learly see from (11) that there are two resonane frequenies in the motion: one at! =! =,qb 0 =m, the ylotron frequeny, and another at! L =,qb 0 =m, the Larmor frequeny. For a frequeny equal

3 Brazilian Journal of Physis, vol. 9, no. 3, September, to the rst one (! ) the partile feels the following effetive elds: E ef f =, 1!B 1[zi 0 + xk 0 ] (15) B ef f = B 1 i 0, B o k 0 (16) and for the seond frequeny(! =! L ), E ef f = 1 [x!b o, z!b 1 ] i y!b oj 0, 1 x!b 1k 0 (17) B ef f = B 1 i 0 (1) Now we onsider a partile inident in region of elds B 0 and B 1, at the origin of the oordinate system, with initial veloity parallel to B 0. That is, x(0) = y(0) = z(0) = 0, v x (0) = v y (0) = 0, and v z (0) = v 0. As in usual NMR, one makes the approximation B 0 B 1. In this limit, it is easy to verify by diret substitution the following solutions of (11), for! =! : are as follow: ( 33 U) = 1:; ( 3 U) = 1:37; ( 35 U)=1:31; ( 36 U)=1:6 and ( 3 U)=1:16. For this simulation we set v 0 =10 m/s, B 0 =1T, B 1 =0:01 T. The osillating eld frequeny is tuned to the ylotron frequeny of the isotope 35 U, that is,! =,1:31 MHz. The drawing is in the rotating system. One an have a piture of the trajetories in the laboratory system by rotating the gure about the z- axis. Note that eah isotope, aording to Eqs. (1) and (13), feels dierent eetive elds. This auses the lighter isotopes to deviate in opposite diretions in respet to the heavier ones. x(t) y(t), v o 3! 1 p 3vo p 3!1 t sin( 3! " p # 3!1 t os( ), 1 ),! 1v o! sin(! t),! 1v o! os(! t) (19) z(t) p p 3v o 3!1 t sin( ) 3! 1 where! 1 B 1.For! =! L : x(t)! 1v o! sin(! L t), v o sinh(! 1t! L L ) y(t)! 1v o! os(! L t)+ v o! 1 osh(! 1t! ) (0) L z(t) v o! 1 sinh(! 1t ) The detailed alulation for the obtention of these solutions is given in ref. []. Then, we see that whereas for! =! the solutions are purely trigonometri funtions, for! =! L there is a mixture of trigonometri and hyperboli funtions. This means that the trajetory of the partile will be a losed path in the rst ase, and an open line in the seond. For a general value of!, whih an be very lose to!, there will also be an exponential drift. As an example, Figure 1 shows the trajetories of partiles in a beam ontaining triply ionized isotopes of uranium. The respetive harge-to-mass ratios in MHz/T Figure 1. Trajetories of triply ionized Uranium isotopes in osillating elds \tuned" to the 35 U isotope resonane (-1.31 MHz) in the rotating oordinate system. The stati eld is along the +z-axis, and the osillating magneti eld is on the xy-plane. The initial veloity of the partiles is v 0 =10 m/s, along the diretion of the stati eld. The orbit of the resonant partile is losed, whereas the nonresonant ones drift away. The trajetories in the laboratory system are obtained rotating the piture about the z-axis. These urves were produed for dierent lengths of time in order to make them all visible in the same sale. III Quantum desription In this setion we approah the problem from the quantum-mehanial point of view. The transformation to the rotating frame in this ase, made diretly on the Shrodinger equation, allows the derivation of the laboratory wave funtion and the partile eigenenergies. Using the following straightforward relations:

4 5 V. L. B. de Jesus et al. (i) Xos(!t) +Y sin(!t) =e,i!tlz =~ i!tlz =~ Xe (ii) [Y os(!t), Xsin(!t)] = e,i!tlz =~ Y i!tlz =~ e (iii) X + Y = e,i!tlz=~ (X + Y )e i!tlz=~ (iv) Z = e,i!tlz=~ Z i!tlz =~ e (v) L x os(!t)+l y sin(!t) =e,i!tlz=~ L x e i!tlz=~ L z = e,i!tlz =~ i!tlz =~ L z e where, L x, L y and L z in Eq. (5) beomes 1 : are the omponents of the anonial angular momentum of the partile, the hamiltonian H e i!tlz=~ H(t)e,i!tLz =~ H 0 = P m + m B0 (X + Y )+ m B1 (Y + Z ), m B 0 B 1 XZ, B 0 L z, B 1 L x (1) This hamiltonian represents a harged partile moving in a stati magneti eld B = B 1 i+b 0 k. The above operation an be interpreted as the quantum-mehanial transformation of the hamiltonian to the rotating oordinate system. Dening the wavefuntion 0 through the relation: and replaing into the Shrodinger equation one obtains: = e,i!tlz=~ 0 (H 0,!L z ) 0 = H0 0 ef f () Sine H 0 is time-independent, the solution of () will be: ef f and onsequently the wavefuntion in the laboratory system will be 0 (t) =e,i(h0,!lz)t=~ (0) (3) (t) =e,i!tlz =~ e,i(h0,!lz)t=~ (0) () Note that sine [L z ; H 0 ] 6= 0, the two exponential operators in Eq. () annot be gathered into one. Now we shall analyze the properties of : = P m + m B 0 (X + Y )+ m B1 (Y + Z ), m B 0 B 1 XZ, B L z, B 1 L x (5) where B B 0 +!=. By adding and subtrating the quantity m! +!B 0 (X + Y ), mb 1!XZ 1 For the quantum treatment wekeep apitals throughout the setion.

5 Brazilian Journal of Physis, vol. 9, no. 3, September, the eetive hamiltonian an be re-written as: H 0 = P + m B ef f (X + Y )+ m B1 (Y + Z ) m, m BB 1 XZ, B L z, B 1 L x + q!! B 1 XZ, + B 0 (X + Y ) whih, in turn, has the general form: (6) = 1 m (P, qa ef f ) + q ef f (7) where the eetive salar potential is again given by ef f =!! B 1 XZ, + B 0 (X + Y ) () A ef f being the eetive vetor potential. The omponents of A ef f an be obtained by ommuting X, Y and Z with, and omparing the result with the denition of the anonial momentum P = m _R + qa. For instane: Consequently, i~ _ X =[X; ] = 1 m i~p x + B i~y P x = m _X, q B Y A ef f;x =, B Y Repeating the proedure for the other omponents, one obtains: A ef f;y =, B 1 Z + B X A ef f;z = B 1 Y These results are the same as those obtained in ref. [6], the only dierene being a fator `' in the denition of B. Written in the form of Eq. (6), the hamiltonian exhibits the eets of the eletri eld. Contrary to what happens when this is negleted [6], it shows two resonane frequenies. At the Larmor frequeny, B = 0, and the hamiltonian beomes: = P m + m B1 (Y + Z )+ B 1 L x, m B 0 B 1 XZ + m B0 (X + Y ) (9) whih represents a partile in a stati magneti eld along the x diretion, plus an eletri eld potential. The eigenstates of the partile in this ase annot be easily found. On the other hand, at the ylotron frequeny, the seond term of ef f in Eq. () vanishes, and the hamiltonian beomes: = P m + m B0 (X + Y )+ m B1 (Y + Z ), m B 0 B 1 XZ, B 0 L z + B 1 L x (30) whih represents a partile moving in a stati magneti eld B ef f = B 0 k,b 1 i. This hamiltonian an easily written in a diagonal form by dening the angle = tg,1 B1 B 0 between the z-axis and B ef f, and writing the operators of H 0 in (30) in terms of the new oordinates, ef f X0, Y 0 and Z 0, where the eetive eld is axial.

6 56 V. L. B. de Jesus et al. The partile's eigenenergies are given in this ase by: E n = P 0 z m + n + 1 ~! 0 (31) p where! 0 = B ef f = B0 + B 1 is the ylotron frequeny about the eetive eld in the rotating system. From this one sees that the quantization axis an be rotated ontinuously by hanging the angle through the hange in the ratio B 1 =B 0. IV Conlusions In this paper wehave studied the lassial and the quantum dynamis of a harged partile in osillating magneti and eletri elds whih are related through the Faraday law. The equations of motion show two resonane frequenies, one at the Larmor frequeny (! L ) and another at the ylotron frequeny (! ). When the eld frequeny equals!, the partile is onned to a simple losed trajetory, but when! =! L, it drifts away, the same happening to o resonane partiles whose frequenies are very lose to!. The use of a \rotating oordinate system" suh as used in onventional nulear magneti resonane allows the derivation of analytial solutions for the equations of motion. By using the orresponding quantum-mehanial transformation, one nds the exat wavefuntion of the partile in the laboratory system. When! =!, the eetive hamiltonian orresponds to that of a harged partile in an stati magneti eld. In this ase the partile eigenenergies are derived in the rotating oordinate system, and it is shown that the diretion of the axis of quantization an be ontinuously rotated by hanging the ratio between the intensities of the elds. On the other hand, when! =! L, the hamiltonian is a mixture of eetive magneti and eletri elds, and the eigenenergies annot be easily derived. The Hamiltonian (30) predits the existene of \urrent ehoes", and therefore is in aordane with the results of referenes [6] and [7]. There is, however, one important dierene whih appears in the present ase where the eletri eld is onsidered. Contrary to what happens in [6], the stati eld term does not vanish at resonane. Thus, in order to desribe properly the formation of a urrent eho in the present situation, one must onsider B 1 B 0 when the pulse is \on", and obviously B 1 = 0 when it is o. Having this in mind, the alulation for the urrent eho amplitude an be arried out in the same way as desribed in [6]. Aknowledgements The authors are in debt to Prof. W. Baltensperger, Prof. N. V. de Castro Faria and to Dr. S. A. Dias, for their useful suggestions. V. L. B. J. aknowledges the support from CNPq, Brazil. Referenes [1] D.G. Swanson, Rev. Mod. Phys. 67, 37 (1995). [] J. Valat, Nu. Instr. Meth. Phys. Res. A30 50 (1991). [3] P. Diament, Phys. Rev. A3, 537 (191). [] K.W. Gentle, Rev. Mod. Phys. 67, 09 (1995). [5] C.P. Slihter, Priniples of Magneti Resonane (Springer-Verlag, Berlin, 1990) 3rd ed. [6] I.S. Oliveira, A.P. Guimar~aes and X.A. da Silva, Phys. Rev. E 55, 063 (1997). [7] I.S. Oliveira, Phy. Rev. Lett. 77, 139 (1996). [] V.L.B. de Jesus, A.P. Guimar~aes and I.S. Oliveira, J. Phys. B: At. Mol. Opt. Phys. 31, 57 (199).