Propagator for a Charged Particle in Time-Dependent Electromagnetic Field and Quadratic Potential

Transcription

1 Propagator for a Charged Particle in Time-Dependent Electromagnetic Field and Quadratic Potential BIN KAN CHENG Departamento de Fhica, Universidade Federal do Paraná, Caixa PosD>I 19081, Curitiba, PR, Brasil Recebido em 12 de janeiro de 1987 Abstract Through a t ime-dependent l inear transformation and the time substitution, we can evaluate exactly the propagator for a charged particle in a time-dependent electromagnetic f ield subjected to a time- -dependent quadratic potential. 1. INTRODUCTION It is well known that for a quadratic Lagrangian, the propa- gator is related to the classical action through the Van Uleck-Pau l i f~rmulal'~. However, the evaluation of the classical action is not always simple. The time-dependent linear coordinate transformations with new-time have been used by Junker and lnomata3, and by cheng4 to transform the original quadratic action into a new quadratic action whose classical action can be evaluated exac t 1 y. Later severa1 a~thors~'~ der ived such transformat ions in a broader sense by apply ing a non-l inear superposition law of Ray and ~eid'. In this paper we are able to deduce them from a Feynman path integral by considering the mid point expansion for each short-time action, and to obtain the pro- pagator for a time-dependent harmonically bound charged particle in a time-dependent electromagnetic field. ' For a time-dependent harmonically boundcharged particle of charge q and massmsubject to a time-dependent e1 ectrornagnet ic field -+ E(t) and $(t) (along the z direction), the Lagrangian has the form This work was supported by CNPq undertheresearch fellowship (Proc. n? I/FA).

2 Revista Brasileira de Física, Vol. 17, no 3, where w(t) = q~(t)/rnc is the cyclotron f requency, r, and E, (t) denote the 3 components of and Ê(t) perpendicular to ~ ( t. ) Here wx(t), ;(t) and w (t) are, respectively, the oscillator frequencies along x, y and z diz 3 rections. Since the z coordinate is separated from the r, (z and y) co- ordinates in eq. (I), the propagator is of the form being the propagatorll of a t irne-dependent harmon ic osci 1 lator. In eq.(5) the functions f (t) and g(t! satisfy the following differential equations. -, Nowweareonly left to evaluate the propagator of the Lagrangian (3), which wi 11 be carr ied out by using t ime-dependent 1 inear coordinate transformation with new-time for a special case. 2. SPACE TRANSFORMATION AND TIME SUBSTITUTION path.integral For the Lagrangian (31, the propagator can be expressed as the (8)

3 where ~ x ( t ) ~ ~ is ( t the ) usual two-dimensional Feynrnan differential measure. Us i ng Feynrnan's pol ygonal paths, the propagator (8) becomes =F(;.) (t. = For later convenience we set ~ ~ = t ~ and - t F.=l"(t ~ - ~.) and F 3 3 i 3 3 = (tj+tj-l)/2) for any function F(t). Introducing the time-dependent linear transfarmations of space and the time substitution we obtain the following relations1 5.-x = s.ax. + E.;.?, yj-yj-i = s.ay. + E.;.? (I)) 3 i - 1 x3 J J X J i Yd 3 J Y I j ' by expanding eq.(ll) about the rnid-point t in the time interval j [ti-,,t$ to terms of order E.. Here we have ler AX = X(T.) - x(t~-~), 3 J' 3 n. = (x(~.)+~(~~-~))/2 and similar for AY. and Substituting eq.(l)) i' into eq.(lo), we have

4 AGj,j-i = (1 5) 2s yj- 1 after s irnpl i f icat ions. In order to get rid of the X Y J' i tem in eq. (Ib), we rnust let We now choose. + w 2 Z = O and +w2.s = O. x~ xj xj ~j YJ ~j In order to satisfy both eq. (16) and eq. (17), we have to consider the following special case hereafter: Using the time substitution eq.(12) or õ = -r. - T = Ü E we obtain j J j-l j j' from eqs. (14) and (17) Choos i ng wi th w being a constant, eq. (19) becornes

5 Revista Brasileira de Física, Vol. 17, n? 3, 1987 measure is given Using eqs. (I]), (12) and (i8), the Feynrnan path differentiai by symmetrizing about the end points in the t irne interval [ti-l,t$. Combining eq.(19) with eq.(22), we obtain our principal result where K(R;,T~~;R;,T') is the propagator of a charged particle in a con- stant rnagnetic field with the cyclotron frequency o, and in a time dependent electric field &(.r) = ~(t(-r))s~(t(-r)), which has been evaluated by us13. Without loss of generality we now consider the case ofe (t)=o 5 or E(T) = E (7) hereafter. We then have13 (T = TI' - TI) Y

6 Combining eq. (5) wi th eq. (23) we have the propagator eq. (4) as our f i- na1 resul t. As a final remark we should mention that since s(t) =s,(t) = s (t) and R(t) = wx(t) = w (t), we have in the continuum case Y Y s2(t)u(t) = ~ g ç2(t)u(t), = 1 and Ç(t) + ~'(tjs(t) = O (27) as we e ~ ~ e c t Unfortunately, ~ ' ~. the present method can not be appl ied to the case of w (t) # w (t), which wi I1 be studied in the near future. x. Y We would like to thank Dr. A.B. Nassar for introducing us to his interesting papers and for sharing some of his unpublished results. REFERENCES 1. *~.~.~e~nman and A.R.Hibbs, (McGraw-Hill, New York, 1965). Quantwn Mechanics ahd Path IntegraZs 2. L.S.Schulman, Techniques and AppZications of Path Integration (Wiley, New York, 1981). 3. G.Junker and A.lnomata, Phys. Lett. IIOA, 195 (1985). 4. B.K.Cheng, Phys. Lett. 113A, 293 (1985). 5. A.B.Nassar, J.M.F.Bassa10 and P.S.Alencar, Phys.Lett. 113A, 365 (1985). 6. A. 8. Nassar and R.T. Berg, Phys. Rev. A34, 2462 (1 986). 7. J.R.Ray and J.L.Reid, J.Math.Phys. 22, 91 (1981). 8. J.Gervais and A.Jevicki, Nucl.Phys. BIZO, 53 (1976). 9. N.K.Pak and I.~8krnen, Phys. Rev. A30, 1629 (1984). 10. C.C.Gerry, J.Math.Phys. 25, 1820 (1984). 11. B.K.Cheng, Rev. Bras. Fís. 13, 220 (1983) ~&men, Phys. Lett. 115A, 6 (1986). 13. B.K.Cheng, Phys. Lett. IOOA, 490 (1984). Resumo Achando a transformação linear dd coordenada dependente dotempo e a sybsti tu ição do tempo, podemos calcular exatamente o propagador para uma particular carregada, no campo e1 etromagnét ico dependente do tempo, e com um potencial quadrático também dependente do tempo.