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On the Correspondence Principle for Rotations 760 Brazilian Journal of Physics, vol. 26, no. 4, decemb er, 1996 On the Corresp ondence Principle for Rotations 1 2 1 S. G. Mokarzel , M. C. Nemes and A.F.R. de Toledo Piza 1 Instituto de Fsica, Departamento de Fsica-Matematica UniversidadedeS~ao Paulo, Cx. Postal 66318, 05315-970, S~ao Paulo, SP, Brasil 2 Departamento de Fsica, Universidade de Minas Gerais, Cx. Postal 702, 31270-970, Belo Horizonte, MG, Brasil Received Octob er 20, 1996 In the presentwork we calculate the radiation emitted from a charged rotating ellipsoid in the context of Classical Electro dynamics. The results are compared with its Quantum Mechanical version. An explicit derivation of the corresp ondence principle is given, showing that for angular momenta of the order of 60~ the classical regime is reached within 12% accuracy. I. Intro duction ation can b e written as The corresp ondence principle states that the results 1 D (1) A = quadr of Classical Physics should b e contained in the Quan- 2 6c r tum Mechanical results as limiting cases. The limit where r is the distance b etween some origin O and the should b e reached for \large quantum numb ers". This observation p oint, c is the lightvelo city and idea has b een a p owerful guide for theoretical conjec- X r tures and the construction of the Quantum theory. (2) D = D n n = ; r Historically the most imp ortant and successful in- ; vestigation based on this Principle is the spatial struc- with ture of the hydrogen atom. Many other applications Z 0 in several contexts followed to our knowledge. Not 0 0 2 0 3 0 D = [3x x r ](x )d x (3) ; ; 0 V as much attention has b een devoted to the Corresp on- dence Principle in the context of the radiation emitted 0 (x ) b eing the charge density which generates the elec- by rotating charged b o dies. tromagnetic elds. It is enough to calculate the mag- In what follows we set up a classical mo del for the netic eld, emitted radiations of a rotating charged ellipsoid. We ::: compare the results with the quantum mechanical ver- 1 n (4) H = D 3 6c r sion of the mo del and answer the following question: What is the order of magnitude of the angular momen- in order to obtain Poynting's vector tum of a rotating rigid b o dy for which the classical c 2 regime has b een reached? ~ = H n (5) 4 In section I I we p erform the classical calculations of and the radiation intensity in the solid angle d the emmitted radiation of a rotating ellipsoid. In sec- tion I I I we construct its quantum version and compare the results. Conclusions are given in section IV. dI = j~ jdS (6) c 2 2 H r d : (7) = I I. Classical calculation 4 The integrated intensityisthus given as The vector eld asso ciated to the quadrup ole radi- Work partially supp orted byCNPq. S. G. Mokarzel et al. 761 0 1 X ::: 1 2 1 0 0 ( I = ) : (8) D ; ze 5 0 2 2 180c @ A D = (a b ) 0 1 0 (9) ; ; 5 0 0 2 where ze corresp onds to the total charge. In order to obtain D ; ie, eq.(9) in the lab oratory system, one ; needs to calculate 0 0 D = T T D = T D T (10) ; ; ; ; ; ; ; where the matrix T e ects such transformation and can b e written as 1 0 T T T 11 12 13 A @ T T T (11) T = 21 22 23 ; T T T 31 32 33 where T = cos cos cos sin sin 11 T = sin cos cos sin cos 12 T = sin sin 13 T = cos sin +cos cos sin 21 T = sin sin +cos cos cos 22 T = sin cos 23 Figure 1. Classical Mo del. T = sin sin 31 Now consider the ellipsoid in Fig. 1, which rotates T = sin cos 32 with constant angular velo city ~! around an arbitrary T = cos 33 axis. In the b o dy's rotating frame the quadrup ole ten- sor D is given by Using the ab ove expressions weget ; c 1 0 2 2 2 1 3 sin sin 3 sin sin cos 3sin cos sin ze 2 2 2 2 2 A @ 3sin sin cos 1 3sin cos 3 sin cos sin (12) (a b ) D = ; 5 2 2 3sin cos sin 3sin cos cos 3sin 2 d It is an interesting exercise in classical mechanics to where K is the pro jection of the angular momenton obtain for the present case, from Euler's equations the the rotor axis, following imp ortant relations J is the total angular moment, and | is the moment of inertia K J = cte cos = = t (13) It is now a simple matter to obtain J | 762 Brazilian Journal of Physics, vol. 26, no. 4, decemb er, 1996 c i h X 2 ::: ze 2 4 2 2 2 3 _ (32 62cos + 30cos ) ( ) = (a b )3 D ; 5 ; 6 2 4 K K J 3 2 2 2 96 186 (14) [ze(a b )] +90 = 2 4 25 | J J Inserting this expression into (8) weget 6 2 4 1 3 K J K 2 2 2 I = 96 186 (15) [ze(a b )] +90 5 2 4 180c 25 | J J d In order to compare with the quantal result we " # r 5 rewrite the ab ove expressions in terms of the average R = b = R 1+ ( ) (16) min 0 16 radius R and deformation parameter according to 0 Assuming small (not an essential hyp othesis) we get (ref. [1]) r 5 3 2 2 2 (17) (a b ) R = 0 2 " # r according to which the classical expression for the emit- 5 R = a = R 1+ ted radiation is given by (2 ) MAX 0 16 c 2 6 2 4 J K 3 K 2 zeR 96 186 +90 (18) I = clas 0 5 2 4 75c 4 | J J d tor with axial symmetry I I I. Quantum calculation 1 1 1 2 2 The essential ingredients for the corresp onding H = J + J (20) 3 2| 2| 2| 3 quantum calculation are the following: and | stands for the moment of inertia in x y direc- a) the system's hamiltonian tions and | for the z direction. J is the angular mo- 3 mentum op erator and J its z {pro jection on the prop er 3 H = H + H + H (19) R rad int axis. The eigenfunctions and eigenvalues of H are well R known (refs. [2-4]) where H describ es the nucleus and mo dels a rigid ro- R c 2 ~ J (J +1) 1 1 J 2 2 J ( )= H D + ~ K ( ) (21) D R MK MK 2| 2| 2| 3 where ^ ^ ^ J i=~ J i=~ J i=~ J z y z D ( )=hJmKje e e jJM Ki MK S. G. Mokarzel et al. 763 b) The Quantized Radiation Field H (refs. [5]) rad Z 1 1 2 2 3 _ A +(r A) d r (22) H = rad 2 8 c with r Z X 2 dk ~c y (kr w t) (kr w t) k k p A(r;t)= [e b (k)e e b (k)e ] (23) k k 2 2 2w k d where describ es the p ossible p olarization of the eld nal state jf i where the nucleus is de-excited bymeans y and b , b are the usual photon creation and annihilation of emitting a photon op erator (see ref. [5]) 2 2 k 2 dw (i ! f k )= jM j d (25) fi c) The interaction hamiltonian H , int ~ ~c Z with 1 H = j Adr (24) int M (k;) hf ; k jH i;00i (26) fi int c where j stands for the nuclear current op erator. The calculation now is standard and contained in many d) Fermi's Golden rule whichgives the transition prob- text b o oks. For details the reader is referred to (ref. ability p er unit time from an initial state jii, in our case [5]) whose notation wefollow. There the calculation is the rotor in a given excited state and zero photons, to a p erformed step bystepuntil the nal result c 2 e j +1 2j +1 w (Ej : i ! f )=8c k B (Ej : i ! f ) (27) 2 ~c j [(2j +1)!!] with 1 (E ) 2 B (Ej : i ! f )= (J jjQ jjJ ) (28) f i j 2 [2j +1]e i d (E ) (E ) the reduced matrix element(J jjQ jjJ ): The factors In eq.(28) (J jjQ jjJ ) stands for the reduced ma- f i f i j j 2j +1 involving the photon angular momentum j and k trix element of the electric multip ole op erator, de ned come from the usual expansion of the electromagnetic as: eld for large wavelengths as compared to the system's Z (E ) j sizes involved. Q drr Y (29) jm jm The calculation of B (E 2) for a rigid rotor can b e In the ab ove expressions the dynamics of the transi- found in ref([1]) with techniques of ([2]) and is given by tion is contained in B (E 2:i ! f ), in particular in c 2 2 3 6 (J K + 1)(J + K +1) 2J +1 1 2 zeR (30) B (E 2:J +1 ! J )= 0 1 2 e 4 4 2J +3 J + (J + 2)(J +1)J 2 2 1 3 3 (J K + 2)(J + K +2)(J K +1)(J + K +1) 2 B (E 2:J +2 ! J )= zeR (31) 0 3 5 2 e 4 8 J + (J + 2)(J +1) J + 2 2 which gives for the transition probability 764 Brazilian Journal of Physics, vol.
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