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. 26, no. 4, decemb er, 1996
2 6 2
2J +1 3 (J K +1)(J + K +1) ~
2 2
(32) zeR (J +1) K 6
T (E ) =
2 J +1!J
0
1
5
75c 4 | 2J +3
(J +2) J + (J +1)J
2
2 6
3 ~ (J K + 2)(J + K +2)(J K + 1)(J + K +1) 3
2
(33) T (E ) = zeR (2J +3)
2 J +2!J
0
3 5
5
75c 4 | 2
J + (J + 2)(J +1) J +
2 2
d
We are now in a p osition to investigate the Cor- First Case: K =0
resp ondence Principle, i.e, to compare Quantum and
In this case there will b e no J +1 ! J transition,
Classical results in the limit of total large angular mo-
since it prop ortional to K . It b ecomes then a simple
menta
matter to verify that eq.(33) tends, in this limit, to
c
2 6
J 3
cl 2
T (E ) ! 96 = T (E ) (34)
zeR
2 J +2!J 2 J +2!J
0
5
75c 4 |
, ie the rotating system is contained in which corresp onds precisely to the classical expression eq.(18) when =
2
the x y plane (see Fig 1).
Second Case: K 6=0
This corresp onds to having arbitrary . In this case there will b e a contribution from the transition J +1 ! J .
K
when p erforming the classical limit in the eqs. (32 and 33) we shall consider K nite, K 1 and 1. We nd
J
" #
2 6 2 4
3 K K J
cl 2
T (E ) ! T (E ) 6 6 (35) zeR
2 J +1!J 2 J +1!J
0
4 | J J
" #
2 6 2 4
J 3 K K
cl 2
T (E ) ! T (E ) 96 192 +96 (36) zeR
2 J +2!J 2 J +2!J
0
4 | J J
d
IV. Conclusions Finally, the Corresp ondence Principle in this case
can b e written as
In the presentwork wehaveinvestigated the cor-
resp ondence principle in the context of the radiation
emitted by a rotating ellipsoid. We found that the clas-
cl cl
T (E ) + T (E ) = I (37)
2 J +1!J 2 J +2!J clas
sical limit is reached for angular momenta of the order
60~ within 12% accuracy. In fact angular momenta of
where I as given in eq.(18)
clas
the order 60~ haverecently b een found exp erimentally
The accuracy of the classical limit for J 60~
=
in sup erdeformed nuclei pro duced by means of the fu-
can b e studied by comparing the classical and quan-
sion of several combinations of nuclei. Their masses lie
tal radiation intensity as function of J , for the case,
in the range 160 - 170 and their sp ectra are found to b e
K =0.InFig.2weshow the p ercentage of discre-
in excellent agreement with those predicted bythepro-
cl
jT (E ) I j
2 J +2!J clas
100 as a function of J . pancy
I
late rigid rotor with axis ratio verycloseto2:1.Such
clas
We see that when J structure covers a spin interval from 34~ to 58~ (refs. 60~ this discrepancy is
=
[6-8]). around 12%.
S. G. Mokarzel et al. 765
2. D. M. Brink and G. R. Satchler; Angular Momen-
tum Oxford, Clarendon (1971).
3. M. E. Rose; Elementary Theory of Angular Mo-
mentum N. Y., Wiley (1957).
4. A. R. Edmonds; Angular Momentum in Quan-
tum Mechanics Princeton, Princeton Universithy
Press, (1960).
5. A. de Shalit and H. Feshbach; Theoretical Nuclear
Physics Volume I: Nuclear Structure N. Y., John
Wiley and Sons (1974).
6. R. Chapman, J.C. Lisle, J.N. Mo, E. Paul, A.
Simco ck, J C. Willmott, J.R. Leslie, H.G. Prince,
P.M. Walter, J.C. Bacelar, J.D. Garrett, G.B.
Hagemann, B. Herskind, A. Holm and P.J. Nolan;
Phys. Rev. Lett. 51, 2265 (1983).
7. J.C. Bacelar, M. Dieb el, C. Ellegaard, J.D. Gar-
rett, G.B. Hagemann, B. Herskind, A. Holm, C.X.
Yang, J.Y. Zhang, P.O. Tjom, J.C. Lisle; Nucl.
Phys. A442, 509 (1985).
Figure 2. Percentage of discrepancy b etween quantum and
8. J. Recht, Y.K. Agarwal, K.P. Blume, M. Guttorm-
classical result as a function of the total angular momentum.
sen, H. Hub el, H. Kluge, K.H. Maier, A. Ma j, N.
Roy, D.J. Decman, J. Dudek, and W. Nazarewicz;
References
Nucl. Phys. A 440, 366 (1985).
1. A. Bohr and B.R. Mottelson; Nuclear Structure V.I e I I N. Y., Benjamim (1969).