On quantum quadrupole radiation

IL NUOVO CIMENTO VoL 63 A, 1W. 4 21 Giugno 1981

On Quantum Quadrupole Radiation.

L. FON])A

International Centre /or Theoretical t)hysies - Trieste, I taly Seuola Internazionale Superiore di Studi Avanzati - Trieste, Italia

tZaeulty of _hTatq~ral Sciences and Technology and J . Ste]an Institute University of L]ubl]ana - L]ubl]ana, Yugoslavia

(ricevuto il 9 Febbraio 1981)

Summary. - - In this paper it is shown that for the electromagnetic decay of a quantum system in a coherent rotational state the total quadrupole radiation is proportional to "Q'" Q* + e.c. For the radiation flux out of a sphere of large radius, a different quantity, closer to the classical expres- sion Q~, is found.

1 . - I n t r o d u c t i o n .

We s tudy in this paper the qu~drupole radia t ion power for the electro- magnet ic decay of coherent ro ta t ional states. The T-decay of these states has been considered in previous papers wi th par t icular a t t en t ion to the pulsed behaviour of thei r t ime evolut ion (1). Wi th in the f ramework of ref. (1), we

evalua te in subsect. 2"1 the to ta l quadrupole radia t ion power and find t h a t i t

is proport ional to "0"" Q* + e.c. I n subseet. 2"2 the power emi t ted out of a sphere

of very large radius is eva lua ted and found to be propor t ional to IQ~+)I 2, where Q(+) is the operator obta ined f rom Q via suppression of the off-diagonal elements

(*) On leave at Techniehen Universitgt, Physik Department, Miinchen. (1) N. MA~Ko~-BOR~TNI](, M. ROSlNA and L. I~o~])A: 2Vuovo Cimento A, 53, 440 (1979); 56, 229 (1980). See also 0. DU~ITRESCU, L. FO~DA and N. MANKO~-BO~TNIK: _WUOVO Cimento A, 58, 105 (1980).

4 8 3

484 L. FO~DA and x. MANKO6-BOR~TNIK

<Q>,,, with I ' < I . This result is, therefore, ve ry close to the classical expres-

sion e). We shall consider the e lectromagnet ic decay of a coherent ro ta t ional s ta te (*)

(1.1) = a7 exp [ - iE, t] ]q,,>, I

which has at t ime t = 0 the following characterist ic propert ies: a) the ab- solute values of the ampli tudes a~ are peaked around a mean value of the angular m o m e n t u m I, b) the phases of a~ are roughly equidistant , c) the ener- gies E~ of the states ~ obey the rule coo I(I ~ 1) to a very good approximat ion . The states ~ are eigenstates of Ho defined as the sum of the nuclear and the free electromagnet ic Hamil tonians . The to ta l Hami l ton ian of the sys tem is then given b y

(1.2) H = Ho + H,, Ho = H .o,o~. + H.o...,

where HI is the quadrupole in teract ion:

1 3

By the symbol of in tegra t ion we mean bo th the in tegra t ion over m o m e n t a k and the sum over polarizat ion vectors e~. Throughout the paper, whenever k appears as a subscript, i t denotes bo th m o m e n t u m /~ and polarizat ion ek of the photon. A cut-off for large k will be always unders tood in the integrals over the variable k, in order to be consistent wi th kRn<< 1, where R~ is the radius of the nucleus.

The quadrupole m o m e n t Q, is given b y

z (1.4) Q~-- ~ 3e (k .r,)(ek .r,)

I n order to get (1.3), we have assumed tha t the nuclear force is veloci ty independent and we have dropped the magnet ic pa r t of the interaction, which is supposed to be small.

2. - Evaluat ion o f the radiation power.

We shall denote b y ~ the init ial coherent rota t ional s ta te of our decaying system. One- and two-photon states will be denoted b y ~p~ and ~ , , respec-

(2) L .D. LANDAU and E. M. LIFSmTZ: Classical Theory o/.Fields (Oxford, 1971), p. 188. (*) Here and in what follows the index I stands for both angular-momentum quantum numbers I and M. We use natural units ~ = v = 1.

ON QUANTUM QUADRUPOL:E RADIATION 4 8 5

tively. Since we shall use the per turbat ion theory in the interact ion picture, we shall write the evolution of these states via the unper turbed Hamil tonian Ho:

[~,~(t)> ~ exp [-- iHot] j~> = ~ a7 exp [-- iEzt] kv~>, I

(2.1) IW~z(t)> --~ a~ exp [-- ikt] I~p~(t)>,

We shall not consider the contr ibut ion of states having more than two photons. The solution of the complete SchrSdinger equat ion

(2.2) HIT(t)> ---- i diT(t)> dt

is then expanded on a complete or thonormal set of 0, 1 and 2 photon states:

(2.3) IT(t)> ---- ~ c~(t)ly, z(t)> + Z~d~ke~(t)l~(t)> +

+ ~ffd~kcl~k'e~,~,(t)lw~,(t)>.

For the coefficients, to second order, we get

(2.4) e~(t)

(2.5) e~(t)

(2.6)

t

= ~ - ~ ~ fdt'fd.k e~,~(t')e~p [-- ikt']@,~(t')]H.a~l~t,~,(t' )>,

t

= - - i ~ f d t ' e~,(t') exp [ikt']@,~(t')lakHxlv~,(t')> , fl' o

t

eZ~,(t) = - - i X f d t ' f d " k " %,k,,(t') exp [i(k + k ' - - k")t ']. f l ' o

2"1. Emiss ion out o] a sphere o] nuclear radius. - Conservation of energy

tells us tha t

(2.7) f = { f ..~ + f dv + f z, dv}l (t)> = 0

where ~ , 5/r , .... JFs~ and JF I are the t tamfl tonian densities corresponding to H, H~ t l .... and HI, respectively. In (2.7) the integrations are of course

486 L. rONDA and ~. :~{A~KO~-:BOlg~TNIK

performed over all the co-ordinate space. Since ~o ~ .~ and o%f~ are nonvanishing only within the small sphere of nuclear r a d i u s / ~ , from {2.7) we get

(2.8) d ,f -~ <F(t) /d dv[T(t)> =

with in R n

- dtd outslde ~ a s l l space

tha t is the change in energy of the system within the sphere of nuclear radius R= manifests itself as the change of energy of the free electromagnetic field in all the space.

The evaluat ion of the r ight-hand side of {2.8) yields

(2.9) d <W(t)lfd~k ka~a~lT(t)>, P ( , ) - m. =

a l l sP~ce

which, to second order, turns out to be

(2.1o) P(t) = -~d ~ fd~klez~(t)l~ k

In order to evaluate this quant i ty , we follow just the same procedure used in ref. (~) for the evaluat ion of W~_.a. One finally gets

d 1 (2.11) P(t) -- dt 36z ~ ,,'LL' ~ (A~rAz~,O(A.,) q- A~L,A. ,O(A~,)} .

�9 QI,,Q~,~ sin (A ,i, - - ALL') t/2 exp [i(ALL,-- A ,v) t/2] = A . , - - A L ~ ,

1 - - * X 72z~ ~ ~ {A~z,AzL,O(Ar,,)q-A~L,A ,,O(ALL,)} QI,,QL'ze P [i(ALL,--An,)t] ,

f l lltLL S

where by QQ* we mean tha t the average over directions and polarizations has

been performed. We have defined Qz,1 and drz as

(2.12) Q,,, = ,,~*,~; <+,,[QI~,>, At,, = E , , - ~ , .

Since the initial state is supposed to be a coherent rotat ional state, the con- t r ibut ion coming from the initial state ~ is the dominant one. We drop then the sum over fl and retain only the fl = e term. One gets a simpler form for (2.11) by exchanging I ' e+ L and I e+ L ' in the sum pertaining to the second te rm in the curly bracket and using the fact tha t for fl = ~ one has Qz'~ ---- Q~*'.

ON QUANTUM QUADRUPOLE RADIATION 487

Using the fact t ha t for the sums appearing in (2.11) O(x) A- O(- -x ) ~ 1, one finally gets

1 (2.13) P(t) - - 72~ ~ AI~,ALL, Q,,1Q*L,Lexp [ i ( A L ~ , - - A H , ) t ] .

I I tLL s

Expression (2.13) can be obtained by taking the average over directions and polarizations of the following quant i ty :

1 ?j'~(t)6*(t)-4- e.c., (2.14) P(t) = 12~7~

where Q~(t) is defined as the mean value of the quadrupole operator for the coherent rotat ional state ~v~(*):

(2.15) Q~(t) = <~(t ) lQl~f~( t )>.

2"2. E m i s s i o n out o/ a sphere o] large radius. - We shall now evaluate the radiat ion power emi t ted from a sphere of large (but finite) radius R. Equa- t ion (2.8) will read now

(2.16) d <T(t)l f dvlT(t)> = <T(t)] f dvlT($)>. with in R outBide

Using on the r ight-hand side of (2.16) the cont inui ty equation for the photon field, one gets for the radiat ion power

(2.17) PR(t) : - - <T(t)[ ( J . n dslT(t)>, SR

where S~ is the surface of the sphere of radius R and J is the Poynt ing vector :

(2.18) J = - - N - ~ - • .

Here N means normal product for the creation and annihilation photon op- orators.

For J we easily get

(2.19)

where

( 2 . 2 0 ) Jo(r) = (2~)-3ff__

J(r) = J o ( r ) - k J ~ ( r ) ,

) dSk dSk' V/kk ek X e~, X l

2

�9 (a~,a~ exp [i(k - - k ' ) . r] ~- h .c . ) ,

48~ L. FONI)A and ~r ~ANKO~-BOR~TNIK

(2.21) s = ( 2 ~ ) - ~ f f - d ~ k d 8 k'

~ / ~ e ~ X X ek, �9 2

�9 (-- akak, exp [i(k q- k ' ) . r] § b.c.).

The operator Jo does not change the n u m b e r of photons in the field, while J2

changes t ha t n u m b e r by two. We shall first show tha t J2 does not give contr ibutions to the emi t t ed power.

Le t us t ake its average on the s ta te IT(t)> and use (2.3) and (2.4)-(2.6):

(2.22) J~(r, t) =-- <T(t)IJ~(r)]kg(t)> =

�9 e~(t) e~kk,(t) exp [i(k -k k ' ) . r] exp [ - - i(k + k') t] .

Since cpk ~, is at least 0(H~), c~ mus t be t aken to the zeroth order, i.e. e~(t) ~_ Opt. Then, apar t f rom a constant , for J2(r, t) one gets

(2.23)

�9 exp [i(k ~- k ' ) r ] exp [- - i (k -~ k')t]aT,*a~a~*,a~ALL,A,,,"

{ l [ e x p [ i ( k + k ' - - A H , - - A L L , ) t ] - - i �9 <Qk>L'L<Qk'>I"k--AL~," k + k ' - - A H , - - A L L ,

exp [i(V--A,,),]-- 1] } - - k ' - - A H , ~- (k<-+k') q- C.C.

We evaluate now (2.23) as follows: if ALL. is negative, we do not have any contr ibut ion to the integral (this is of course approximate) . The same holds for AL~, = 0, since A~L, appears as a fac tor in the integral. I f ALL' is positive, we add the in terva l (-- ~ , 0) and evaluate the integral b y contour in tegra t ion in the k complex plane (this is possible since a cut-off funct ion is unders tood in the integral). Wi th these approximat ions , one just gets the contr ibut ions a t the poles k = ALL" and k = A~L, ~ AII, - - k', which are bo th zero:

(2.24) s t) _~ 0 .

Let us now come to the evaluat ion of Jo. I t s average on the s tate IT(t)>

is given by

) Jo(r, t) ~ <T(t)tJo(r)lT(t)> = (2re) -3 3kdak'%/k~e~><. ~TXek, "

�9 Re [e;~,(t) c~(t) exp [i(k - - k ' ) . r] exp [i(k' - - k) t]] .

ON QUANTUM QUADRUPOLE RADIATION 4 8 9

Using (2.5) and (2.17) one finally gets

a~, a,a~,a~,exp -- (/b~,+ A ~ ) t �9 (2.25) _pR(t)- ~ ~: ~* ~ 8, l l ' t L

sin (k'+ A,,,)t/2 sin (k--ALL,)t/2 ~r k'p') + c.c. k' + A H, k - - d ~L,

where J~(kp, k'p') is given b y

(2.26) f r , '

where we have explici t ly shown the sums over polarizations p and p ' . I n order to proceed with the calculation, one takes advan tage of the slow dependence of the quadrupole ma t r ix elements on polarizations and angulur directions. One then ext rac ts these ma t r ix elements f rom the integrals and sums by t~king their average values (Q}z'z'(Q}L'~.

The evaluat ion of the spat ia l angular integral, summed over polarizations and in tegra ted over angles in k und k ' m o m e n t u m spaces, for R large (recall the presence of eut-offs in the k and k' integrat ions), gives

(2.27) R l~rge 2kk 'R~ exp [ikR] exp [ - - i k ' R ] ,

where, b y the a rgumen t of the s ta t ionary phuse (note t ha t t > 0 ) , we have dropped all t e rms of the type exp [ ~=ih(R + t/2)], where h here stands for ei ther k or k' , which give v~nishing small contr ibut ions when in tegra ted over k or k'. I n order to in tegra te on k and k', one can follow the sume analyt ic pro- cedure used for the evaluat ion of ,/2. We get

(2.2s) 1

PR(t)-- 72~ I , r ~ z L'~JZ~'~LL'"

�9 0(- - A H,)O(ALu)<Q}rz<Q>L,L exp [i(A ~,~- ALu)(R-- t)] + c .c . .

Again, since the initial s ta te is supposed to be a coherent ro ta t ional state, the contr ibut ion of the ini t ial s ta te ~ is dominan t ; we drop then the sum fl and re ta in only the fi ~ ~ term. I t is then immedia te ly seen tha t the complex conjugate t e rm equals the first t e rm on the r ight -hand side of (2.28) (to see this interchange 1 ~-~ L ' and L +-+ 1' in the c.c. term). I f we define an opera tor

Q(+) such t ha t

(2.29) (Q(+)}z,z ---- (Q}l,~ O(Al,z),

32 - l l Nuovo Uimento A .

490 L. FONDA and ~r. MABIKO~-BO~TNIK

we can finally write the radiat ion power out of a large sphere of radius R as

(2.30) 1

where the average over directions and polarizations is taken for the operator Q ~ J ( t - - R ) given by

(2.31) Q~)(t - - R ) = < ~ ( t - - R)IQ'+)IVJ~(t - - R)>

and QC+) is obtained from Q~ (as defined by (2.15)) by suppression of the off-

diagonal terms (Q)~,~ with I ' < I . The result (2.30) is very close to the clas-

sical expression Q~ for the radiat ion power (~)(*).

* * *

We thank Profs. R . E . PEIERLS and M. RosI~A for discussions. One of

the authors (/~. MA~Ko~-BOR~TNIK) would like to thank Prof. A. S A L ~ , the

Internat ional Atomic Energy Agency and UNESCO for hospitali ty at the In ternat ional Centre for Theoretical Physics, Trieste.

(*) One easily gets

= �89 Re

One expects that the second term on the right-hand side vanishes in the classical limit.

�9 R I A S S U N T 0

In questo lavoro si mostra che, nel deeadimento elettromagnetico di un sisr quan- tistieo in uno stato rotazionale coerente, l'emissione totale di radiazione di quadrupolo

proporzionale a "Q'Q* ~- c.c. I1 flusso delia radiazione attraverso una superiieie sferica di raggio molto grande ~ invece dato da una diversa espressione, pifi simile a quella

classica "~2.

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