IL NUOVO CIMENTO VOL. 25 A, N. 4 21 Febbraio 1975

On the Fermi Golden Rule.

L. FO~DA

In terna t iona l Centre ]or Theoretical Phys ics - Trieste I s t i tu to di ~ i s i ca Teorica dell' Univers i th - Trieste

G.C. GHIRARDI and A. RIM~NI

I s t i tu to di F i s ica Teorica del l 'Univers i th - Trieste I s t i tu to Nazio,t~ale di ~'isica Nuclea te - Sezione di Trieste

(rieevuto il 21 Ottobre 1974)

Summary. - - The derivation of the Fermi golden rule for the transition rates of an unstable system is reconsidered within a recently proposed Omoretical framework to describe decay processes. It is shown that the obtained transition rates contain a dominant term which coincides with the standard result, plus correction terms whose features are discussed.

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

In this paper we want to reconsider, from a new point of view, the problem

of determining the transi t ion rates induced by a weak static per turbat ion. The

problem is usually t reated in first-order per turba t ion theory. By making use

of some further approximations, one gets a t ime-independent t ransi t ion rate

usually known as the Fermi golden rule (1). The t ime independence of the

transit ion rate is a necessary feature of any sensible t r ea tmen t of a decay process

and is u s u a l l y o b t a i n e d j u s t by m e a n s o/ the a b o v e - m e n t i o n e d a p p r o x i m a t i o n s .

Recently, a new description of the decay process has been given, in which

it has been taken into account tha t an unstable sys tem does not evolve un-

disturbed but is subjected to repeated measurements , occurring at random

times, which ascertain whether the sys tem has decayed or not by means of

(i) See, for example, A. I~¢[ESSIAII: M~canique quantique (Paris).

35 - I1 Nuovo Cimento A. 5 3 7

L. FONDA, G. C. GHIRiRDI aIl(1 A. RIMINI

a localization of the decay products (2). Such a description enforces a purely exponential decay law for the unstable system which immediately leads to time- independent transition rates without the need o] any approximation. I t is there- fore natural to derive the transition rates induced by a weak static perturbation within the new framework and compare the results with the :Fermi golden rule. Of course, in order to get an explicit result one has to solve the dynamical problem, and this will be done through first-order perturbation theory as in the standard approach.

The final result for the transition rates contains a leading term which is just the Fermi golden rule, plus corrections whose features are discussed.

2. - The Fermi golden rule for a system subjected to measurements.

Let us have a quantum system whose Hamiltonian is

(2.1) g = H o + v = ~r~...+ g ~ . , . + v ,

where Hd.,. has a purely discrete spectrum and describes the decaying system, Hd.p. has a continuous spectrum and describes the decay products, while V is a small perturbation. We write

(2.2a)

(2.2b)

Hol~o, o) = Bolqo, o>,

~o1~,, x=> = E,=I~,, x~>.

The state [~0o, 0> is the direct product of the eigenstate ~o of Hd.,., whose decay is under study, times the vacuum state for the decay products; similarly Iq~*, Z~> indicates another eigenstate ~ of Hd... times a free state Z~ of the decay products. The energy

(2.3) Ei~= s t + e~

is the sum of the eigenvalues ei of Hd.,. and e~ of Hd.p. The state evolved from the state 1~3, 0>, to first-order perturbation theory, is

(2.4) exp [--iZ~t/~][~o, 0> =

= exp [--iEotl~][I+., o> + ~',; d3~V':exp[-i(E"-Eo)t/~]~,: -- E. - - 1 ]¢f~, Z">],

where V , ~ : <~0,, X, lV[~o, 0}.

(2) L. FONDA, G. C. GHIRARDI, h . RII~INI and T. WEBER: N~bOVO Cimento, 15 A, 689 (1973); 18A, 805 (1973).

ON T H ~ F~RMI GOLDEN RUL~ ~ 3 ~

According to the treatment of the decay given in ref. (2), the decaying system is subjected with mean frequency ~ to repeated measurements which ascertain whether the decay products are inside or outside a given space region. If the system is found undecayed, the wave function is reduced by the pro- jection operator Po ~ P<, where / ) o : I~o, 0} (~0, 0[ and P< sets to zero the wave function of the decay products outside the above-mentioned space region. The reduction process leads to an essentially unique state which is identified with the unstable state [~p~}. The decay rate is then (compare formulae (2.20) and (2.22) of ref. (2))

(2.5)

where

---- )~f] exp [-- it] [1 -- P(t)] dt, O

(2.6) P(t) -~ [<F~[ exp [-- iHt/?~] [~p~}l 2

is the quantum nondecay probability and

(2.7) ), = )~ -- v.

The material system playing the role of measuring apparatus is charac- terized by the two quantities 4, the mean frequency of measurements~ and R, the linear dimensions of the space region on which P< projects the decay pro- duct wave function. The quantum state whose decay is under study is charac- terized among others by the quantities F~ the decay rate as given by the Fermi golden rule, vo~ and ko~, the velocity and the wave number of the decay prod- ucts in each channel i corresponding to the decay energy E o - ~. The fre- quency 2 is supposed to be large with respect to F, i.e.

F (2.8) 0~ : ~ <4 1.

Moreover, it is supposed that in the mean interval between two meas- urements the decay products travel a distance much greater than R, i.e.

(2.9) 02i ~-- ~ ~ 1 . Voi

The relationship between the measuring apparatus and the quantum system is further characterized by the large number

1 (2.10) 03i koiR ~> 1

~ 4 0 L. FONI)A~ G. C. Gtt l I~ARDI DJIld A. R I M I N [

of wavelengths of the decay products contained in the distance R. I t follows from the analyses of ref. (~.3) tha t in practical cases the quantit ies 01, 0~ and 08 are very small compared with uni ty, even though they vary within wide ranges. F rom condition (2.9) it follows tha t the measurement process, when the decay products are found inside the reduction region, essentially leads the system back to the state [~o, 0>. In fact, the norm of the first t e rm in (2.4) is equal to one, while, for times near 1/2, the square norm of the second term is of order F / 2 ~ 0~, which is much smaller than one. When P0 + P< is applied, the first t e rm is left unchanged, while P< roughly reduces the square norm of the terms under summation by the addit ional factor R2/voi-~ 0~. Therefore, for the moment we shall evaluate the decay rate by means of (2.5), using for _P(t) the expression

(2.11) P(t) = 1<~o, ol exp [ - i n t / ~ ] Iqo, °>l ~.

Using the completeness of the states I%, 0> and [~, Z=> we get

(2.12) 1 -- P(t) = ~fd3~l((p~, Z~,lexp [-- iHt/h] [%, 0>I ~ ,

so tha t , introducing the transit ion rates v~, we have

(2.13) v : ~ v i , i

(2.14) , , ,= z~.j~e~p [ - it] dtfd"o,l<q~,, x=lexp [ - ¢Ht/h] IVo, o> 1~. 0

Using (2.4) in order to evaluate the matr ix element, we have then

(2.15) cos (( ~,,, - Eo)t/~))

(Ei¢, - - Eo)2

I~itegration over the t ime variable yields

(2.16) vi : 22 fd~,x ]EV~: [2 j (.E~,-- o) + (2~)~"

In order to compare this result with the s tandard one we make use of the ffSUal approximation tha t I Vi~l ~ can be taken out of the energy integral; this

(s) A. DEGASPERIS, ]L. FONDA and G. C. GI-IIRAttDI: N~OVO Cimento, 21A, 471 (1974).

ON TtIE FERMI GOLDEN RULE 541

is sensible since the integrand is peaked around E0. We get

(:~.]7) co

f f 22 d~IV,~(E ~] 2 dE,~ dE~ [(E,~ Eo):@ (ih)2] - t ,

where d2c~ is the element of solid angle contained in d3c~. To perform the energy integral one must either take dlc~/dE~ out of the

integral or specify the kinematics. I f the second al ternat ive is chosen, using nonrelativistic kinematics ( d ~ / d E ~ = mk~/h") the integral can be performed explicitly giving

~,=:~0 I f d~~] V~:(E°) P ] V i1[ - 227~ d ~ Re l @ i E o _ e ~ (2.18) v~--2 h d E ~

- - E ~ S i "

Since . = f is the dl~/dE,~]~,a ~o- d2~ density of final states in channel i, / ~ i is the usual expression for v~ given by the Fermi golden rule. The factor 2/2 can be cxp~nded in terms of the small qum~tity 0~ = F /2 ~ F~/2, and we obtain

t

2 1 (2.19) ~ -- 1--v/~2 -- 1 @ 0,@ 0(.0~).

The quant i ty -2h/(Eo--s~) appearing in the square root in eq. (2.18) is easily seen to be

E0 -- e~ -- 2 020~, (2.20)

so tha t

(2.21) Re

I t follows tha t

(2 22)

VI+ ,~o~- ]+ ~(~)(0~ 0~)2 + . . . .

[ ] ~ = F~ 1 + ~ + ~ ~ E o - - ~ j + "'" "

We see tha t the decay rates given by our description of the decay process are, in zeroth-order approximation, again given by the Fermi golden rule. This derivation is more satisfactory t han the usual one, since it takes into account bet ter the actual physical si tuation and exhibits explicitly the value of the corrections (note, however, tha t a more accurate t r ea tment of the cor- rections is necessary, as will be shown in the next Section). We also stress

5 4 2 L . F O N D A , G. C. G H I R A R D I a n d A. R I M I N I

t h a t first-order per tu rba t ion theory, which is used here as well as in the s tandard derivat ion, is valid only for t imes sufficiently small to satisfy a condit ion which is usually wr i t ten as (1)

(2.23) F t << 1 ,

being the decay ra te given by the Fermi golden rule. Due to the presence of the factor exp [-- ~t] in (2.5), in our t r e a t me n t this condit ion is automat ica l ly satisfied, since we use (2.4) for t imes of the order of 1/~, whieh satisfies (2.8). In the s tandard t r ea tmen t the condit ion (2.23) gives rise to the unsat is factory s i tuat ion t ha t the result obta ined for the decay ra te is valid only for t imes

much smaller t han the lifetime.

3 . - F u r t h e r r e f i n e m e n t s .

The t r e a tmen t of ref. (2), which was followed in Sect. 2 of the present paper, is based on the assumption tha t , when the system is found undecayed, the re- duct ion process leads the system back to a unique state, so t h a t the quan tum nondecay probabi l i ty P(t) is a unique function. This assumption has a l ready been discussed in ref. (2) and in Sect. 2 of the present paper and is cer ta in ly so well verified in actual eases tha t the preceding t r ea tmen t is sufficient for most pract ical purposes. However , if one is in teres ted in evaluat ing possible cor- rect ions to the Fermi golden rule, one must take into account t h a t such an assumption is not s t r ic t ly true. In fact , suppose for the sake of definiteness t h a t a t some t ime t ~ 0 the system is prepared in the s ta te l%, 0>. I f the system suffers a reduct ion af te r a t ime v, the small t e rm arising f rom the ap- pl icat ion of P< to the second t e rm in (2.4) depends on v. When this s ta te evolves during a t ime t, the resulting s ta te depends not only on the t ime t f rom the last reduct ion, bu t also on the preceding t ime in terval 3. I t is obvious tha t , in the case in which the prepara t ion of the system in the s ta te ]%, 0> has occurred in the remote past , the dependence of the s ta te on the t ime intervals between

the reduct ions does not go beyond a few steps, due to the effect of the repea ted

evolut ions and projections by P< . The theory of ref. (2) can be modified to take into account the dependence

of P(t) on the t imes of the preceding reductions, provided t h a t such a depen- dence is l imited to a finite number of steps. Wi th this l imitat ion, in fact, the sys tem still loses the memory of the absolute t ime f rom preparat ion, so tha t the decay ra te is independent of t ime. Consider, for example, the case in which the wave funct ion depends, besides t, only on the t ime • a t which the preceding reduct ion has t aken place. Then it is easy to modify the argument , which has been given in ref. (2) leading to eq. (2.5), and get the expression

(3.1) oo ¢o

v ~-- ; t fdv~ exp [ - -~T]fd t ] exp [ - - i t ] [1 -- P~(t)]. 0 0

ON THE FERMI GOLDEN RULE 543

The quan t i t y ), is again re la ted to 2 b y eq. (2.7). Equa t ion (3.1) reduces to (2.5) if P~(t ) is independent of 3. The q u a n t u m decay p robab i l i ty m u s t now be defined as the p robabi l i ty of finding the decay products outside the reduct ion

region, so t h a t

(3.2)

where

1 - - P , ( t ) = Op~(t)lP>Iy~(t)> ,

(3.3) P> = 1 - - B o - - B < .

To see t h a t when the reduct ions lead to a unique s t a te eq. (3.2) coincides

wi th eq. (2.6), one has s imply to observe t h a t in such a case

1 (3.4) iF,,} - <y, ul~(t)> (Po-f- P<)ly,(t)> •

The general izat ion of (3.1) to the case in which m e m o r y of the t imes of several

preceding reduct ions remains in the reduced s ta te is s t ra ightforward. Le t us now look a t the s t ruc ture of the wave funct ion I~v(t)>. Dur ing each

in te rva l be tween two reduct ions the s ta te kvo, 0> emits the wave funct ion of the decay products and this wave funct ion spreads f rom its source towards infinity. The wave funct ion ]~v(t)> a t the t ime t following the last reduct ion

can then be wr i t ten

(3.5) [yz(t)> ~ exp [ - i H o t / h ] ]q~o, O> ÷ Iv(t)> =

=- exp [-- i H o t / ~ ] [q~o, 0> ÷ rind

(") "t'> where W ........... t t has been emi t t ed b y kvo, 0> dur ing the n- th t ime in te rva l before the last reduct ion and has suffered n project ions b y B<. Thus ]v°(t)> is jus t the second t e r m at the r.h.s, of (2.4), while the nex t t e r m is

(3.6)

and in general

(3.7)

IV~ (t)> = exp [-- i H o t / ~ ] B<IV¢°)(T~)},

•(n) t \ ~,.~ ........ ( ),, = exp [-- i H o t / h ] P ~¢,-1) % ,,, < q,,,...,~.L i)/ •

Using (3.5) in (3.2) we get

(3.8) 1 - - P ( t ) = <v ( t ) I v ( t ) } - - <~7(t)IP< 1~7(t)> =

- <v(°~(t)IVc°)(t)> ÷ 2 Re ~ < (o)(t) ("~ "r- - - V V~, .~ . . . . .~ . ( ) ) -q-

(era) "t" (,I ' t '" cm) t B c,~.., t .

5 4 ~ L . FO :N D A , G. C. G t t l R A R D I and A. R I M I N I

We observe now tha t the th i rd t e rm in the last expression using (3.7) can be handled as follows:

(3.9) (m) (n,) (m--l) (n--l) @ . . . . . . . ,,~(t) l~,,.....,,(t)) = ~ @ . . . . . . . . . (~,)lP<l~ ....... , ,(~,)} =

- - X (m) (n) - - @= .......... (v~)[P<l~ ....... , , + ~ ( ' q ) } - ~m~O

When I -- P(t), as given by (3.8), is inserted in the formula generalizing eq. (3.1) for the case in which memory of several reductions remains, the last two te rms of (3.8) cancel due to (3.9) because of the t ime integrations. Therefore, in place of (3.8) we can use the expression

(3.10) [1 -- P(t)].,f---- (~(°)(t)]~(°)(t)} -~- 2 Re ~ 07(°)(t)Iv~)...~.(t)> n = l

for calculating v. We note tha t the cancellation just described allows us to s tudy ve ry s imply the way in which the possible (~ memory effects ~ of the s ta te influence the quan t i ty v, since in this way we avoid the analysis of the overlap propert ies of states which have suffered several and/or different numbers of reduct ions (in part icular , the invest igat ion of the orders of magni tude of t h e various te rms in the sum at the r.h.s, of (3.10) tells us when we can consider t ha t m em ory is lost, for all pract ical purposes).

When (3.10) is inserted into (3.1), the contr ibut ion v0 to v coming f rom the first t e rm (V(°~(t)l~c°)(t)} is the decay ra te a l ready evaluated in Sect. 2, eq. (2.22). The orders of magni tude of the contr ibut ions v~, v2, ... coming f rom the successive te rms of the series on the r.h.s, of (3.10) are easily unders tood to be quickly decreasing, by taking into account the physical meaning of the various [~(~) ...... (t)}. Le t us therefore confine ourselves to the evaluat ion of v~. ~or this purpose, assuming t ha t the reduct ion region is a sphere of radius R, we write

(3.11)

where

(3.12)

We also write

(3.13)

$m

.R

Pl(k~, k~) = 2 fj l(k~r)j ,(k~,r)r2dr .

o

~m

To obtain vl we must first of all write down the explicit expression for [~°)(t)} and ]~( t )} . Since [~°)(t)} is given by the second t e rm on the r.h.s, of (2.4),

ON TIIE F:I~RMI GOLDEN RULE

using (3.13) we h~ve

( 3 . ~ )

where

(3.~5)

5 4 5

I.~(°,(t)> = e x p [ - - i E o t / h ] :~ dE,~ (lE, ~ ~ Y~m(k~)g(t~(E~) ]~(t)]q%, Z~> ,

] ~ ( t ) = (~xp [ - - i ( E , ~ - - E o ) t / l t ] - - 1

E,~ - - Eo

To get [~(~l)(t)> we use (3.6), (3.11) and (3.14), obtaining

(3.16)

so tha t

(3.17) <~ff°)(t) I~(~l)(t)> =

- - ~ gzm (Ei~)gzm(E~) ],~(t)P,.(kn, k~) ]i~(T) . • zm d ¢J ~ J

Performing the t ime integrals appearing" in (3.1) one obtains

r E dlfi /' E dla

(i)* (i • gtm ( E ~ ) g ~ m ( E ~ ) ( E ~ - - E o - - i 2 h ) - l P z ( k n , k ~ , ) ( E , ~ - - E o - - i ~ l i ) -1 .

We make use of tile usual approximat ion tha t the g's can be t aken out o£ the energy integrals and we assume nonrelat ivist ic kinematics. For simplicity, we confine ourselves to the case in which the in teract ion V,-, is sphel'ical, ob- taining from (3.18), as shown in the Appendix,

27¢[ q dlo¢ ) (3.19) ( 1 ) -- 4 ~ 7 ~ - - tWiot(Eo)120(Eo__~i) sil l 2~0i R I

Taking into account (2.22) we have

(3.20) v~ = F~ 1 + 2 - - 2 . E o - - ~ sin2ko~R .

The result (3.20) exhibits the corrections to the Fermi golden rule to first order in the quantit ies 0~ and 0o 0a. We note tha t the correction terms depen4 on the quanti t ies 2 and R which characterize the exper imental set-up. We

546 L. FONDA, G. C. GHIRARDI and A. mMINI

f u r t he r note t h a t the coefficient in f ront of 2~/(E0--e~), due to its rapid va r ia t ion wi th the quan t i t y R which is not defined too sharply, cannot be t a k e n too

seriously. The last t e r m only indicates t h a t corrections of t h a t order of mag-

n i tude are present . I t is also to be kep t in mind t h a t use has been m a d e of the a p p r o x i m a t i o n t h a t the in te rac t ion can be t a k e n out of the energy integrals.

Such an approx ima t ion is necessary if one wants an expression to be com- pa red wi th the Fe rmi golden rule.

I t is easily seen t h a t this k ind of app rox ima t ion influences the coefficient .co~ of eq. (2.17) of the correct ive t e r m in 2~/(Eo -- s t ) . I n fact , if one evaluates ~

b y t ak ing the k inemat ica l fac tor dlo~/dE,, out of the energy integral (an ap- p r o x i m a t i o n of the same t y p e as t h a t of t ak ing out the interact ion) , one would

get in (2.22) an addi t ional t e r m -- (1/:~) 02 0a. When 01 >> 02 03 the t e r m ~ ~ / 2 i

represents a meaningful correct ion to the Fe rmi golden rule, whose dependence

on 2 could perhaps be tested.

4 . - Conc lus ions .

We have reder ived the Fe rmi golden rule, wi thin a descript ion of the decay

processes which we consider closer to the ac tua l physical s i tuat ion t h a n the usua l one. As a l ready underl ined, the present approach avoids some unsat is-

f ac to ry features of the s t andard procedure and allows an eva lua t ion of the correct ions to the Fe rmi golden rule.

Wi th in the context of our t r e a t m e n t of decay processes, we wan t to stress

t h a t the discussion of Sect. 3 is i m p o r t a n t since it shows t h a t the reduct ion to a unique state, which was assumed in ref. (~), is not essential for the va l id i ty of this k ind of approach. I n fact we have seen t h a t the ma in conclusions r ema in t rue even when some (< m e m o r y ~) of the previous h is tory of the s ta te remains .

Moreover, for our specific but ra ther general example , we have made clear t h a t

this m e m o r y cannot ac tua l ly go beyond a few reduct ion processes.

APPENDIX

D e r i v a t i o n o f eq. (3 .15) .

Since V~ is supposed to be spherical

1 (o - (A.1) V, = ~ g o (E,~),

O N T t I E F E R M I G O L D E N R U L E 547

only t he 1 = 0 t e r m surv ives in (3.18). F r o m (3.12)

(h.2) Po(k~, k ~ ) 1 1 ( s i n (k~ - - ] ca )R sin(k~-H__k~)R~ k~k~\ k ~ - - k a k ~ + k~ ]"

Us ing in (3.18) eqs. (A.1) a n d (A.2), t o g e t h e r wi th t he k i n e m a t i c a l r e la t ion

dl~ m (A.3) dE~, -- h 2 k~,,

we ob ta in

(A.4)

co co

o 0

k~ - - k~

sin + k )R] ]

whe re b~ = Eo- -e~ a n d e~ = h"k~/2m. P e r f o r m i n g t he in tegra ls , we ob ta in

(A.5)

where

] /.,mO i . ih

, - I Im ~

I m S~ > 0 .

Since ?i~l~= (21~)0203 is a smal l q u a n t i t y , we can e x p a n d S~, o b t a i n i n g f r o m (A.5)

(1) 8 7 ~ ° " m 2 2 [ ~ R ] s i n 2 k o ~ R . (A.7) v ~ - h'ko~ IY'~'(E°)]2exp--v0~J

E x p a n d i n g the e x p o n e n t i a l in powers of the smal l q u a n t i t y ~R/vo~ = (~/2)0.,, we can wr i te

(A.8) 27~ } , m 2

v~{1) _ _h 4~--h~ko ~ IV~,(Eo)12sin2ko~R,,

which is jus t (3.19).

• R I A S S U N T 0

Si riconsidera il problema di derivare la (~ golden rule ~) di Fermi per i tassi di decadimento di un sistema instabile, utilizzando lo schema teorieo recentemente proposto per descri- vere i proeessi di decadimento. Si mostr~ che i ~assi di dec~dinmnto cosi ottenuti con- tengollO un termine dominante che coincide col risultato usuale, pih termini eorrettivi di cui si discutono le caratteristiche.

5 ~ L. FONDA, G. C. GHIRARDI ~tnd A. RI1V[INI

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