Temporally-quantized theory of exponential radioactive decay: Resolution of Zeno's paradox of quantum theory

ELSEVIER

Physica A 215 (1995) 181-200

Temporally-quantized theory of exponential radioactive decay: Resolution of Zeno's paradox of quantum theory Sidney G o l d e n 8614 North 84th street, Scottsdale, AZ 85258, USA

Received 12 October 1994

Abstract

As characterized experimentally by Rutherford, an essential feature of radioactive decompositions is their being constituted of randomly occurring events in terms of which the decomposing systems exhibit exponential temporal decay behavior with associated characteristic half-lives. This feature is rigorously accounted for generally by the recent temporally-quantized dynamical theory of strictly-irreversible evolution of isolated and localized non-relativistic quantum systems, which theory also obviates the celebrated Zeno's paradox of conventional quantum theory.

1. Introduction

The radioactive process discovered just about a century ago by Becquerel [1] and extended and elaborated by others shortly thereafter [2] has served as a cornerstone so to speak - of our present understanding of the atomic and nuclear structure of matter. Many questions concerning the fundamental nature of radioactivity arose following its discovery [3]. With the exception of an exponential temporal decay behavior of apparently randomly occurring events ascribed to it, which early on was determined experimentally by Rutherford and his associates [4-7], these questions were soon regarded as essentially answered. The exception stimulated later independent investigations [8-9] which served to support its validity so that, within quite narrow limits of experimental uncertainty and throughout time-lapses of many half-lives duration, Rutherford's characterization of radioactive decompositions now does appear to be confirmed. Prior to such confirmation, the question had become: Can ostensibly random

radioactive decompositions which appear to exhibit exponential temporal decay behavior with associated characteristics half-lives be rigorously accounted for in general quantum-mechanical terms? And it still remains, since an unqualified affirmative answer to it has not been obtained from conventional quantum theory even though considerable attention has been directed for some time now to determining just what 0378-4371/95/$09.50 ~(?~, 1995 Elsevier Science B.V. All rights reserved SSDI 0 3 7 8 - 4 3 7 1 ( 9 4 ) 0 0 2 6 4 - 9

182

Sidney Golden/Physica A 215 (1995) 181-200

sort of temporal behavior should occur in both radioactive and non-radioactive quantum processes [10-32]. 1 In fact, significant departures from the experimentally-observed exponential temporal decay behavior are to be expected according to the theory, both for very short and for very long lapses of time following the initiation of the processes. Thereby, more appropriately, the question would appear to be: To what extent can seemingly random radioactive decompositions which appear to exhibit exponential temporal decay behavior with characteristic half-lives be accounted for rigorously in general quantum-mechanical terms? As will be shown here presently, however, this question no longer requires any answer since an unqualified affirmative one to the earlier question now has been o b t a i n e d - the result of an important and necessary modification of conventional quantum theory. Because of the properties of the products which are emitted in radioactive decompositions, individual disintegrations and the time-instants at which they essentially occur can be recorded by detectors that are appropriately placed relatively remote from the radioactive systems themselves. Prior to their decompositions, the systems are then virtually free of any interaction with the detectors. In such cases, an absence of response to the radioactivity by the detectors can be regarded as an observation of the radioactive system involved, without influencing its behavior, which indicates that it had not decomposed at the time that the observation was made. This "non-influencing" aspect of observation is evidently restricted to those processes in which products are emitted from a system and can be remotely detected when they are [29]. Together, a decomposing radioactive system and its associated detecting system can thus be considered as comprising an isolated quantum system which undergoes the same intrinsic decompositional changes and temporal decay behavior whether it is being observed via the detectors or not. How it is to be described theoretically is another matter, however. The probability that any such radioactive system could be observed at any later designated time-instant to be in the same non-stationary state in which it was prepared originally can be formulated in conventional quantum-mechanical terms [10-19]. This conditional probability may be called a survival-probability, one that allows for the possibility that a change from the original state may occur at any time exclusive of the designated instant at which such a change presumably could be observed not to occur. Similarly, the probability that the system could be observed to be in the same non-stationary state at each of a designated sequence of later time-instants also can be formulated in conventional quantum terms [14, 22, 23, 25, 26, 32]. This temporally-sequenced conditional probability may be called a multisurvival-probability, one that allows for the possibility that a change from the original state may occur at any time exclusive of the designated instants at which such a change presumably could be observed not to occur. Upon taking the evolution that the system must undergo into account, the two probabilities then generally will have values which are less than unity, in accord with the expectation that there always should be some t Many more referencesthan those listed can be obtained from the latter.

Sidney Golden / Physica A 215 (1995) 181-200

183

non-zero probability of observing a change from the original state occurring by the time the designated time-interval has elapsed. They will differ from each other because of the different observational information that will have been incorporated into them. When all the time-intervals between successive observations throughout a fixed lapse of time become vanishingly small in duration, while increasing in number without limit, the probabilities of change from the original state which presumably could be observed to occur during each of them also become vanishing small. Moreover, the limiting multisurvival-probability - to be termed here a persistence-probability - then appears to attain a value of unity, implying that a change from the original state certainly will not occur during the entire lapse of time simply as the result of the foregoing conceived sequence of observations. Contrasting with an intuitive expectation that merely observing the radioactive system as described should be without physical influence on its disintegration and that some change from the original state then should still be possible, this behavior has been termed "The Zeno's Paradox in Quantum Theory" 1-18, 22-31]. It also has been described as "The Watched-Pot Never Boils Theorem" [28] and "The Watchdog Effect" [28, 31], reflecting a view that such a paradox actually does not occur 2 in quantum mechanics. Regardless of what this behavior may be called, its major implication is clearly bizarre: Experimentally occurring decompositions of a radioactive system would be completely unaccounted for theoretically whenever it is subjected to quasi-continuous 3 observations (which would apparently fail to reveal them)! As a result, much attention has been given to determine possible limitations of the quantum Zeno's paradox [18, 23-31]. Despite the formal rigor and generality of its derivation [22], it appears not to arise in some simple models [25], nor for very short elapsed times 1-23, 24, 26] nor when quasi-continuous observations somehow happen to be forbidden 1,29]. Neither does it appear to arise when time-energy indeterminacy is taken into account 1,18] nor when the evolution of the system happens to be describable by a Pauli-type quantum-mechanical master-equation [31]. It has also been suggested that if time-intervals between successive observations were consciously restricted to non-vanishing durations, the apparent temporal discreteness could serve to eliminate the paradox 1-30]. Nevertheless, the paradox must be resolved in some comprehensive way if a rigorous and general quantum-mechanical support for any sort of experimentally-observed temporal decay of radioactive decompositions - exponential or otherwise - is to be forthcoming. One aspect of the quantum Zeno's paradox that should be noted in this regard is that a persistence-probability is not a conventional time-dependent expectation value of a time-independent observable. It involves, instead, a limiting sequence of 21 am indebted to Professor Michael J. White for pointing our that the behavior does appear to be a special case of what is often known as "Zeno's paradox of measure" 1-331. 3 The use here of the term "quasti-continuous" is intended to emphasize that the observations leading to the quantum Zeno's paradox actually are not temporally continuous but are instead a limit-set of temporally-discrete ones.

184

Sidney Golden / Physica A 215 (1995) 181-200

time-dependent conditional probabilities which is treated by current quantummechanical theory in a manner that may merit further examination in order to possibly resolve the paradox [22]. A further aspect to be noted is that the evolution of quantum systems subjected to quasi-continuous observations may not be properly describable in conventional quantum-mechanical terms, although a radically modified alternative theory has not been regarded to be a satisfactory means of possibly resolving the paradox [22]. There are several reasons which prompt the consideration of an alternative theory, however. Perhaps the primary one is the unquestioned temporal discreteness with which radioactive decompositions occur [5], which is suggestive of a possibly fundamental temporal discreteness prevailing in these processes. For another, it has been known for some time [34] that a probabilistic theory of intermittent forces which are applied to atoms over lapses of time which are long compared to those of the intermittent time-intervals can yield quite the same effects as those which are obtainable from continuously applied forces. In such cases, there would be no need for a temporally-continuous dynamical description of the systems to account for the resultant force-related effects. For still another, as has been noted previously [30], a conscious restriction to temporal discreteness could possibly serve to obviate the paradox. Furthermore, in this regard, it has been shown recently [35, 36] that the strict-irreversibility of isolated and localized non-relativistic quantum systems implies that their evolution must be expressible in terms suggestive of an intrinsic temporally-quantized discreteness. Since radioactive processes give every appearance of evolving in a strictly-irreversible manner, such a theory would seem to be appropriate for their description. One aim of the present paper is to determine the necessary and sufficient temporal conditions for the Zeno's paradox in quantum theory to occur generally and to show how the negation of them implies the aforementioned temporal-discreteness of the requisite evolutionary theory. Another more important aim is to show in some detail how such a temporally-quantized dynamic theory then does provide the sought-after rigorous and general quantum-mechanical support for the exponential temporal decay behavior of seemingly randomly occurring radioactive decompositions with associated characteristic half-lives that is observed experimentally. For this purpose, some preliminary material is collected in Section 2 that pertains to the analysis which is to be undertaken. The necessary and sufficient temporal conditions are determined in Section 3 for the Zeno's paradox of quantum theory to occur throughout the course of a quasi-continuously observed radioactive decomposition of a dynamically-isolated system originally in a single non-stationary state. In Section 4, by negation of the foregoing conditions, it is required heuristically that the quantum Zeno's paradox should not occur for isolated systems in general. As a consequence, it follows that any statistical operator of such a system must then evolve in an abrupt and discontinuous manner suggestive of an intrinsic temporal discreteness. Thereby, the recent theory of temporally-quantized strictly-irreversible

Sidney Golden/Physica A 215 H995) 181 200

185

evolution of dynamically-isolated and localized non-relativistic quantum systems [36] is shown to provide rigorous and general quantum-mechanical support for Rutherford's experimental characterization of radioactive decompositions as consisting of exponential temporal decay processes with characteristic half-lives. His further experimental characterization of them as consisting essentially of Poisson-distributed randomly occurring events is shown, in Section 5, to be a further consequence of the theory. A summary of the results obtained, together with some remarks regarding them, is contained in Section 6. Appendices that examine certain algorithmic aspects of the theory and exemplify its possible applicability to non-radioactive processes, with some suggestions for testing it, conclude the paper.

2. Preliminaries Each system to be considered here is a member of a Gibbsian ensemble consisting of identical single, dynamically-isolated, localized and non-interacting, radioactivelydecomposing non-relativistic quantum systems. For simplicity, the properties of the ensemble often may be expressed as being those of an average one of its constituent systems. The system's time-dependent statistical operator at time t, p(t), is of particular interest. For some purposes it suffices to be expressed in a simple form involving the initial statistical operator at time to, p(to), from which it has evolved, as O(t) = J(t

-

to:p(to)),

to <<. t < ~ .

(2.1)

Here, Y ( t - to;... ) is an appropriate temporal transformer which, determined by its equation-of-motion for the indicated lapse of time, ensures that the statistical operator will be Hermitian, non-negative and of unit uace 1-35]. In this form, whether or not the statistical operator is presumed to evolve reversibly will be indicated as needed. When it does undergo a strictly-irreversible evolution, the statistical operator can be more usefully expressed in terms of those of its matrix elements which are evaluated in the basis of energy-eigenfunctions of the system's time-independent Hamiltonian H, {1~ )}, viz., all quantities henceforth being in atomic units, p(t) = ~ I~,.><~,.I o(t)l~'.><~.l,

(2.2)

trl, n

where < ~,.IHI t/'.> = E,. 6,..,

(2.3)

the E's being energy-eigenvalues. Upon recognizing that the matrix elements can evolve independently of each other when the system undergoes strictly-irreversible

Sidney Golden / Physica A 215 (1995) 181-200

186

evolution, we can show [36] that solutions of its equation-of-motion yield

~.,I p(O)l'/'.)

< (~u=[ p(t)[ g'.) = [1 + i Zmn(Em

-

-

En)] Nmn(t) '

t >>-O,

(2.4)

where r,.. = Z.m > 0

(2.5)

is the basic time-interval between successive changes of the matrix element that depends on the pertinent energy-eigenstates through the quantization relation (z,..) 2 (E. - E.) 2 = F 2,

(2.6)

with F a universal constant [36]. The integer exponent Nr,.(t) is given by Nm.(t,=

k=~ O ( @ - k ) ,

(2.7,

with O (...) being the Heaviside unit-function of its argument, so that the set of successive time-instants (not exceeding tu(mn), the designated final time-instant) at which the matrix element changes is [t(mn)] - I N z,,.],

N = O, 1,... ,N,..(t).

(2.8)

The set of all time-instants at which the statistical operator of the system changes is the union of the preceding set for all pairs of energy-eigenstates, viz., [ t N ] - U [t(mn)],

(2.9)

m, n

but exhibiting their successive values explicitly is not feasible without a knowledge of the values of the ~,...

3. A Persistence-probability theorem and the quantum Zeno's paradox We now proceed to determine the necessary and sufficient temporal conditions for the Zeno's paradox of quantum theory to prevail generally. For this purpose, we suppose [t,] to be a stipulated set of successive time-instants from each of which intrinsic changes in the statistical operator of a system can originate, and that t. + 1 -- t. -- z. - z(t.)/> 0,

to = 0 ~ n < N -- 1,

(3.1)

with N-1

Y', ~k = tN, k=O

(3.2)

Sidney Golden / Physica A 215 (1995) 181-200

187

the time-interval of interest. No restrictions as in Eqs. (2.6)-(2.9) are either imposed or implied at this stage. There can be additional successive time-instants intermediate to those indicated from which intrinsic changes in the statistical operator also may originate, but these will be specified as necessary. With no undue loss of generality for the purpose at hand, we first imagine the system-ensemble being prepared initially to be in an arbitrary non-stationary pure state, 0., corresponding to the original statistical operator of the system 0(0)

= O',

O" = 0 .2 =

0.*,

Tr(0.) = 1.

(3.3)

Then, immediately after the time-interval tN has elapsed, the survival-probability of observing the system to be in that state again, despite the evolutionary changes of its statistical operator, is given by s(tN) = Tr{0.~--(tN; 0.)}.

(3.4)

No strictly-irreversible evolution restriction is required or implied here. We next imagine observing the original system-ensemble at each time-instant t, and immediately thereafter selecting that subensemble of the system which would have been observed to manifest no change from the original state. For these, we have the statistical operators

O(t.)=o, o = o 2 = o t, T r ( o ) = l ,

0~
(3.5)

each of which then proceeds to evolve throughout the following time-interval r,. Immediately after the total time-interval tN has elapsed, the multisurvival-probability of observing the system to be in that state again, despite the successive evolutionary changes of its statistical operator, is given by N-1

S(tN) = H s(z.).

(3.6)

.=0

Note that 0 < s(z,) ~ 1,

(3.7)

so that 0 < S(tN) ~< I,

(3.8)

the equality holding only when the indicated time-lapses vanish. Again, no restriction here to strictly-irreversible evolution is required or implied. In view of Eqs. (3.7), (3.8), we may take the logarithm of the multisurvivalprobability and obtain N-1

In S(t~)= ~ lns(z.). fl=O

(3.9)

Sidney Golden / Physica A 215 (1995) 181-200

188

Then, presuming the survival-probability, s(z,), to be a continuous function of its arguments, but allowing, by Eqs. (3.1), (3.2), that the elapsed time, t, can be a discontinuous function of the zn's, we express Eq. (3.9) in terms of a Stieltjes integral, tN

lnS(tN) = f dt lns(z(t)) z(t-~--<~O,

(3.10)

lo

the bound resulting from the fact that the integrand is always non-positive. As required for the quantum Zeno's paradox, the persistence-probability then will attain the value of unity, P(tN) = lim S(tN) = 1, ~(t)~O

tN fixed,

(3.11)

if and only if the corresponding logarithms vanish, so that lns(z(t)) (ds(T)~ lim = \ dz ]~=o : 0 ' ~(t)-~o z~

O~t<~tN,

(3.12)

which evidently implies Eq. (3.11) in turn. The general occurrence of the Zeno's paradox of quantum theory is thus ensured as the consequence of a Persistence-probability theorem which we have just established: An evolving dynamically-isolated quantum system will be observed quasi-continuously to persist in any original non-stationary state with certainty throughout all finite non-zero intervals of time: If and only if the probability of the system remaining in any non-stationary state has vanishing temporal rates at all time-instants,from which its changes can occur only continuously.

4. Quantum Zeno's paradox obviation and exponential radioactive decay

Since there appears to be no direct physical evidence attesting to its actual presence, we shall heuristically suppose that the Zeno's paradox of quantum theory cannot arise in any dynamically-isolated quantum system of interest here. There evidently can be no such paradox if these systems evolve in a manner that negates Eqs. (3.11), (3.12), so that its absence can be ensured as a consequence of a nonpersistence-probability theorem which follows from that negation: An evolving dynamically-isolated quantum system will be observed quasi-continuously not to persist in any original non-stationary state with certainty throughout all finite non-zero intervals of time: I f and only if the probability of the system remaining in any non-stationary state has non-vanishing temporal rates just at time-instants from which its chan#es can occur only discontinuously. The time-dependence of a survival-probability, e.g., Eq (3.4), is entirely that of the temporal transformer. The foregoing conditions which serve to obviate the quantum

Sidney Golden/Physica A 215 (1995) 181--200

189

Zeno's paradox for arbitrary non-stationary states must then pertain to the temporal transformer alone and apply, therefore, to the evolution of any statistical operator of the system. These conditions - and more, in fact - are satisfied by statistical operators that conform to Eqs. (2.2)-(2.9) in the course of their strictly-irreversible evolution. Although such statistical operators then do not change at all at time-instants intermediate to those at which they do change abruptly and discontinuously, and quasi-continuous observations which may be conducted throughout any such open interval will yield survival-probabilities of unity pertaining thereto, the paradox then does not arise. For it to do so, the system's statistical operators must be able to undergo changes throughout the intervals - which they cannot, under the current temporally-discrete circumstances. When the strictly-irreversible equations are supplemented by the information that is provided by quasi-continuous observations of the system, they yield the exponential temporal decay behavior which we seek for radioactive decompositions, as we now show. For generality, we associate the original statistical operator of a decomposing radioactive system with a manifold of mutually orthogonal "unstable" states that spans an appropriate Hilbert-subspace of the system, viz.,

o'=~a,, n

~'1 < oo,

(4.1)

n

where G"n ~

2 ~ rr.*, Tr((r.)

G"n

1, Vn,

(4.2)

and rr.(rm = a. fro.,

(4.3)

p(0) = cr/Tr(cr),

(4.4)

with

so that Tr{cr p(0)} = 1.

(4.5)

The matrix elements of the statistical operator, Eqs. (2.4)-(2.7), may evolve independently of each other [36] and make individual contributions to the multisurvivalprobability of interest here as they do. Although it is possible that some matrix elements, apart from their complex-conjugates, may not evolve independently of others, we shall assume that they all do, for reasons of simplicity; this entails no undue loss of generality, as we later point out. Thus, presuming that the system has been observed not to have changed from its original state (manifold) immediately after both ( ~u,,[p(0)l ~,> and < ~,lp(0)l ~,.> have changed at zm,, the basic survival-probability at

Sidney Golden / Physica A 215 (1995) 181-200

190

that time-instant is S(Zm.) = Tr{tr~--(Zm.; a/Tr(a))},

(4.6)

which yields, after straightforward manipulation, 2F 2

s(rm,) = 1

1 + F2

I( ~ , . l a l ~ , . ) l z Tr(a)

(4.7)

Because of temporal quantization, Eq. (2.6), the basic survival-probabilities due to successive changes of just these matrix elements will be the same for each of their succeeding basic time-intervals that elapses. As a result, their combined contribution to the multisurvival-probability will be Sm.(t) = [S(Tm.)]Nm""),

(4.8)

where Nm.(t) is given by Eq. (2.7). Since no changes can occur in the (open) intervals between the temporal discontinuities of the matrix element, the persistence-probability that we seek now becomes P(t) -= 1-[ Sm.(t) = I-[ [S(Zm")]Nmn")" m>~n

(4.9)

m>~n

Upon taking logarithms, we obtain In S(Zm.) In P(t) = ~ Nm.(t) " ~ m n m >~n

(4.10)

Tran

Since it is readily verified from Eq. (2.7) that 1 >1 Nmn(t )

"~mn/t > 1 -- Zmn/t ,

Vm, n

(4.11)

the equality being attained only at those time-instants when the pertinent statistical operator matrix element undergoes an abrupt and discontinuous change, it is evident that the persistence-probability is a discontinuous function of the elapsed time. From Eq. (4.7), the survival-probabilities do not exceed unity. As a result, it is readily established that In P(t) = - Kt - D ( t ) ,

(4.12)

where the decay constant, K -

-

~

lns(rm.) > / 0 ,

m>~n

(4.13)

Tran

is evidently characteristic of the system and is independent of the elapsed time, and D(t), 0 ~< D(t) < -

~ lns(zm.), ra>n

(4.14)

Sidney Golden / Physica A 215 (1995) 181-200

191

is a discontinuous function of the elapsed time characteristic of the system, in accord with Eqs. (4.10) and (4.11). The bounds are temporally independent and, as shown in Appendix A, establish that the extent to which the discontinuous behavior may intrude upon the exponential temporal decay behavior will be relatively small. Eqs. (4.12)-(4.14) clearly provide the rigorous and general quantum-mechanical confirmation of Rutherford's original characterization of radioactive decompositions as exponential temporal decay processes with associated characteristic half-lives that we have sought. It remains to establish their constitution of seemingly random occurring events, to which we now turn.

5. Essential randomness of radioactive decompositions

When some of the basic time-intervals of a system happen to be incommensurable with each other, the successive time-instants at which its statistical operator can change will occur irregularly with the passage of time. In radioactive systems, there is a probability that the system can be observed to decompose at each of these instants so that the ensuing decompositional behavior can have an appearance of "randomness" as the system evolves. When the basic time-intervals involved are few in number or commensurable, however, there may be no apparent irregularity to be observed. Indeed, in marked contrast, changes could appear to occur quite regularly, as in the extreme situation of a two-state system described in Appendix B. However, apart from appearances, an essential randomness underlying radioactive disintegrations in more general circumstances can be inferred from the present theory. For this purpose, we determine the probability-distribution of the number of distinct radioactive decompositions which may be observed to occur in a fixed interval of time of appropriately short but observable duration. From Eqs. (4.7) and (4.13) and some straightforward manipulation, e.g. like that leading to Eq. (A.5), the decay constant can be expressed adequately as r2 K ~ 1 + F2

1 l( ~ . l t r l ~ . ) l z,~. Tr(~)

2

(5.1)

Since the matrix-element factors sum to unity, K evidently is inversely proportional to an appropriate harmonic mean of the basic time-intervals of the system. When the time-lapse, t, greatly exceeds each of the basic time-intervals, a related such quantity is an effective-time between successive changes of the statistical operator, ( r ) , which we can define with the aid of Eqs. (2.7), (2.8) as t Em>~. Nm.(t)'

(z)-

-~

1

Y~m~>.(1/~'mn)'

0
Vz..<
(5.2)

Sidney Golden / Physica A 215 (1995) 181-200

192

assuming that the sums exist. Because of possible coincidences when changes of the various matrix elements occur, the effective-time cannot exceed and may even be considerably less than the mean-time between successive changes of the statistical operator, ( ( z ) ) , viz., ( z ) ~< ( ( z ) ) - t/N(t),

(5.3)

where N(t) is the number of time-instants indicated in Eq. (2.9). Ordinarily, however, we may expect them both to be of comparable magnitude. Upon identifying the decay-time of the process, T1/~, with the reciprocal of the decay constant, I/K, the ratio of the effective-time to the decay-time is F2 1 I(~mlal~)12(~,~.~) (T1/~ z) = (z)K ~ ~ Tr(tr) . = 1 + F 2m, n T~mn

-1

(5.4)

Presuming that the matrix-element contributions are each much less than unity and that the value of F can be estimated to be about 0.21 [36], we can expect that ( z ) <<0.084 T1/e,

(5.5)

so that ( z ) will be considerably less than the decay-time of the system T1/e. As a result, time-lapses that are much smaller in duration than the decay-time can still include a significant number of time-instants at which the system's statistical operator may change from its original state. In accord with Eqs. (3.1), (3.2), we now choose a time-interval tN, consisting of N consecutive time-instants, Its], when the statistical operator can change, such that tN -- ( ( z ) ) N < < I / K

I<
= T1/e,

(5.6)

to a good approximation. This means that few changes of the system's statistical operator from its original condition probably could be observed to occur during such an elapsed time-interval. Because the time-instants which are included therein are final termini of basic time-intervals of the system, they can be associated with the sequence of basic survival-probabilities, [s(tn)], that implicitly correlate with the basic time-intervals involved, e.g., Eq. (4.6). They can likewise be associated with a sequence of conditional basic decomposition-probabilities, [p(tn)], for the decompositions that could be observed instead to occur immediately following a sequence of survivals. These are expressible in terms of the associated multisurvival-probabilities of Eq. (3.6) as

[p(t,)] = ~ s(t"-x) (1 c s(t.)] _ [1 -- s(t.)].

1

s(tn_ 1)

(5.7)

1

In terms of the foregoing, the conditional probability that a distinct decomposition of the system-ensemble could be observed to occur immediately following each of

Sidney Golden/Physica A 215 (1995) 181-200

193

n time-instants (among the total of N in the time-interval) is

the notation indicating a summation over permutations of the factors [37] that are distinct. All time-intervals of the same duration may not have the same set of time-instants associated with them, so that this probability can depend on their actual choice. Guided by what is done experimentally, whereby large numbers of time-intervals of the same duration are employed to determine it in some sort of average sense, we shall restrict our attention to those radioactive systems for which it can be justified to be essentially independent of any particular time-interval of the fixed duration that may be chosen. In that case, since the time-instants in any one of them could have basic time-interval termini that differ from those in another, it would follow that all the basic decomposition-probabilities must then have essentially the same value, ( p ) , ascribed to them, regardless of when they originate. With this simplification, we obtain W(N, n) ~

(N)! ( N - n)!n!

(p)n.

(5.9)

For n<< N, after straightforward manipulation, Sterling's approximation yields ln(N-

n)~ ~ n l n N + (9 ~

,

(5.10)

so that w ( g , n) ,~

(N(p))" n!

(5.11)

n<
In order to eliminate the need to know the number of time-instants, N, which are involved, we normalize W(N, n) so that some decomposition should be observed to occur within the chosen time-interval with certainty, viz., we take N

y~ C~W(N, n) = 1,

(5.12)

n=0

where CN is a normalization constant. Then, for sufficiently large N, the sum can be approximated with little loss of accuracy by extending it indefinitely and we may take CN ~ exp( - ( n ) ) ,

with

( n ) - N(p><
(5.13)

where ( n ) now is the mean number of "successful events" - distinct radioactive decompositions in the present case - probably to be observed as occurring in the

194

Sidney Golden/ Physica A 215 (1995) 181-200

chosen time-interval. As a result, we finally obtain the distribution that we seek, viz., CN W(n) ,~

" exp(--(n>) n! '

(5.14)

which evidently no longer exhibits explicit dependence on N. Since any collection of several identical, non-interacting radioactively-decomposing systems itself can be regarded as a subensemble of the individual constituents, the probabilistic behavior of the collection should be precisely the same as that of an individual constituent system. As a consequence, the foregoing distribution should be applicable to more than single radioactive systems, upon which the foregoing derivation has been based. As it turns out, Eq. (5.14) is just what Rutherford and Geiger found experimentally with a sample of polonium for the radioactive decompositions occurring in time-intervals of extremely short duration as compared to that of its half-life [6]. This equation is the familiar Poisson distribution formula for equally probable randomly occurring events [38, 39]. Although Bateman derived it theoretically on the basis of a temporally-continuous theory [6], it likewise characterizes the randomness which derives from the present temporally-quantized theory for the aforementioned experimentally-restricted time-intervals. Thereby, it brings our search for a rigorous and general quantum-mechanical confirmation of Rutherford's characterization of radioactive decomposition processes to a successful conclusions.

6. Summary and remarks In addition to obviating the celebrated Zeno's paradox of quantum theory in the course of having done so, the primary objective of finding a rigorous and general quantum-mechanical treatment of radioactive decompositions which is in full accord with their observed behavior has been achieved. There are, however, several aspects of this accomplishment that merit comment. (1) It is to be emphasized that the recently developed temporally-quantized dynamical theory of strictly-irreversible evolution of isolated and localized non-relativistic quantum systems [36] has yielded results which are fully in accord with Rutherford's characterization of radioactive decompositions as exponential temporal decay processes of randomly occurring events with no supplementary adjustments havin# been introduced into it to do so. An essential contribution to the success of the theory has been its focussing on the persistence-probabilities of the processes rather than on their simple survival-probabilities. An additional contribution to its success has been its regarding radioactive decompositions to occur only at time-instants comprising sets of measure zero in any observable time-interval. Although the heuristically assumed negation of the necessary and sufficient temporal conditions for the occurrence of the quantum Zeno's parodox does imply the abrupt and discontinuous evolution of the pertinent systems, neither

Sidney Golden / Physica A 215 (1995) 181-200

195

temporal-quantization nor evolutionary-independence of the energy-eigenfunctionbased matrix elements of the system's statistical operator appears to be implied similarly. Were both these features of the theory not intrinsic to strict-irreversibility which they actually are - it seems likely that the structural form for the persistence-probability expressed in Eq. (4.9) might not be obtained. In such a case, both the temporal-quantization and the cited evolutionary-independence would appear to be necessary for deriving the exponential temporal decay behavior of radioactive decompositions and thus would comprise further essential contributions to the success of the theory. This aspect of the theory merits further examination. (2) If taken literally, the simplication preceding Eq. (5.9) leading to the Poisson distribution strictly implies that all the energy-eigenstate-evaluated off-diagonal matrix elements of the original statistical operator determining the survivalprobabilities must have the same magnitude. This ensures that these probabilities at all time-instants will be the same, regardless of when they are determined. In such cases, the original statistical operator requires its sole non-stationary contribution to be proportional to an appropriate pure state having such matrix elements. If the Poisson distribution should prove to be a necessary feature of all radioactive decompositions, an important aspect of these processes could well be that the initial statistical operators of the decomposing systems must be the consequence of contributions which arise solely from single non-stationary pure states that are characteristic of them. To what extent the Poisson distribution may be found to be necessary in general thus merits further examination from both the experimental and theoretical viewpoints.4 (3) Some idea of the algorithmic aspects which may be involved in the theory and the sort of information that is needed to carry out theoretical calculations of decay behavior which may be compared with experimental results is given briefly in Appendix A. Although radioactive disintegrations can be expected to occur discontinuously with the passage of time, an upper limit estimate of the extent to which that behavior may intrude upon the exponential temporal decay behavior also is given there which shows it to be so small that it may be difficult to observe. (4) The assumption made to avoid possible multiplicity of various energyeigenvalue differences in Eqs. (4.6), (4.7) can be eliminated by replacing the squares of the matrix elements there by appropriate sums of them which have the same energy-eigenvalue difference, with no alteration of the conclusions which have been reached. In the context of Eq. (A.5) they are already included automatically. Another assumption involves the restriction to a discrete formalism for the theory. In the sense that a sufficiently large finite enclosure for the systems can adequately approximate a continuum of interest, there appears to be no need for concern that the results do not involve a continuum of energy-eigenstates. Nevertheless, this aspect of the theory also merits further examination. -

'*See Ref. [39] for a treatment of several aspects pertaining thereto.

196

Sidney Golden/Physica A 215 (1995) 181-200

(5) Apart from their assumed localized behavior, no explicit properties of various eigenfunctions of the system, either those of an original "unstable" manifold of states or those of its energy-eigenstates, have been employed in the foregoing theory. As a result, whether or not the theory needs to be restricted to radioactive decomposition processes can be questioned. In fact, since the persistence-probability actually amounts to "the conditional probability that a dynamically-isolated and localized non-relativistic system will remain in its original condition throughout a designated lapse of time", it would appear not to be so restricted. Accordingly, in order to illustrate the precise temporal behavior that may be expected from the theory in perhaps the simplest of situations, it can be useful to consider a system which has only two energy-eigenstates and which is prepared initially to be in some non-stationary pure state. This is done in Appendix B and shows that such a system does indeed exhibit the expected exponential temporal decay, with discontinuities that are small in effect. This behavior contrasts markedly with that determined by conventional quantum theory for a model of a two-level atomic system in the course of its spontaneous emission of radiation [21], which gives long-time deviation from exponential temporal decay. (6) Finally, we note that the measuring process in the present theory has not taken on anything like its traditional role in quantum mechanics [40]. In fact, we have concerned ourselves exclusively with observations which are to be made on a system that is dynamically isolated rather than with the distinctly different measurements which are to be made on it when it is in interaction with an appropriate measuring apparatus [29]. Moreover, the only observations that have been considered here are those of a purely theoretical sort which have had to do just with designating certain possible states in which the system might be distributed in its evolving ensemble; in particular, in its original one or in its complement [28, 29]. Because of the probabilistic character of quantum mechanics, the effect of these theoretical observations has been merely to focus attention on those subensembles of the system which could be described as indicated and to make a determination of their fractions, identified as the probabilities of their being observed as such. That focus does not alter an evolved ensemble at all since the various subensembles are still available-or can be produced - for calculating the probable results of other such observations and for the determination of other properties of the entire evolved ensemble, as well. The theoretical situation bears some similarity to the experimental one in which a radioactive system is separated from its decompsition products. Both separated portions are then available for further observation or measurement, if desired, to determine their individual properties or those that relate to the entire evolved system. If the separation happens to affect the properties of the system, its unseparated counterpart must be obtained to determine them, which may not always be feasible experimentally. In the case of the theoretical separation, however, such a counterpart - the entire evolved ensemble - always is.

Sidney Golden/Physica A 215 (1995) 181-200

197

Appendix A

To obtain some idea of the algorithmic aspects of the theory, we examine here the sort of information that is needed to carry out calculations which may be compared with experimental results. An upper limit can be estimated of the extent to which the temporal discreteness of individual radioactive disintegrations will intrude upon their exponential temporal decay behavior. From the form of the survival-probability in Eq. (4.7) and from a well-known bound [41], we can determine that 2/- 2

I(~l~rl~.)l

1 + F2

-Ins(z=,)< 1

2

Tr(o)

2F 2

i(~mlal~, )12

1 + F2

Tr(tr)

(A.1)

Presuming that

2/-2 I<~lal~">12<< 1, 1 +/-2

(A.2)

Tr(a)

we can further obtain by appropriate summation /-2

-

~ lns(z..)< /-2" =~. 1+

(A.3)

A precise value of F is not yet available but an estimate of it is 0.21 [36], so that with this value and Eq. (4.14) we get D(t) < 0.042.

(A.4)

Accordingly, it can be anticipated that the maximum deviation of the persistenceprobability from strict exponential temporal decay may not even be noticeable experimentally since radioactive decay-rate determinations invariably involve measurements which are made over time-intervals that include many individual disintegrations. Some sort of average value of D(t) would be more appropriate and that value can be expected to be somewhat smaller than the foregoing upper-bound estimate. A value of the decay constant, K, is evidently needed in order to compare theoretical and experimental radioactive decay results quantitatively. To the extent that Eqs. (A.1)-(A.3) are adequate approximations for the quantities involved, we can obtain from Eq. (4.13) by straightforward manipulation that F K ~ ~1 + F

~ IEm - E,I

m, n

I< ~ . laJ~,>l 2 Tr (a)

(n.5)

Sidney Golden / Physica A 215 (1995) 181-200

198

Thus, in addition to the energy-eigenvalues and the energy-eigenstate matrix elements of the manifold of "unstable" states of the radioactive system, a reliable value of F is needed to determine the decay constant. The matrix elements appear in what can be described as a "binary distribution of energy-eigenstates" for the initial statistical operator of the system. In that case, the sum can be described as a mean value of the magnitude of the energy-eigenvalue differences occurring in the system, (AE>, in terms of which the decay constants is F 1+

K~ ~

.

(A.6)

The conclusion to be drawn from this expression is that radioactive decay constants should be proportional to an appropriate mean of the magnitude of the energy-eigenvalue differences occurring in the system, the proportionality constant being universal. In the simplest case of systems that consist of just two energy-eigenstates, which we consider in Appendix B, there is a single energy-eigenvalue difference. If that could be altered in a determined manner simply by appropriate external time-independent fields, Eq. (A.6) might provide a possible means of testing the present theory.

Appendix B To illustrate possible non-radioactive decay behavior of a system in a non-stationary state, we consider the extreme situation of a two-state system and suppose that it is describable initially by the non-stationary wave function 14 > so that the original statistical operator is 2

0(0)= ~

l'/'m><~ml~><~l~,><~,,l,

(B.I)

m, n = 1

in accord with Eqs. (2.2), (2.3). By Eq. (4.10) only a single survival-probability is needed, viz., s(z12) = 1

2F 2 1 + F 2f(1 - - f ) '

(B.2)

where f = 1< ~Ull¢>{ 2

(8.3)

is the probability of the system being originally in the indicated energy-eigenstate. By Eqs. (2.6), (2.7) and (4.10), the persistence-probability is In P(t) = In s(z12) ~ O ( t / z , 2 - k ) . k=l

(B.4)

Sidney Golden/Physica A 215 (1995) 181-200

199

It is thus evident that P(t) is an exponentially decreasing step-function of time, the discontinuities occurring regularly at multiples of the basic time-interval z12 • Despite such regularity, the Poisson probability-distribution of changes from the initial state can be expected to prevail as an expression of their randomness. F r o m Eq. (2.6) and the previous estimate of 0.21 for F [36], the survival-probability in Eq. (B.2) becomes In S(Zl2) ~ - 0.084f(1 - f ) ,

(B.5)

so that the decay-time estimated from Eq. (B.4) is T1/e "~ 12 z12/f(1 - f )

>~ 48 z12,

(B.6)

which indicates that the discrete changes of the persistence-probability that can occur regularly with succeeding changes of the statistical o p e r a t o r will be quite small c o m p a r e d to the decay time and thus m a y be difficult to observe. However, in accord with Eq. (2.7), changes in z12 could be effected by changing the energy-eigenvalue difference, and by m a k i n g them of adequately long duration it might then be possible to observe discrete changes - albeit possibly irregular - and thus to test the present theory directly. Alternatively, the decay-time itself could be affected by changing the energy-eigenvalue difference in a k n o w n m a n n e r so that its changes might then be used to determine an independent value of F, which would serve as an indirect test of the theory.

References [1] H. Becquerel, C.R. 122 (1896) 420. [2] See, for example, A. Pais, Inward Bound (Clarendon Press, Oxford, 1986) Chs. 2, 3. [3] Ibid. Ch. 6. [4] E. Rutherford, Philos Mag. 49 (1909) 1. [5] E. Rutherford and H. Geiger, Proc. R. Soc. A 81 (1908) 141. [6] E. Rutherford and H. Geiger, Philos. Mag. 20 (1910) 698, with a Note by H. Bateman. [7] E. Rutherford, Akad. Wiss. Wien. 120.2a (1911) 303. [8] R.G. Winter, Phys. Rev. 126 (1962) 1152. 19] D.C. Butt and A.R. Wilson, J. Phys. A 8 (1972) 1248. 1,10] V. Weisskopf and E. Wigner, Z. Phys. 63 (1930) 54. [11] V.A. Fock and S.N. Krylov, JETP (USSR) 17 (1947) 93. 1,12] M. L6vy, Nuovo Cimento 14 (1959) 612. 113] R.D. Levine, J. Chem. Phys. 44 (1966) 2029. [14] H. Ekstein and A.J.F. Siegert, Ann. Phys. (NY) 68 (1971) 509. [15] M.V. Terent'ev, Ann. Phys. (NY) 74 (1972) 1. 1,16] L. Fonda, G.C. Ghirardi, A. Rimini and T. Weber, Nuovo Cimento 15A (1973) 689. 1,17] P.I. Knight and P.W. Milonni, Phy. Lett. A56 (1976) 275. 1,-18] G.C. Ghirardi, C. Omero, T. Weber and A. Rimini, Nuovo Cimento 52A (1979) 421. 1,19] A. Peres, Ann. Phys. 129 (1980) 33. 1,20] K. Wan and R.G. McLean, J. Phys. A 17 (1984) 825, 837. [21] J. Seke and W.N. Herfort, Phys. Rev A 38 (1988) 833: A 40 (1989) 1926. 1,22] B. Misra and E.C.G. Sudarshan, J. Math. Phys. 18 (1976) 756.

200

Sidney Golden / Physica A 215 (1995) 181-200

1-23] C.B. Chiu, B. Misra and E.C.G. Sudarshan, Phys. Rev. D 16 (1977) 520. 1-24] A. Peres, Am. J. Phys. 48 (1980) 931. [25] K. Kraus, Found. Phys. 11 (1980) 547. 1,26] C.B. Chiu, B. Misra and E.C.G. Sudarshan, Phys. Lett. B 117 (1982) 34. [27] M.D. Srinivas, in: Quantum Probability and Applications to the Quantum Theory of Irreversible Processes, L. Accardi, A. Frigerio and V. Gorini, eds. (Springer, Berlin, 1984) pp. 357, 358. [28] A. Sudbery, Ann. Phys. 157 (1984) 512. 1-29] T. Sudbery, in: Quantum Concepts in Space and Time, R. Penrose and C.J. Isham eds. (Clarendon Press, Oxford, 1986) p. 65. 1-30] P.H. Eberhard, in: Quantum Theory and Pictures of Reality, W. Schommers ed. (Springer, Berlin, 1989) p. 202. 1-3l] E. Joos, Phys. Rev. D 29 (1984) 1626. [32] E. Wigner, in: Foundations of Quantum Mechanics, Proc. Int. School of Physics "Enrico Fermi," Course IL, B. d'Espagnat ed. (Academic Press, New York, 1970) pp. 1-19. [33] B. Skyrms, in: Physics, Philosophy and Psychoanalysis, R.S. Cohen and L. Lauden, eds. (Dreidel, Dordrecht, 1983) pp. 223-254. 1,34] J.J. Thomson, Proc. R. Soc. Edinburgh 46 (1925) 90. [35] S. Golden, Phys. Rev. A 46 (1992) 6805. [36] S. Golden, Physica A 208 (1994) 65. i-37] G.H. Hardy, J.E. Littlewood and G. Prlya, Inequalities (Cambridge Univ. Press, Cambridge, 1978) p. 44. 1-38] S.D. Poisson, Recherches sur la Probabilit6 des Jugements (Bachelier, Paris, 1837) pp. 205 207. 1-39] H. Jeffreys, Theory of Probability (Clarendon, Oxford, 1961) pp. 68-70, 135-137, 319-322. 1,40] J. von Neumann, Mathematical Foundations of Quantum Mechanics, translated by R.T. Beyer (Princeton Univ. Press, Princeton, NJ. 1955) pp. 417-445. I-4l] G.H. Hardy, J.E. Littlewood and G. Prlya, Inequalities (Cambridge Univ. Press, Cambridge, 1978) p. 106.