ON THE DERIVATION OF THE COLLISION LIFETIME MATRIX

TI~XN TKONG G I E N ~ Depart~irczt of Physics, Ohio Uniuersity, Athens, Ohio

Received March 26, 1965

If the lifetime of a collision is defined as the difference between the tirnes in which a particle moves in free space and under an interaction from a scattering center to the outside (or in the reverse direction), this paper shows that exactly the same relation between the lifetime operator Q and the scattering matrix S as was found by Slnith (1960) can be derived. This can be done in a natural way provided that the time operator is defined as + = -in a/aE and the manifold of the asylnptotic outgoing (or incoming) states is chosen to be the set of cor- responding eigenstates. 'The part of the time needed for the particle moving freely from the scattering center to R is equal to R l v .

INTRODUCTION

In a recent paper, Sinith (1960) has successfully formulated a lifetime matrix to build an operator theory for the lifetime of a collision. With his theory, inetastable states of a collision can be pictured as eigenstates of a lifetiine operator having long lifetime eigenvalues.

Since the energy is not well enough defined in a decay process of a metastable state, the observed decay lifetiine is shown to be nothing but the average of the eigenlifetinles over the not-well-defined energies. When the lifetime of a illetastable state is long enough, it is believed that finer details of the lifetime can be measured by studying steady scattering lasting much longer than 7 = fi/217, where l' is the familiar resonance width (Smith 1960). The fact that the lifetime of a long-lived collision may be more accessible to measurement than other scattering properties ma1;es the lifetiine matrix theory significant. The lifetime matrix and its associated eigenfunctions can becoine a suitable framen-ork for discussing illetastable states. Let $,(E) be an eigenfunction of the lifetiine operator Q having an eigenvalue equal to qii. Then i t represents a collision state with a definite internal state of the colliding coinpound par- ticles, \\-it11 definite values of other quantum numbers of the collision system, and ~vith a definite lifetiine eigenvalue. If $k(E) is a state describing a collision process A, it can be written as the superposition of the lifetime eigenstates $z(E) :

If all the #i, except one, represented by #,, have very short lifetimes, the probability that the collision is of the simple type is Xi+, a*xiaxi, while the probability that the collision passes through a resonance intermediate state of lifetiine qkl, is a*xkaxr. The mode of decay of this metastable state mill be described by the outgoing asymptotic form of #,. If the compound metastable

'Present address: Laboratoire de Physique Corpusculaire ?i Haute Energie, Centre d ' ~ t u d e s Nuclkaires de Saclay, BoEte Postale No. 2, Gif-sur-Yvette (S. e t O.), France.

Canadian Journal of Physics. Volume 43 (Xovember, 1965)

1978

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GIEN: COLLISIOX LIFETIME MATRIX 1979

particle in the same state is formed by a different collision mode, represented by +,, the probability of forming this inetastable state could be different ( ~ * , ~ a , ~ ) , but its lifetiille and mode of decay ~vould still be the same.

If there exist two or more long-lived metastable states, different modes of collision will, in general, cause different contributions of these lifetimes to the total lifetiine of the collision process. Since the lifetiines can be identified by half-lives of the decay states, careful analysis of the decay curves could identify them as the superposition of many pure exponential decays corresponding to different eigenlifetimes 1vit11 the contribution coefficients proportional to U * ~ , U ~ , .

The relation between the Q inatrix and the \\ell-lmolvn S matrix has been found to be:

This relation means that Q and S both give the saille information on the scattering process, from quite different points of view. One can consider relation (I) as a definition of one inatrix in terms of the other and vice versa.

With a suitable boundary condition for Q and S, relation (I) call be written in an integral form :

I t has also been shown that relations (I) and (2) still hold for relativistic scattering by Dirac and Klein-Gordon particles (Gien 1965).

In the present paper, we attempt to seek another approach for the lifetime operator Q. We recall that the relation between Q and S (eq. (I)) was derived by Smith (1960) through a definition of the lifetime matrix Q in terins of the steady collision states. Here, we shall show that this relation can be derived directly from a time operator defined as 4 = -iha/dE, provided that the outgoing states are used as corresponding eigenstates. The lifetime of a collision is considered as the difference between t \~.o times, one for a free motion of the particle from the scattering center to the outside and the other for a motion under an interaction.

The one-dimensional case will be considered first. In this case, the lifetime Q will be shown to be equal to fidrl/dE.

Next, relation (I) will be derived for the general case. When applying the time operator 4 = --ifid/aE to an eigenstate, nre shall get the part of the time corresponding to the free motion of the particle from the scattering center to a point x equal to x/v. This time will be called the free collision time to simplify the language. I t is equal to the classical time the particle uses to travel the distance x with velocity v.

I t will be shown that in the forn~ulation of the lifetime matrix, Smith also adopted R/v as the free collision time, although this cannot be clearly seen a t first. The adoption of R/v as the free collision time is not too seriously in conflict with the quantum theory for the case where the lifetiine of the metastable state is long enough.

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In the case of one-dimensional elastic collision, the asjmptotic outgoing waves of a particle n~oving in free space or ~ ~ n d e r an interaction are respectively

zRz ine ikz

If the time operator is defined as 4 = -ifia/aE, n-hen it is applied to the free asyinptotic wave eikr, me get:

x/v is the classical time the particle uses to travel freely from the scattering center to a point x with velocity v. I-Ience, -iiia/aE is the correct time operator for the set of asynlptotic outgoing waves and to = x/v.

When the interaction is present, applying the time operator 4 = -ifia/aE to the corresponding outgoing states eiwe'" gives a free collision time to plus a delay time Q, as will be seen in the following. We get:

Since x/v = to is a free collision time, the lifetiine Q should be given b ~ r :

I t should be noted that when applying -iiia/aE to the asymptotic incoming wave e-ih'z, nTe get:

The minus sign in front of the free collision time x/v means this is a time in the past. (The scattering center is chosen to be the particle position a t time t = 0.) We want to call attention to the fact that 4 = -ifia/aE gives a free collision time only when it operates on the asymptotic incoming \\-ave eci"; this is because we have chosen an asymn~etric form for the asymptotic total wave function :

,$, = e- iRz - ineiRz

In this, the delay effect is given totally by the outgoing part containing the tern1 ein. This term is the source of the delay effect.

With an asymptotic forin of the total collision wave syinmetrical in the incoming and outgoing parts, i t is expected that the t\vo parts will share this delay effect equally. For instance, if the asymptotic forin is chosen to be

the time operator 4 = iiia/aE will give -x/v - +aq/aE and x/v + %aq/aE operating on e-iKze-'?l? and eih'" e %I?.

I n the inore general case, equations (3) and (4) can be written as:

(7a) -ifia+(+)(E, x) /aE = ( 4 0 + Q)+(+)(E, x),

(7b) - i i i a 4 ( + ) ( ~ , x ) / ~ E = &'@+)(E, x),

where $,(+) denotes the asymptotic outgoing states in the presence of a n inter- action and 4(+) is the asyn~ptotic outgoing state of the free motion. &, is the

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G I E S : COLLISIOS LlFhTIME MATRIX 1981

operator corresponding to the free collision time. With the set of states +(+),

the free collision time operator is identical \ ~ i t h -ifid/dE. $0' is a diagonal matrix, \vhose elements represent the free collisioil times. Q is the lifetiille operator.

In the next section, 11-e shall show from eqs. (7a) and (7b) that the relation between Q and S (eq. (I)) call be derived in an obvious \Yay.

TI-IE RELXTIOSSIIIP B E T W E E S T H E Q X S D S MATRICES

In the general case of a many-channel collision, an asj,n~metric asynlptotic forin of a collision state is:

Thus, the asymptotic outgoing- state is

(9) + A ( + ) = Cr SAP+&,(+).

We are going to show that from (7a), (7b), and (9), relation (I) can be derived. In illatrix form, (9) can be written as:

where

+,(+I is the free collision asyinptotic \vave. Now it is necessary to express the operator +o in (7a) in terms of + = ifid/dE. The unitary transformation +(+) = ,!?+(+I changes the operator 40 into ST+&', which is the operator of the free collision time in the new representation (Wigner 1959). But the operator of the free collision time in the new representation is simply -ifid/dE (eq. (7b)). Hence

or equivalently

Equation (7a) can noiv be written as:

Hence the follo~ving operator equation is derived :* *Sote that relation (45) in Smith's paper (19GO) should be written as (14) instead of

Q = -StS.

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1982 C.-\S:\DI=\I*! JOURSAL OF PIIYSICS. VOL. 43. 1SG5

or equivalently

Note that when the asymptotic forin of #(+I is sl-inmetrical in the incoining and outgoing parts, (7a) is to be written as:

Ho\vever, it is not clear how to handle the S matrix analysis for this case. Following are some reinarlts on how 40 and Q operate on #A(+) in eq. (7a). Applied on #h(+), the operator 40 gives C,SA,~O,&,,(+). If the collision is

inelastic, C , Sh,,tO,,, is the average of the free collision times corresponding to all possible channels of reaction. If the collision is elastic, then

In this case, the free collision tiines of all channel reactions are equal to each other:

The operator 40 lvill give a free collision time to, when applied on I t can be seen by (15) that Q is a Hermitian operator provided that S is

unitary. Since Q is a I-Iermitian operator, there could be a set of eigenstates of Q having eigenvalues qii. Equation (13) can be written for one of these eigenfunctions :

If, moreover, the collision is elastic, (17) can be written as:

(18) -ifid#,(+)/dE = ( q i i + to)#i(+).

Note that there is a two-by-two correspondence betlveen #,(+I and S~nith 's lifeti~ne eigenfunction #i. In our approach, a lifetime eigenstate of the collision is represented by an as)~mptotic outgoing wave #,(+). This is not considered as a questionable point, since, in a scattering process, one usually observes all the scattering properties a t the periphery.

CONCLUSION AND DISCUSSION

We have just shown that if the lifetime Q of a collision is defined as the difference between two times, one for a free motion of the particle from the scattering center to outside, and the other for a motion under an interaction, then the expression for Q in terms of the S matrix can be derived in an obvious \\lay, using a time operator defined as 4 = -ifid/dE, provided that the asymptotic outgoing waves are used as the corresponding eigenstates. The metastable states will be represented by eigenstates of a Hermitian lifetime operator Q having sufficiently long lifetimes.

In our approach, the free collision time is a result of the operation -ifid/dE

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GIEN: COLLISIOS LIFETIME MATRIX 1983

on the eigenstates and is equal to R/v. This is the classical time which the particle uses to cross a distance R with a velocity v. This point should be studied to see whether or not it seriously conflicts with quantum philosophy.

I t will be sho~vn that this was also assumed indirectly in the for~~lulation of the lifetime n~at r ix by Smith (1960). We simplify matters b17 considering the one-dimensional case :

We shall show that the two averaging processes involved in this formulation of Q are to approxirnate the free collision times as their classical values 2R/v.

The term

LRrn++,+dr + I Flux I

has the dimension of time and is equal to 2R/v - (fil2E)sin 2kR. The factor 2R/v is the free collision time the particle uses to cross a distance R toward the scattering center and back to the starting point, while (fi/2E)sin 2kR makes the free collision time fluctuate around 2R/v by an amount equal to fi/2E. This fluctuation is believed to be due to the fact that when the position R is fixed, the free collisioll time is not well defined within this range. Averaging over R makes this oscillating term vanish, which means one has considered only the classical free collision time 2R/v.

Similarly, the remaining term,

is equal to Tza?/aE + 2R/v - (fi/2E)sin(2kR + ?). The first term is the delay time, the second term is the classical free collision time, and the third term is oscillatory. Averaging over R makes this last term vanish. This term is not important if the lifetime fiaq/aE is long enough in comparison with fi/2E.

Averaging over R means that \\re do not specify a certain fixed value of R ; rather, we consider only the average values of these quantities over a range of R.

Hence, the lifetime Q should be written as:

where (2t) and (2to) indicate respectively R { I R + + + d x + \Flux and 1 (m++rn+)dx i IFluxl.

In the forinulation of Q, the approximation of the free collision times is not apparent, since in Q these times cancel each other.

I t has just been shown that even in the forillulation of the Q matrix by Smith, the free collision tiines were also taken equal to their classical values of 2R/v. This adds support to our approach, since we have also made this assu~llption concerning this part of the time.

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1984 CAXADIAS JOLRS.\L 01; PIIYSICS. 1'01,. 43, l9G;

Therefore, as \\;as discussed by Smith (1960), the collisions can be classified into three tj7pes depending on nillether Q >> fi/2E, 1Q1 < fi/2E, or Q << -fi/2E. If IQ) < fi/2E, we are in the domain of the special quanta1 resonance effect; if Q << - f i /2E, these are rapid collisions across the iilteractioil range; and i f Q >> fi/2E, i t becomes meaningful to discuss the collision in terms of the metas- table states and then it is safe enough to approximate the free collision time by its classical value 2R/v.

The author wishes to thank Dr. Lawrence J. Gallaher for some helpful discussions and Mr. Thomas EIess who helped in revising the manuscript.

REFERENCES

B O H ~ I , D. 1951. Quantum theory (Prentice I-Iall, Sew York). GIEN, TRAN 'I'. 1965. J. Math. Phys. 6, 671. GOLBEKGER, M. L., LEWIS, I-I. W., and LVATSON, I<. M. 196:3. Phys. Rev. 132, 2764. GOLBEIIGEK. M. I-. and \VATSON. K. M. 1964. Phvs. Rev. 134. B919.

1964n. Collision theory (John Wiley and sons, Inc., Sew York). -- 1964b. Phys. Rev. 136, B147'3. -- 1965. Phys. Rev. 137, B1396. S ~ I I ~ H , FELIX T. 1960. Phys. Rev. 118, 349. \\;lGNEtI, E. P. 1959. Group theory (.Academic Press, Inc., Sew Yorli). \\TATSON, K. M. 1960. Phys. Rev. 118, 886. \VU, TA-YOU and OHAIUIIA, T. 196'3. Quantum scattering (Prentice Hall, Engle\vood

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