delay time and phase of the initial state
TRANSCRIPT
Canadian Journal of Physics VOLUME 48 MARCH 15, 1970 NUMBER 6
Delay time and phase of the initial state
TRAN TRONG GIEN' Departn~etzt of Physics, Metnorial Ut~iuersity of Nelvfouncllat~cl, St. Jolztz's, Ne~vfoundlatld
Received August 12, 1969
The study presented in this paper is intended to show how to calculate the average delay time of a scattering within the framework of the delay time operator Q in order to prove that the average delay time is independent from the phase of the initial state. This conclusion differs from Ohmura's assertion in a recent paper that in the three-dimensional scattering, the average delay time depends on the phase change of the initial state. Our result saves the hope that the delay time of a collision can be known in ternis of the S matrix alone. The reason for the difference between our result and Ohniura's is also discussed. We also investigate the commutation of the & and S matrices and search for the interesting consequences of this commutation.
Canadian Journal of Physics, 48, 639 (1970)
I. Introduction
There have been attempts to search for a new theory to replace the conventional description of the dynamics of a physical system where an equation of motion is used to represent its evolution. Many physicists believe that such a conventional description of a scattering process often proves itself to be inadequate. For instance, while the Schrodinger equation
determines the change of the state function Y for an arbitrarily small time interval dt, it is, however, quite impossible to verify [ l ] in such an infinitesi- mal time interval, since the macroscopic process of measurement may prevent our ability to do so.
Heisenberg proposed that the new dynamics of the scattering may be found in a theory yielding directly the scattering matrix S which relates the states of the system in the remote past and future (Heisenberg 1943). Many theories have been developed in order to obtain the S matrix without the Schrodinger equation (Chew and Goldberger
'Supported by Operating Grant A-3962 of the National Research Council of Canada.
1961). One tries to calculate the observed quanti- ties in terms of the S matrix done. Efforts have been made in obtaining the delay time and spatial separation induced by scattering in terms of the S matrix (Smith 1960; Gien 1965a; Goldberger and Watson 1962; Goldberger, Watson, and Froissart 1963). Although the complete success of these efforts still remains to be seen, such attempts would be nice, since one may calculate, for instance, the delay time of the scattering without using the concept of the delay of the wave packet motion which is the by-product of the equation of motion theory (Wigner 1955; Bohm 1951). Since the Smatrix does not contain the space and time variables, if the relations between the S matrix and quantities concerning the space-time development of the collision phenomenon can be found, this should-be a very interesting theoretical achievement.
The delay time of a collision may also be regarded as an observable physical quantity which is represented by an operator in quantum mechanics (Gien 19658, 1969). The average delay time can, therefore, be obtained by calculating the expectation value of the operator Q. We have been successful in finding the explicit form for the delay time operator. It turns out that Q can be expressed in terms of the S matrix alone.
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640 CANADIAN JOURNAL OF
In a recent paper, Ohmura (1964), using the concept of the delay of a wave packet, finds that the calculation of the average delay time in terms of the S matrix alone might be impossible in the three-dimensional collision, since in this case the average delay time might depend on the phase change of the initial wave amplitude. If this should be right, all the above efforts would lose much interest. I t is, therefore, interesting to see within the framework of the delay time operator (Gien 1965b, 1969) whether the average delay time depends on the phase of the initial state or not. One of the aims of this paper is to answer the above fundamental question. In the following study, we find that within the framework of our delay time operator, the average delay time induced by scattering does not depend on the phase change of the initial state. We also criticize the way of calculating the average "time of motion" of a scattering by using only the scattered parts of the outgoing states as Ohmura (1964) did. The possibility of the commutation between the delay time operator Q and the S matrix will also be investigated. Useful consequences of this commutation will then be discussed.
In Section 11, we shall present the method of calculation of the average delay time in the one- dimensional scattering case. This part serves as a guide for the calculations of the average delay time in the more con~plicated cases. The mean life of the resonance scattering is also calculated. Section I11 is devoted to the proof of the coMmu- tation between the delay time operator Q and the Smatrix. Due to this commutation, the resonance may be defined with a set of definite quantum numbers such as spin, isotopic spin, parity, etc., if the resonance is assumed to be a lifetime eigen- state with an eigenvalue qi(E) in the resonance form (Section IV). We also show that the scattered parts of the outgoing scattering states may be used to represent the linear space of state vectorsin which the delay time operator is defined, but for a linear space of state vectors in which the "time of motion" operator ia/aE is defined, one must use either the precollision states or the out- going scattering states (Section V). In Section VI, we shall show how to calculate the average delay time of a collision in the general cases. The three- dimensional collision will be studied in particular to show that its average delay time is indeed independent from the phase of the initial state.
PHYSICS. VOL. 48, 1970
11. Average Delay Time of the One-Dimensional Elastic Scattering
The average delay time of a collision is given by (Gien 1969)
[2 I <Q> = ( y , Q y )
= j Y+(E)Q(E)Y(E) d~
where Q(E) is the lifetime (or delay time) oper- ator (Smith 1960; Gien 1965a, b, 1969)
and Y(E) is the energy representation of the physical state already under the effect of the interaction
~ 4 1 Y(E) = s(E)@(E)
Y(E) is normalized to be 1,
In the one-dimensional elastic scattering, S(E) = e2'VE). Hence
and
This is an expected result, since 2aFlaE is the delay time induced by scattering of a state with the energy known definitely to be E and the average delay time is obtained by averaging 2a6/aE over the energy distribution IY(E)I2 of the collision state.
One may, however, be interested only in the average delay time of the scattered states, i.e., the scattered parts of the outgoing states Y(E). These scattered states Y S ( ~ ) represent the particles which have been scattered by the interaction:
In the one-dimensional scattering
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GIEN: DELAY TIME AND PHASE OF THE INITIAL STATE 641
v ~ ( E ) = (e2iS'E) - l)@(E) I- I-2 dE
= 2ieis sin F@(E) (E - E ~ ) ~ + - One obtains, 4
S a8 or IYYE)I'~ aE dE I-
[91 (Q), = S IYS(~)I2 dE I-2 dE
(E - ~ 0 ) ~ + - The average delay time of the scattered states in
4
the one-dimensional scattering is given by The precollision state QReS(E) is believed to contain a factor
1 1 1 [lo] (Q)s = sin2 F I ~ ( E ) I ~ dE I-
L The resonance scattering is a case of special i.e., it must be prepared in such a way that it is
interest. In a resonance scattering, an interme- favorable for the formation of a metastable state diate metastable state is formed before it is (Goldberger and Watson 1964). Coldberger and disintegrated into outgoing particles. The corre- Watson believe that the desire of representing a sponding S matrix is in the following form spatially confined decaying system requires the
L
The lifetime QRes(E) is
In order to calculate the mean life of the reso- nance, one may use either [7] or [lo], depending on how one defines a resonance. On the one hand, if the resonance is regarded as an eigenstate of the lifetime operator Q(E) which has an eigenvalue q(E) in the special form
presence in the asymptotic wave amplitude mRe,(E) of a factor which will permit such a description and the factor 1/D *(E) is sufficient to insure the possibility of describing a localized state and to reflect the presyce of a resonance. mRCs(E) should therefore be in the form
where I-
D*(E) = E - Eo - i - and lg(E)j2 2
is a very slowly varying function of E. The uncertainty AE of lg(E)I2 must be very great compared with I- so that the delay time of the metastable can be observed. The mean life of the metastable is
the resonance state can be represented by I- . -
where SRcs (E) is in the form of [l 11 and a,,, (E) ' 4
is the corresponding precollision state in the @R,,(E) is normalized to be 1, energy representation. The mean delay time of the resonance scattering is equal to the average of [19] S 'g(E)'2
dE = 1 q(E) over the energy distribution of the state (E - ~ 0 ) ~ + - VRes( E), 4
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642 CANADIAN JOURNAL O F PHYSICS. VOL. 48, 1970
Since lg(E)I2 is a very slowly varying function of representing the collision states in which the time energy, one may approximate it to be constant. of motion operator ialaE is defined should be One obtains formed by those like Yi(E) but not by the
1 scattered parts Y is(E). Consequently, one cannot r2 dE = 1 calculate the mean "time of motion" of the
(E - E0)' + P scattered states by using such an expression as
which yields
[20 I lgI2 E r /2n
With this value of lgI2, one obtains The story is quite different for the delay time operator Q. The scattered parts YiS(E) can be
[21 I (Q) = 22 = 2 / r used to represent the eigenstates of Q. The average delay time of the resonance scattering can there-
As expected, (Q) is equal to twice the mean life fore be given by of the decay resonance.
The above method of calculation of the average [24] (12) = (yS(E), Q(E)yS(E)) delay time of the metastable state conforms to the usual way of calculating the mean value of an operator in quantum mechanics. There is, how- ever, a slight weakness in this method of calcula- tion; that is, one cannot specify by any strong argument that the factor of (D,,,(E) is required to be exactly in the form 1/DXz(E), except that l/D"(E) is a reasonable factor reflecting the presence of a resonance in the scattering.
On the other hand, one may also obtain the mean life of the metastable state by averaging the lifetime q(E) over the weight factor of the meta- stable state (Tobocman and Celenza 1968). Using [lo] and with the resonance weight factor
one obtains again
t23 I 22 = (Q) = 211-
The use of the resonance weight factor to calcu- late the average delay time is equivalent to the hypothesis that the metastable state is represented by the scattered part of the lifetime eigenstate Yi(E). In this case, the precollision state (Di(E) is no longer asked to contain the resonance factor
The resonance distribution comes from the
If the eigenstates of Q are represented by y i S ( ~ ) , the second method of calculation of the average delay time still conforms to the usual way of calculation of the mean value of an operator in quantum mechanics. Moreover, in the case of a resonance scattering, it also yields directly the energy distribution
1 r2
(E - E O ) ~ + - 4
without having to put forward any hypothesis for @Res(E).
111. Possibility of the Commutation between the Lifetime Operator Q and the S Matrix
In this Section, we shall show that the delay time operator Q may commute with the energy operator and the S matrix.
(I) Comnzutation of tlze Lifetime Operator Q and the Energy Operator E
By definition (Gien 1969),
It is easy to show that Q commutes with the energy operator E. In fact,
factor T(E) = S(E) - 1. In Section V, we shall Since S commutes with the energy operator discuss this point in detail. Here, we only want to (Goldberger and Watson 1963) and (ia/aE, E) = note briefly that the linear space of vectors i, one obtains
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GIEN: DELAY TIME AND PHASE OF THE INITIAL STATE 643
(Q, E) = i - iSSt One deduces that S(E) and aS(E)/aE also com-
Hence,
[28 I (Q, E) = 0
mute with each other
\
provided S is unitary. This is an expected result, The expression of the delay time matrix in terms since [28] means that the energy eigenstate might of the matrix is, also be an eigenstate of the operator Q(E). That is why the energy representation can be used for [37] Q(E) = -ia*)S'(E) the eigenstates of Q : aE
(2) Commutation of the Delay Time Operator Q and the S Matrix
In the following, it is interesting to see that the delay time operator Q(E) and the S matrix may commute with each other provided that S(E) is on the energy shell (Goldberger and Watson 1963).
In the energy representation (Gien 1969), S(E) is a matrix with its element SZp(E) = (alS(E)I P). By definition, the matrix S1(E) = aS(E)/aE has its element
If S(E) can be diagonalized, let us consider a special set of state vectors Yi(E) which diagonal- ize S(E). With these states, an element Sij(D) of the diagonalized S(E) matrix is
The Sr(E) matrix will then also be diagonalized. In fact,
Hence, a S ( D ) / a ~ and S ( D ) ( ~ ) commute with each other,
The S matrix in the general case can always be obtained from S ( D ) ( ~ ) by a unitary transforma- tion whose unitary matrix of transformation is U,
[341 S(E) = US(~)(E)U+
U is energy independent, if S(E) and S ( D ) ( ~ ) are matrices on the energy shell. Hence,
It is then easy to see that Q(E) and S(E) commute with each other
1381 [Q(E), S(E)I = 0 if [36] and the unitarity condition for S(E),
~391 s~(E)s(E) = s(E)s~(E) = I are used.
Since Q(E) and S(E) commute with each other, there will be a complete set of orthogonal state vectors Yi(E) which diagonalize simultaneously Q(E) and S(E). These states are eigenstates ofthe delay time operator Q(E).
IV. Consequences of the Commutation between the Q and S Matrices
Some useful relations which are the conse- quences of the commutation between the Q and S matrices will be derived in this part. Later, they will be used in the calculation of the average delay time of the collision.
( I ) General Case . . The complete set of states Yi(E) which diagon-
alize simultaneously Q and S matrices are eigen- states of the delay time operator Q. Let Ya(E), Yp(E), etc. be the collision states in the energy representation which are specified by a , P, etc. The unique notation a or P, etc. may represent directions of the scattering angles 0+, helicity states of the particles, other conserved quantum numbers in the collision such as isospin, parity, charge conjugation, etc., as well as labels of the inelastic channels. An element Sap on the energy shell of the S matrix is
[40 I S,p(E) = (alS(E)I P) Let Y i be an eigenstate of the S(E) matrix; one may write
[41] Y,(E) = 1 (ali)Yi(E) i
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644 CANADIAN JOURNAL OF PHYSICS. VOL. 48, 1970
and
[''I Sap(E> = (uIS(E) I P )
plete set, so that
= C (uli)(ils(E)li')(i'IP> Hence, ii' as:@) hi* Since the nondiagonal elements Si i . are zero, one [48] Q ~ ~ ( E ) = i X f a i s i ( ~ ) 7
obtains i
[43I Sap(E) = CfaiSi(E)fpi* = Cfaiqi(E)fpi* i
i
where Si (E) is an eigenvalue of S ( E ) ; ( i I P) = fp i * where
and ( u J i ) = fui are elements of the matrices U t [49] qi(E) = i s i (E) aS,*(E) --- and U mentioned in the last part. From the aE definition
~ S ? ( E ) is an eigenvalue of the Q matrix. Through [43]
[441 Q(E) = iS(E) --- and [48], again, it is obvious that Q ( E ) and S ( E ) aE can be diagonalized simultaneously provided
an element Qap(E) on the energy shell of the Q that S ( E ) and aS(E)/aEcommute with each other.
matrix is From
[45I Qap(E)=(aIQ(E)IP) [501 Y ( E ) = S(E)@(E)
asp,* where Y ( E ) and @ ( E ) are column matrices, one
= I C Say(E) deduces that Y
Using [43], one deduces that P I ( i I y ( ~ 1 ) = C (ilS(E)lj)(j l@(E))
j - - -
y i k
Of course, the states Y y ( E ) must also form a com- where one may set Si (E) = e2i6i(E)
(2) Spin less Particles In the case of a collision of spinless particles in the three-dimensional space, one has the following
equation
[53 1 ( E ; 841Y) = jd a ' d E f ( E ; 841SIE'; 014 ' ) (E ' ; 8'4'1Q)
With
[541 ( E ; 841SIE'; 8'4') = (O~IS(E)~O'I$')F(E - E')
the equation for the matrix elements on the energy shell is
[55 I ( 9 4 l Y ( E ) ) = j df i ' (94 Is(E)l0'4')(e'4'I@(E)>
where a ' is the solid angle corresponding to 0'4' which are the direction angles of the relative momen- tum of the particles. Of course, one should use the center-of-mass system to simplify the formulas. The matrix element (04IS(E)10'4') represents the transition from an initial state with its momentum direction 8'4' to a final state with its momentum direction 84. The energy of the collision is conserved, i.e., E = E'.
Since the angular momentum must be conserved in the collision, there will be no transition between states with different quantum numbers I and m. The S ( E ) matrix is diagonalized in a basis of angular momentum state vectors
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GIEN: DELAY TIME AND PHASE OF THE INITIAL STATE
[56I (lmlS(E)ll'ml) = Slm(E)GlltFmm*
It is possible to expand (O+(S(E)lO1+') in terms of Slm (E) as follows,
P I (O+ls(~)le'+') = 1n1,I'm' C (O~~lm)(lm~~(~)~l~m~)(llm'~O'~')
= C Ylm*(Q+)Slm(E) Ylm(el+') Iin
If the scattering is symmetric around the initial direction of the relative momentum of the particles,
[581 SJE) = Sl(E) = e2i61(E)
Hence
To obtain the last line of [59], we have used the relation
where ct is the angle between the directions 8+ and €It+'. The unitarity of the S matrix is expressed by
[61al j(e+ IS(E) ~el+ ' ) (e~+~ IS+(E) lel+") dnr = (e+ lell+") = s(n - n t r )
and
[62 I SlmSlm * = Slm*Slm = 1
Between Y(E; O+) and Ylm(E), there is also the relation
1631 (Q+p"YE)) = C 1 nl (e+llm)(lmJy(~))
[641 Y(E; e+) = C im ylrn(e+)ylm(~)
The equation which is similar to [52] is
It is worth mentioning that in a collision where the momentum directions of the initial state concen- trate around a mean direction P, the amplitude (Q+l@(E)) will be very small if the direction €4 is outside this range of concentration. In this case, it is obvious that
1661 WE; 0 ) = j dfi1(0+ IS(E) le'+'>(~'+'I@(E)>
= @+l@(E)> + j dnl(e+l T(E)lel+'>(e'+'l@(E>> reduces to
where T(E) = S(E) - 1. In terms of the orbital angular momentum states, one obtains
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646 CANADIAN JOURNAL OF PHYSICS. VOL. 48, 1970
[68 I Yln,(E) -- Tl(E)Ql,(E)
As for the delay time operator Q(E), it is easy to derive
where
~ 7 0 1
is the delay time of the lth partial wave. When the collision particles are spinless, the orbital angular momentum state is also a lifetime eigenstate with eigenvalue q,(E). Hence, the lifetime eigenstate has a definite orbital angular momentum.
In the general case, one may always expand an element Sap of the Smatrix in terms of the eigenstates of a physical quantity A. If this physical quantity is conserved under the collision (such as isospin, parity, etc.), the S matrix will be diagonalized by the eigenstates of A. Then the Q matrix will also be diagonalized by these eigenstates. If a resonance is regarded as a lifetime eigenstate with a special form for its eigenvalue, according to the above results, a resonance may be defined with a set of definite quantum numbers which are conserved in the collision.
(3) Particles with Spins In the case of particles with spins, similar results can easily be obtained by using the helicity
formalism (Jacob and Wick 1959). In this case, [55] becomes
An element of the S matrix can be expanded in terms of the total angular momentum states as follows,
[72] (0$h3h41~(E)10'+'; h,h,) = C (8$h3h41 JM~~'~,')(JM~,'~,'(S(E)(J'M'~~'~~') As'A~'JM
Al'A2'J'Mr
x (JIM'; hl'h2'(0'~'hlh2) With
and
[73'1 (J'M'hlfh2'l8'$'h1h2) = NJ,6,,,,,6,,,,, 9~ , , ( $ ' , el, -4')
where NJ and NJ, are normalization constants, we obtain
[74] (8$h3h4ls(~)[e'$'hlh,) = C NJ '~M,* (4, 8, -8)g~,($ ' , Q', - $')(h3h41SJ(E)lhlh2) J M
On the other hand, we also have the other following relations
[751 (0$13h4ly(~)) = JM 1 (8$h3h41 J M ~ ~ ~ ~ ) ( J M ~ ~ ~ ~ I ~ ( E ) ) 0 r
[75'1 Y(E; ~ h 3 h 4 ) = C N J C ~ ~ * ( $ , 8, -$)Q(E; JMV~) J M
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GIEN: DELAY TIME AND PHASE OF THE INITIAL STATE
and
[77] S J + + = ( J M + I S J ( E ) I ~ ~ + ) , and
1 . 1 S J - - = ( J M - ISJ(E)IJM- ) r . i
The submatrix S J ( E ) has not been diagonalized be represented by column matrices with only one yet. To diagonalize S J ( ~ ) , we have to make use of nonvanishing element Y i ( E ) and Q i ( E ) ,
But the conservation of parity reduces these four [gob] Qi(E) = @.(E) to two independent matrix elements l o i I
the other conservation properties of the collision such as parity conservation, time reversal invari- ance, etc. These conservation properties make S J ( E ) become symmetric, and a symmetric unitary S matrix can always be diagonalized. As an illustrative example, we take the simple case of [80n] Y ( E ) = an elastic meson-nucleon scattering (Jacob and Wick 1959). Here, there are altogether four matrix elements
[781 J S J + + = S - -
sJ+- = S J - +
If the so-called eigenstates of the "orbital parity" The relation between Y i ( E ) and cDi(E) is
- -
0
y i ( E ) 0
1791 Y ( E ; J M , ( - = [gl] Y i ( E ) = S(E)Qi (E)
( J M + IY(E) ) + ( J M - IY(E) ) where the S matrix is diagonalized. With S ( E ) = I + T ( E ) , one obtains
are used to represent the collision, the submatrix S J ( E ) will be diagonalized. An eigenstate of the [82] Y , ( E ) = Q ~ ( E ) + Y , ~ ( E ) delay time operator Q will, therefore, have a definite orbital parity ( - 1 ) ' = i.e. a Y i S ( E ) is the scattered part of the lifetime eigen-
definite parity (- 1)'q 1 q 2 , where q l q 2 are state Y i ( E ) . Later on, Y i S ( E ) will be called
intrinsic parities of the collision particles. In the scattered state. By definition, case of a meson-nucleon scattering, q l q 2 = - 1. [83] Q ( E ) y i ( E ) = q i (E)Y i (E)
Since Q ( E ) commutes with S ( E ) , Q i ( E ) is also an lm~oss ib i l i t~ Of the Calculation Of the eigenstate of Q ( E ) Hence the scattered state Average Time of Motion with the
Scattered States Y i S ( E ) is also an eigenstate of Q ( E ) with the same eigenvalue q i (E) ,
Another interesting consequence of the com- mutation between the lifetime operator Q ( E ) and Lg4] Q ( E ) y i S ( E ) = qi(')yiS(E) the matrix should be mentioned here. Let Y , (E) We shall assume that the resonance which is be an eigenstate of Q ( E ) . In the representation observed in the collision cross section is repre- which diagonalizes S(E) , Y i ( E ) and a i ( E ) might sented by the scattered states Y i S ( ~ ) . This
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648 CANADIAN JOURNAL OF
assumption is reasonable, since the resonance is formed by the wave amplitudes which have been scattered by the collision. With this assumption, the mean life can be obtained by averaging the delay time qi(E) over the energy distribution
It should be noted that the correspondence between Yi(E) and Y i S ( ~ ) is two-by-two. There- fore, the scattered states YiS(E) also form a com- plete set of vectors spanning the linear space 9 in which the operator Q(E) is defined. In this linear space, the state vectors Yi(E) and Y,"E) just have different norms.
It is quite a different story for the time of motion operator i a/aE. In fact, let us consider a com- plete set of precollision states cDA(E) which are eigenstates of the operator ia/aE. The states YA(E) which are obtained from the states cDA(E) by a unitarity transformation
of course, also form a complete set of vectors spanning the linear space 9' in which i a/aE is defined. But the scattered states YAS(E) = T(E)cD,(E) cannot form a complete set of vectors to span the linear space 9'. Since
187 I YA(E) = W E ) + YAS(E) one obtains
Y,(E) and cDA(E) always have different "times of motion", because i a/aE and S(E) do not com- mute with each other (Gien 1969). There will never exist a scattered state Y,~(E) which can be an eigenstate of ia /aE. In other words, the scattered states YAS(E) cannot be used to span the linear space 9'. Hence, such a mean value as
; PHYSICS. VOL. 48, 1970
VI. Average Delay Time of the Collision
In the following, we shall calculate the mean delay time of a collision in the general case and within the context of the delay time operator. The case of a three-dimensional collision of spinless particles will be of interest in particular to prove that even in this case, the average delay time is independent from the phase change of the initial state. This result contrasts Ohmura's (1964) assertion and saves the hope that the average delay time can be expressed in terms of the S matrix alone.
( I ) General Case Let Y,(E) be an outgoing scattering state in the
channel a. The notation a represents direction of scattering 04, states with different helicities, and other conserved quantum numbers such as iso- spin, parity, etc. a is also the label of inelastic channels. Y,(E) is a final state obtained by a collision whose initial state is allowed to be in any accessible channels limited only by the conserva- tion laws.
Consider a subspace of collision states Ya(E) which is specified by a definite energy E. We shall represent the state vectors of this subspace by column matrices. Within this subspace of state vectors, we define the scalar products for the life- time eigenstates Yi(E) as follows,
and
Note that the column matrices Yi(E) form a basis of vectors in this subspace. The average delay time of the collision state Y, (over the indefinite energy as well) is given by
C ffai*fai,(Yi(E), Q(E)Y~,(E>)'E - - i i '
C J f a i* f a i , (~ i (~ ) ,~ i , (E ) ) dE ii '
where we used the following expansion
does not make sense. The proof of the dependence 1931 ya(E) = CfctiYi(E) of the average delay time on the initial phase of i
the collision states is based on the average of the to obtain the last line. Using the definitions of the time of motion over the scattered states. The scalar products for the basis vectors Yi(E), we result is therefore questionable. obtain
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GIEN: DELAY TIME AND PHASE O F THE INITIAL STATE
fa, are assumed to be independent from the is the probability that the final state Ya(E) be in energy. Hence the delay time eigenstate Yk(E). The result
1 Ifail ' J 4i(')Iy~(')I~ d~ obtained in Section I1 may be used to calculate i
Lg51 = 1 1jail2 J I Y ~ ( E ) / ~ d~ the average delay time of the state Ya(E) in the case when Y,(E) is a metastable state,
i
Using the normalization [lo51 (Q>a = aa,2/r,
[961 J ~ Y ~ ( E ) ~ ~ d~ = 1 (2) Collision of Spinless Particles in the Three-
and the definition of the average delay time of the . Dimensional Space lifetime eigenstate Y ,(E) over the energy In the case of collision of spinless particles in
[97 I 22, = J ~ , ( E ) I Y ~ ( E ) ~ ~ d~ the three-dimensional space,- a represents the direction 0+ of the relative momentum. Equation
we obtain [7 1 ] becomes
This is an expected expression for the average where Q' is the solid angle in the 0'+' direction. In
delay time (Q),. The average delay time of a this case, the orbital angular momentum eigen-
collision whose final state can be in any accessible states diagonalize S and Q matrices and
channel a is given by [lo71 WE; = z yln1(0+) y l r I I ( ~ ) 2 11 / f a i l 2 2 i IIII
[991 (Q) = One obtains the following equations which are 1 a i i ] f a i l 2 similar to 1901 and 1911
With the normalization 11081 (YL,,,(E), yi,lll,(E)) = ~ ~ ~ ~ ~ j l I l 1 1 ~ ~ ~ ~ n j ( E ) ~ 2
[loo] 1 J IYa(E)I2 dE = 1 and
a
one obtains [log] (YL,,~,,(E), Q(E)Y,m(E)) =
[loll 11 l f a i l 2 = 1 6~t,6mm,q~m(E)I~im(E)12
a i The mean delay time of the collision state Hence Y(E; 04) is . .
[lo21 (Q) = 1 a i l fa i12(Qi) J (y(E; Q+), Q(E)y(E; Q+) dE
where (Qi) is the average delay time of the eigen- ['lo' ('), = J(Y(E; Om), Y(E;0+))dE state Y,(E). When only ( Q , ) is predominant, or (Q) reduces to
[lo31 (Q) - 1 a lfak12(Q,)
and (Q,) reduces to
[lo41 (Q)a I f a k I 2 ( Q k )
1 \ f a i l 2 i
If IY,,(E)~~ are independent from m, i.e.
11 121 l'I'l,,l(E)12 = l'I',(E)12
Yi(E) is normalized to be 1 and one obtains
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650 CANADIAN JOURNAL OF
c I yl11,(04)12 J S~(E)IY~(E)~' d~ I."
where the following relation was used,
(Q(E)),+ may also be reduced to
where Y ,(E) is normalized to be 1 and
is the average delay time of the delay time eigen- state YI(E). Through [113], one remarks that the average delay time of a collision is independent from the direction 04 of the final momentum. It is known that if the energy of the collision is low, only a few partial waves of lowest order are predominant in the scattering (Schiff 1955), and (Q),+ can be approximated to be
If only one of the delay time eigenstates, say Yl,(E), has a long mean delay time 2.sl,, one may approximate (Q),+ to be
where 21' + 1
a,, = C (21 + 1)
I S L
is the probability that the state Y(E; 04) be at the metastable state Y,,(E). As the collision energy increases, channels of higher partial wave will be open, hence a,, becomes smaller. Since only a few metastable states with long average delay time are
) PHYSICS. VOL. 48, 1970
observable, it is expected that (Q(E)),+ also becomes smaller. This is an anticipated result. At higher collision energy, the average delay time should be shorter.
In order to calculate the average delay time 2.sl of the metastable state Yl(E), we use [68],
where Nl(E) and @,,(E) are smooth variation functions of energy (Goldberger and Watson 1964) and
[I201 r Dl(E) = E - Eo + i - 2
Using the normalization f(Y1(E)l2 d E = 1, one obtains
11211 IYl(E))' = Tl
- E,)' + c] 4
With the above expression for IY,(E)(', eqs. [I 161 and [118] yield
Through the above calculation, the average delay time of a collision is seen to be independent from the phase of the initial state even in the three-dimensional scattering. The average delay time of a collision can therefore be calculated in terms of the Smatrix alone. This conclusion saves the hope that the Smatrix can be used to describe the dynamics of a collision and the observable physical quantities can probably be calculated in terms of the S matrix. The reason for the differ- ence between our result and Ohmura's has been discussed in Section V. Within the context of the delay time operator, we are able to find the independence of the average delay time from the phase of the initial state without having to per- form the explicit calculation of the average delay time. In fact, let us consider two initial delay time eigenstates Qi and xi which differ from each other by a phase ai,
The state xi(E) is obtained from (Di(E) by a unitary transformation whose matrix of trans- formation is eiar. Under this transformation, the delay time operator Q(E) becomes (Wigner 1959)
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GIEN: DELAY TIME AND PHASE OF THE INITIAL STATE 651
[I241 Q1(E) = e i " iQ(~) e- '"I
The new average delay time
[I251 (Q1(E)) = J(xi, Q'(E)xi> d E
is obviously equal to the average delay time
( Q(E)) = J(@i, Q(E)@i) d E
(3) Collision with a Known Initial State The situation which is often met in the experi-
ment is a collision where the initial state is known to be in a defini-te channel P while the final state can be in any channel ci provided that the conser- vation laws are respected. The number of open channels also depends on the mean energy of the collision. In this case, we shall use the initial states (instead of the final states) to represent the collision. The initial state in a channel P is related to the final states in a channel ci according to
Since the states ap are obtained from the states Y, by a unitary transformation with the matrix of transformation S t , the delay time matrix in the Qp representation is
[I271 Qf(E) = St(E) .Q(E) .S(E)
Q and S commute with each other. Hence
With the similar steps of calculation, we easily derive
where @(E) is normalized to be 1 and (Q,) is the average delay time of the lifetime eigenstate @,(El,
[1301 <Qi) = Sqil@i(~)I' d E
= Sqi(~)l\I.'iYE>l' d E
is the probability that the delay time eigenstate Qi(E) can be formed by a collision state Op(E).
VII. Conclusion In this study, we have shown how to calculate
the average delay time of a collision within the context of the delay time operator in several cases. We found that in all circumstances, the average delay time does not depend on the phase of the initial state. Even in the three-dimensional scattering, the average delay time was shown to be independent from the phase change of the initial wave. This contrasts with the result obtained re- cently by Ohmura (1964). We have tried to look for the source of the difference between our re- sult and Ohmura's. We have found that it is im- possible to use the scattered parts of theoutgoing scattering states to form a linear space of state vectors in which the "time of motion" operator i a/aE is defined. One may not therefore use these scattered waves to average the time of motion of the collision. Such a quantity as
does not make sense. The proof of the dependence on the initial phase of the average delay time is based more or less on the average of the time of motion over the scattered parts of the scattering waves. The result is therefore questionable. We have also shown that on the contrary, we may use the scattered parts of the outgoing scattering states to form a linear space of state vectors in which Q(E) is defined. This property holds only for the delay time operator Q(E) because Q(E) and S matrices commute with each other. Our result saves the hope that observable physical quantities such as the delay time can probably be calculated in terms of the S matrix alone without having to know the details of the collision state.
Since the Q and S matrices commute with each other, the lifetime eigenstates may be defined by a set of definite quantum numbers which are conserved in the collision. The metastables are regarded as the lifetime eigenstates with their eigenvalues qi(E) in the resonance form; they may therefore also be defined by a set of definite quantum numbers such as parity, spin, isotopic spin, etc. This is an interesting point, since to our knowledge, so far there has not been any sound reason for the resonances to be defined with such a set of definite quantum numbers.
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652 CANADIAN JOURNAL OF PHYSICS. VOL. 48, 1970
Acknowledgments - 19656. Can. J. Phys. 43, 1978. 1969. Can. J. Phys. 47, 279.
The author wishes to thank Profs. C. Bloch and GOLDBERGER, M. L. and WATSON, K. M. 1962. Phys. Rev. M. Froissart for the hospitality extended to him 127y 2284.
1963. Collision theory (John Wiley & Sons, Inc., at the Department of Theoretical Physics, CEN- New York), p. 213. Saclay, France, during the summer of 1969, when - 1964. Phys. Rev. 136, B1472.
a part of these ideas was clarified. H~ is also in- GOJ-DBERGER, M. L., WATSON, K. M., and FROISSART, M. 1963. Phys. Rev. 131, 2820.
debted to the National Research 'Council of HEISENBERG, W. 1943. Z. Physik, 120, 513. Canada for a Special Travel Grant which made JACOB, M. and WICK, G. C. 1959. Ann. phys. (N.Y.), 7,
404. the Saclay possible and for Operating OHMURA, T. 1964. Progr. Theoret. Phys. (Kyoto), Suppl. Grant A-3962 under which this work was 29, 108. performed. SCHIFF, L. I. 1955. Quantum mechanics (McGraw-Hill
Book Company, Inc., New York), p. 110. SMITH, F. T. 1960. Phys. Rev. 118, 349.
BOHM, D. 1951. Quantum theory (Prentice Hall, New TOBOCMAN, W. and CELENZA, L. 1968. Phys. Rev. 174, York), pp. 257-261. 1115.
CHEW, G. F. and GOLDBERGER, M. L. 1961. Dispersion WIGNER, E. P. 1955. Phys. Rev. 98, 145. theory and elementary particles (John Wiley & Sons, - 1959. Group theory (Academic Press, New York), Inc., New York). p. 50.
GIEN, T. T 1965a. J. Math. Phys. 6, 671.
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This article has been cited by:
1. Vladislav S. Olkhovsky. 2011. On time as a quantum observable canonically conjugate to energy. Uspekhi Fizicheskih Nauk 181,859. [CrossRef]
2. V. S. Olkhovsky. 2009. Time as a Quantum Observable, Canonically Conjugated to Energy, and Foundations of Self-ConsistentTime Analysis of Quantum Processes. Advances in Mathematical Physics 2009, 1-83. [CrossRef]
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