on the operators for time of motion and delay time induced by scattering
TRANSCRIPT
On the operators for time of motion and delay time induced by scattering1
TRAN TRONG GIEN Depart~nent of Physics, Metnorial University of Netvfocmdland, St. John's, Net$ifo~mdland
Received August 13, 1968
Recently, Lippmann showed that a time operator might exist in quantum theory and that its explicit form is i a/aE in the energy representation. In this paper, we wish to stress that i a/aE actually represents only the time which the particle used to move from the scattering center to a certain point at the outside. (Later on, we shall call this time the "time of motion.") We also show that, although it is possible to define an operator for the "time of motion" of the particle in the energy representation, it is necessary to introduce a new conjugate representation of the energy representation, the "time representation," to overcome difficulties which may arise in the evaluation of the mean time of motion (T). If the position representation is used for the calculation instead, we shall meet these difficulties. The reason is that time of motion T, position x, and energy E are not two-by-two compatible variables. Bearing in mind that i a/aE only represents the time of motion of the particle, we can then deduce the explicit form of the delay time operator Q in terms of the scattering matrix S directly from the time of motion operator by simple reasoning. This form is Q = - i(aS/aE) . ST as was found previously. Some interesting features of the delay time operator will also be sketched at the end of the paper.
Canadian Journal of Physics, 47, 279 (1969)
I. Introduction
There have been attempts (Smith 1960, 1962, 1963a, b, c ; Gien 1965a, b) to formulate a lifetime operator (or delay time operator) in quantum mechanics to replace the usual indirect method of calculation of the delay time induced by scattering (Wigner 1955). The explicit form of the delay time operator in terms of the S matrix can be derived either by a direct formulation of the Q matrix in terms of the wave functions (Smith 1960; Gien 1965a) or by an extrapolation from the "time operator" defined as a differentiation with respect to the energy (Gien 19656). This relation was found to be Q = - i(aS/aE) . S t . Although these attempts are successful to some extent, there are still things to be clarified, such as, (a) what the representation of the states on which the delay time operator operates is and (b) how one can develop a theory for the delay time operator without contradicting the uncertainty principle of quantum mechanics.
In one of our previous works (Gien 1965b), we used the manifold of asymptotic outgoing states in the x representation for the time operator and assumed that the time operator is T = - i a/aE. With these assumptions, we can show that the relation between the delay time and the S matrix mentioned above can be derived. However, in this derivation of the Q matrix, we have to approximate the time of motion to be a classical quantity, i.e. the ratio of a distance x over the
'Work s~~ppor t ed by Operating Grant A-3962 from the National Research Council of Canada.
velocity v. This seems to contradict the un- certainty principle in quantum mechanics which states that the position x and the momentum (or velocity) of the particle cannot simultaneously be known precisely; hence, the definition of a quan- tity such as x/v seems to be absurd. An attempt to solve this difficulty has been made by reasoning that this approximation is acceptable if the life- time of the scattering is long compared with a length of time equal to 1/2E. In the case of a resonance scattering, the state of resonance usual- ly has a sufficiently long lifetime to satisfy the stated requirement. To confirm that this argu- ment makes sense, we showed that this approx- imation was also made, although indirectly, in the formulation of the lifetime matrix by Smith (Smith 1960) and that this condition of a long lifetime of the scattering is equivalent to the con- dition for having a sharp peak in the resonance cross section (Gien 1966). These arguments are not, however, considered as completely satisfac- tory, since some assumptions are still to be made.
Recently, Lippmann (1966), in an attempt to make the existence of a "time operator" become legitimate in quantum theory, showed that the operator for time might exist and that its expres- sion T = i a /aE can be derived by a conven- tional method from the action and angle variables in classical theory. He argued that classically, if the action variable J, is identified with the energy, the conjugate variable 4, will be the time. The equation of motion is then
[I1 a H $1=-=1 aJ1
Can
. J. P
hys.
Dow
nloa
ded
from
ww
w.n
rcre
sear
chpr
ess.
com
by
MC
GIL
L U
NIV
ER
SIT
Y o
n 11
/19/
14Fo
r pe
rson
al u
se o
nly.
280 CANADIAN JOURNAL O F PHYSICS. VOL. 47, 1969
or equivalently,
12 I 4l(t> = t + 4(0)
He showed that the quantum-mechanical analog of eq. [I] is the conventional commutation rela- tion for conjugate variables T and E, and that of eq. [2] is the relation between the "time operator" T and the time parameter t,
13 1 [T, H,] = i and
t41 ~ ( t ) = eiHot ~ ( 0 ) eWiHot
= t + T(0)
Equation [3] means that outside the range of an interaction potential, the explicit form of T is i a/aE if the energy representation, where H, is diagonal, is used for the scattering states.
Lippmann's ideas suggest that we should re- investigate the problem of the derivation of the operators for time and delay time induced by scattering. We accept Lippmann's ideas about the derivation of the explicit form for the time operator T = i a/aE, but we want to stress that Lippmann's time operator actually only repre- sents the time of motion of the particle, i.e. the time interval during which the particle moves from the scattering center to the outside. This time will by no means also include the delay time of the scattering. The reason is that T = i a/aE was derived from its definition as the ratio of the distance to the velocity (Lippmann 1966). To be clearer, if i a/aE operates on a physical state of the particle under an interaction, it gives only the part of time corresponding to the time of motion of the particle. Bearing this idea in mind, we may then find the relation between Q and S in an obvious way. The form of the operator Q in the energy representation where the outgoing scatter-
of motion operator, one should use the energy representation and the time representation as its conjugate to represent the physical system, instead of using the energy and position representation as usual. Let $ ( + ) ( E ) be the energy representation of an outgoing scattering state. It can be con- sidered to be the superposition of the eigenstates of the time of motion operator T. These eigen- states are assumed to form a complete set of orthonormal vectors. The coefficient of expansion x(+)(T) represents the outgoing scattering state in the time representation. On the other hand, an outgoing state in the position representation $(+)(x) may also be considered to be the super- position either of the time eigenstates or of the energy eigenstates. The energy distribution of the outgoing scattering state IIJJ(+)(E)~~ has a peak at a certain value Ec which is the most probable energy of the particle. ~$(+)(x)l' has a peak at xc which corresponds to the most probable position of the particle and I x ( + ) ( T ) ~ ~ is the "time distribu- tion". This distribution has a peak at Tc which is the most probable time that a particle having an energy E, uses to move a distance x, froin the scattering center. There should exist the relation T, = xi/v, among T,, x,, and v,. If one accepts that the most probable value of a physical quan- tity is equal to its average, then one inay have the classical energy, time, and position, respectively, as
ing states are used is Q = - i(aS/aE) . S t . A 11. Necessity of the Introduction of the close study shows that it is impossible to calculate the mean time of motion (and mean delay time) Time Representation
without introducing a "time representation" which is considered to be the new conjugate of the energy representation. We show that, if only the position and energy representations are used as usual, we shall meet difficulties in these calcula- tions. These difficulties arise from the fact that time of motion T, position x, and energy E are two-by-two incompatible variables. One may use any two of them as conjugate variables. For the scattering in which one is interested in the time
In this section, we shall show that it is necessary to introduce the time representation in order to evaluate the mean time (T). We shall only discuss the problem within the one-dimensional scatter- ing process to simplify matters. An outgoing particle is assumed to move in the positive x direction. An asymptotic wave which represents a free particle having a definite positive momentum, p, is $,(x) = [I/ J(2n)l eiP". It is assumed that the manifold of these asymptotic outgoing states
Can
. J. P
hys.
Dow
nloa
ded
from
ww
w.n
rcre
sear
chpr
ess.
com
by
MC
GIL
L U
NIV
ER
SIT
Y o
n 11
/19/
14Fo
r pe
rson
al u
se o
nly.
GIEN: OPERATORS FOR TIME OF MOTION AND DELAY TIME 28 1
forms a complete set of vectors in the sense that, at a far distance from the scattering center, any packet of outgoing waves moving in the positive x direction may be considered to be the super- position of these asymptotic outgoing eigen- states :
[51 $'+)(x) = $'+'(~)$dx) dp
The notation (+) indicates that these states are asymptotic outgoing. Here, the particle is assumed to move in the positive x direction. Therefore, only positivep is considered. $(+)(p) is the expan- sion coefficient which is simply the outgoing state in the momentum representation. Since we are interested in the asymptotic outgoing states, x will be defined only in the interval from 0 to + a . Wemay assume that I$'+)(x)l 1. Oforx E (- co,O). This is because l$(+)(x)J2 represents the probabil- ity that an outgoing particle will be at x. 1$'+)(x)J2 is therefore very large only near a position x which is positive and sufficiently large. Inversely, $(+)(p) may be considered to be the superposition of position eigenstates $, (+)(p) = [I/ J(2n)I e- '"" in the momentum space. In fact, from eq. [5], we may have
Here, we have extended the lower limit of integra- tion to - co, instead of 0. (Because of the small- ness of j$(+)(x)J in the interval ( - a , 0) of s, the contribution of $(+)(x) to the integral is small in this interval. The last line was derived by using the expression of the delta function,
which represents the orthonormality of the momentum eigenstates. Inversely, one may obtain eq. [5] from eq. [6] if one again assumes that
is large only near a positive valuep, ofp. Under the effect of an external force, an
asymptotic outgoing eigenstate in the position representation becomes
where q is the phase shift of the asymptotic out- going wave (we use q instead of 26). The asymp- totic outgoing wave packet becomes
Jbd J 4'' )(P) ei" eipx [9] +'+ '(x) = - d P
+(+)@) = eiS $(+)(p) is the conjugate of +(+)(x). +(+)(p) is therefore the momentum representatioil of the outgoing scattering when an external force is present. In a manner similar to the derivation of eq. [6], one may also obtain
if again the condition of sharp localization for +")(x) and \I,(+)@) is assumed.
If the energy representation (Kemble 1958) (instead of the momentum representation) is used, a set of energy eigenstates a,(s) is intro- duced such that
and r m
Equation [12] expresses the orthonormality of the energy eigenstates while eq. [I 31 expresses the closure relation. Each energy eigenstate is de- generated to two momentum eigenstates, a,'+'(x) - [I/ J(2.n)l eiPx and a,'-)(,u) - [I/ J(2n)l e-IPx corresponding respectively to p = J(2nzE) and p = - J(2mE). Equations [I21 and [I31 can be written as
Can
. J. P
hys.
Dow
nloa
ded
from
ww
w.n
rcre
sear
chpr
ess.
com
by
MC
GIL
L U
NIV
ER
SIT
Y o
n 11
/19/
14Fo
r pe
rson
al u
se o
nly.
282 CANADIAN JOURNAL OF
A wave packet $(x) can be expanded in terms of the energy eigenstates cE(x),
If one is interested in only the asymptotic out- going wave packet, one may have
In writing eq. [I71 we have assumed that, in this case, [$(+)(E)I is large only near a positive momentum p,, and ~$(+)(x)l is large only near a positive x,. Inversely, by using the orthonormality of the energy eigenstates, one may obtain
The last line was obtained by assuming that ~$(+)(x)l is large only near a positive x,.
By comparing the closure relations of the energy and momentum eigenstates to each other, one may deduce easily the explicit form of
and by comparing eq. [18] to eq. [6],
On the other hand, the position operator xop may be shown to have the following explicit form in the energy representation,
In fact, from the classical relation x = (p/ii~)T, one may have the operator sop as
With p = J(2nlE) and T = i a/a E, we deduce that xop is given by [21]. It is also possible to verify that [a,(+)(x)]", given by [19], is a position eigenstate of sop given by [21],
' PHYSICS. VOL. 47, 1969
Under the effect of an external force, the asymp- totic outgoing energy eigenstate becomes eiq x cE(+)(x) and the asymptotic outgoing wave packet $(+)(x) becomes
1241 +(+)(XI = lo+m $ ( + ) ( E ) ei' c j +) (4 dl3
We still assume that the asymptotic outgoing packet +(+)(x) is localized at a certain positive and large position x,'. The energy representation of the asymptotic outgoing state becomes
In the following, we shall see that it is impossible to calculate with satisfaction the mean time of motion (T) if the position representations are used for the scattering states. By definition, the mean time of motion is
1261 (T) = [$'+'(E), T$'+'(E)l
If $(+)(E) is replaced by eq. [18], we obtain
[27] (T) = j o + m d x j o + m d x l ~ o + m
One cannot go through this calculatioil to obtain the value in(x)/(p) as expected. Therefore, the introduction of the "time representation" is definitely necessary. This is quite natural because position x and time Tare not compatible variables as we shall show in the next section.
111. Incompatibility of the Operators for Time of Motion T, Position x , and Energy H,
In this section, we shall show that T, x,,, and Ho are two-by-two incompatible. Hence, we may conclude that a state of the scattered particle
Can
. J. P
hys.
Dow
nloa
ded
from
ww
w.n
rcre
sear
chpr
ess.
com
by
MC
GIL
L U
NIV
ER
SIT
Y o
n 11
/19/
14Fo
r pe
rson
al u
se o
nly.
GIEN: OPERATORS FOR TIME OF MOTION A N D DELAY TIME 283
cannot simultaneously be eigenstates of operator in the momentum representation, we have T and operator xop and, therefore, a time repre-
[301 Ul a 171
sentation must be introduced. [ T , ~ 1 = - O P + , Classically, it takes a time equal to T = mx/p y 2 a p P
for an outgoing particle having a velocity v = p/m which is different from zero. wi th to move a distance x from the scattering center.
a ;(21?1~j a This time will be equal exactly to the time of I T = i - and xop = --- i - - +
motion of the particle in the case of a free particle, 8E nl aE (2,/2rnE) and will be approximately equal to the time of motion of a article under an interaction effect if in the energy representation,
the range of the potential is so short that the time of motion is only modified slightly compared with a free motion of the particle through the same distance. As customary, if x and p are assumed to be Hermitian, then T = mx/p is not a Hermitian operator. In the quantum-mechani- cal domain, the operator for time of motion T should be defined as,
In the momentum representation, xop = i slap. Therefore, the explicit form of the operator T is
i a irn T = m - - - - Pap 2p2
We may find the commutator [T, H o ] as below,
Equation 1291 means that, outside the range of the interaction, energy H o and time T are conjugate variables and that the time operator T is i a/aE in the energy representation. Note that, by its definition as mxlp, T should be known as the time of motion of the particle, i.e. the time interval which the particle uses to move from the scattering center to a certain point at the outside. T by no means contains the delay time due to the delay effect caused by the interaction. It is possible to show that the time of motion operator T and the position operator xop d o not commute, either in the momentum representation or in the energy representation. In fact, with
[ 3 11 [ T , xop] =
which is also different from zero. According to eqs. [ 3 0 ] and [3 1 1, x,,, T , and H o
are two-by-two incompatible. It is therefore necessary to use either E and T or x and T as conjugate variables for the scattering where the time of motion operator T is concerned. Since we already knew the explicit form of the time of motion operator in the energy representation as T = i a/aE, the energy and time representations should be used if we want to calculate the mean time of motion ( T ) . We introduce the state x ( + ) ( T ) such that I x ( + ) ( T ) ~ ~ represents the "time distribution" of the outgoing particle. One should understand the physical meaning of JX(+) (T)12 as follows. The time that the particle uses to move from the scattering center to the outside cannot be determined completely, since there is no way t o measure this time with precision unless the energy of the particle is completely indefinite. However, what we certainly know is the probability that the particle moves from the scattering center to the outside with a time T; this probably is IX(+)(T)12. The distribution function I X ( + ) ( ~ ) 1 2 has a peak at a certain value T, which is the most probable value of T. x ( + ) ( T ) is normalized to be 1,
Since we are only interested in the outgoing particle observed at the periphery, we may expect that T, is positive and sufficiently large.
One may then consider x ( + ) ( T ) to be the super- position of a complete set of orthonormal vectors cp(+)(T, E) which are assumed to exist so that
i a im a T = ni - - --Z and x,, = i - [ 3 3 ] xli '(T) = Stm ~ E + ( + ) ( E ) ~ ( + ) ( T , E )
P O P 2~ ap 0
Can
. J. P
hys.
Dow
nloa
ded
from
ww
w.n
rcre
sear
chpr
ess.
com
by
MC
GIL
L U
NIV
ER
SIT
Y o
n 11
/19/
14Fo
r pe
rson
al u
se o
nly.
283 CANADIAN JOURNAL OF PHYSICS. VOL. 47, 1969
In the time representation, cp(+'(T, E) represents The right-hand side of eq. [41] represents the an energy eigenstate. The orthonormality of mean value of the time of motion to which (T) cp(+) (~ , E) may be expressed as is equal as expected.
T and x are also incompatible variables. We [34] jo+m[cp(+)(T, E)]*~(+)(T, E') d T may also assume that the state +(+)(x) can be
expanded in terms of a set of orthonormal vectors = S(E-E') c(+)(x, TI,
and their closure relation as, 1421 +(+)(x) = d~ X ( + ) ( T ) ~ ( + ' ( X , T)
~351 jo+m[q(+)(T, E ) I * ~ ~ ( + ) ( T ~ , E) d~ Jo+ *
The states k ( + )(x, T), defined by eq. [42], may be seen as the time eigenstate in the position repre-
= s(T- sentation. They are assumed to exist and verify +(+'(E) is the conjugate representation x(+)(T). the following conditions, The square of its absolute value, I+( ' ) (E) I~ , represents the probability that the particle has an [431 {o+m[~'+'(x, T)]"k(+'(x, TI) dx energy equal to E. It is expected that I + ( + )(E)I2 has a peak at a certain value E,. Using eqs. 1341 = S(T - T')
and [35], we can deduce that and r+m
[36] +(+)(E) = J i m X(+)(T)[cp(+)(T, El]* d T 0
Since T = i a/aE, we may write the following equation in the energy representation,
We also assume that the time eigenstates cp (+ ) (~ , T) exist and by definition (p(+'(E, T) [cp(+)(~, E)]*. We have the following equation for a "time eigenstate"
Inversely, from eqs. [42] and [43], we can deduce that
[45] x(+'(T) = +(+)(x)[k(+)(x, T)]* dx
Although the explicit form of T in the position representation is not given here, it is expected, however, that
By using eqs. [44] and [45], the expectation value of T may be found as below :
where To is an eigenvalue of the time of motion [47] (T) = [+(+ T+(+)(x)] operator. Note that eq. [38] inay be used to find the explicit form of time of motion eigenstate = So+ w[+~'~(x)l* T+(+I (~ ) dx cp" ' ( 4 To),
1391 cp (+ ) (~ , T,) - e-iToE = T ~ X ( + ) ( T ) ~ ~ d T
The expectation value of T may be calculated as follows: This is an expected result. [40] < T) = [+'+'(E), T+'+'(E)l One may also find the relations between the
eigenstates k(+)(x, T), cp(')(~, T), and cl(+)(x, E) a = Jo+m[$(+)(E)l'ki - +(+)(E) d~ defined by eqs. [42], [33], and [I71 From eqs.
a E [36] and [45], we can obtain
Using eqs. [36] and [35], one deduces that [48] +(+)(E) = d T J'Im dx +(+)(x)
0 0
1417 (T) = So+- T ~ x ( + ) ( T ) I ~ d T x [k")(x, T)]*[~(+)(T, E)]:!:
Can
. J. P
hys.
Dow
nloa
ded
from
ww
w.n
rcre
sear
chpr
ess.
com
by
MC
GIL
L U
NIV
ER
SIT
Y o
n 11
/19/
14Fo
r pe
rson
al u
se o
nly.
GIEN: OPERATORS FOR TIME
Comparing eq. [48] to eq. [18], we can deduce that
Using the orthononnality ofthevectors c(+ )(x, T), lp(+ '(T, E), and a(+ )(x, E), we may also obtain
and
The purpose of this section is just to show that we must assume the existence of the time repre- sentation in order to calculate with satisfaction the mean time of motion (T). We do not intend to be very vigorous regarding the question of whether this time representation exists or not. It is evident, however, that in the position repre- sentation, the time eigenstate cannot exist since position x and time of motion T cannot be deter- mined simultaneously unless the uncertainty piinciple is violated. We also always assume that only the asymptotic outgoing states are used to represent the scattered particle. This is reasonable since the outgoing scattered particle is observed at the periphery far from the scattering center. The packets $(+ )(x) and x(+ )(T) may be con- sidered to be localized at large and positive x, and T,. The coiltribution of $(+)(x) and x(+ )(T) to the integrals of the transformation from one representation to another can be neglected for negative values of x and T.
O F MOTION AND DELAY TIME 285
how to derive this relation. To clarify the physical ideas of this section, we shall first consider the one-dimensional scattering, then extend our reasoning to the multichannel scattering.
I. One-Dimensioizal Elastic Scattering An asymptotic outgoing state representing a
particle moving freely in the positive x direction has the following form:
IV. Relation Between the Operator for the Delay Time and the Scattering S Matrix
I t takes a time equal to x/v for this particle to move from the scattering center to a certain point outside. We already know that this time of motion is represented by i a / aE in the energy representation. Hence, we may have
Under the effect on an interaction, $(')(x) becomes
Hence, the state $(+)(E) becomes
Let T be the operator representing the time of motion of the free particle mentioned above in the new representation +(+)(E). The time of motion in the energy representation is still repre- sented by i a/aE for the scattering case. If i a/aE operates on \Ir(+ )(E), the time of motioil of the particle in this scattering case is obtained. This time should be shorter than T. since the article
One of the important features of the discussioil presented in the previous section is that, if we accept the operator for the time of motion of the particle as T = i a / aE in the energy representa- tion where the asymptotic outgoing states are used to represent the outgoing particle, then we may deduce the explicit form of the operator for the delay time in terms of the scattering S matrix by simple reasoning. In this section, we shall show
is delayed by a time Q inside the interactio~; range. Therefore, one must have the following operator relation :
We can interpret the meaning of eq. E55] as follows. If we measure the times of motion of the particles in the two cases, one under an inter- action and the other free, we can see that while we have already detected the particle in the free motion, the particle under an interaction still
Can
. J. P
hys.
Dow
nloa
ded
from
ww
w.n
rcre
sear
chpr
ess.
com
by
MC
GIL
L U
NIV
ER
SIT
Y o
n 11
/19/
14Fo
r pe
rson
al u
se o
nly.
286 CANADIAN JOURNAL O F PHYSICS. VOL. 47, 1969
stays somewhere in front of the counter, due to 6 the fact that the particle has stayed inside the = [ p ( E ) , i $(')(E)] potential range for a while. If we calculate the mean positions of the particle in the two cases, + m (+) E
[ $ ( + ) ( ~ ) ] * i ~ ? k U d E
x,' = [$(+)(E), xop$(+)(E)] (with interaction) o aE
The mean delay time or mean life is given by x, = [+(+'(E), xop $'+'(E)] (free)
we expect to have x,' < x,. As for the time of ~621 .c = <Q) = c + ( + ) ( ~ ) , Q$(+)(E)I
m motion, it should therefore be shorter in the case of the particle under an interaction effect. = So [$'+'(E)]* 2 $(+)(E) dE
Equation [55] can be written equivalently as = (To) - (To')
At the asymptotic region, $(+)(E) = eiq $(+)(E). +(+)(E)can be considered to be the unitary trans- formation of $(+)(E). T is therefore related to T~ = i ala E by
[571 a .
eiq i - e- Iq = T aE
Hence,
We believe that the argument shown above is more reasonable than Lippmann's since the delay time comes out with a correct positive sign, while in Lippmann's derivation, we have trouble with the minus sign in front of aq/aE. The plus sign is believed to be more correct since, in the resonance scattering, aq/aE is positive under the effect of an attractive potential (Wigner 1955).
Usually, these times cannot be known precisely. Their mean values must be calculated. The mean time of motion of the particle under the inter- action effect is given by
The mean time of motion of the particle in the case of free motion is
Through eq. [62], it is concluded that the mean life of the scattering is obtained by averaging the delay time aq/aE over the indefinite energies of the scattering process.
2. Multichannel Scattering We assume that the scattering process is studied
in the center of mass system. States of different channels of the scattering are indicated by p or v. A typical state is represented by a ket IE; p) where E is the energy of the scattering system and p represents other quantum numbers as well as the direction (8, $) of the relative momentum. Let I@) be a state representing a free motion particle, then the energy representation of this state having p as other quantum numbers will be @,(E) = (E ; pi@). If the helicity states (Jacob and Wick 1959) are used for the discussion, then
[631 @A,A,(E; = (E; h1h28$l@)
As we already knew, an outgoing scattered state under the effect of the interaction becomes
where S is an unitary operator which maps I@) into IY). One may write [64] in the energy representation as
Certainly, we must assume that the energy is conserved in the transition, i.e. (E; pISJEf; V)
= (E; pISIE; v)6(E- E'). Equation [65] can be rewritten as
[66a] (E; ply) = (E; ~ ( s I E ; v)(E; vl@) v
Can
. J. P
hys.
Dow
nloa
ded
from
ww
w.n
rcre
sear
chpr
ess.
com
by
MC
GIL
L U
NIV
ER
SIT
Y o
n 11
/19/
14Fo
r pe
rson
al u
se o
nly.
GIEN: OPERATORS FOR TIME (
For the helicity states, we may write
= Sdnl x (E; O+hlh,lSIE; O1+'hl'h,') 0. I
= S dn ' x (E; O+h,h,lSIE; O'+'hllh,') (A')
X @h,'hz'(E;
C(,,). represents the summation over all the possible helicity states. One may write eq. [66b] under the matrix form as
S is now the scattering matrix with Spv as ele- ments. +(E) and +(E) are column matrices with @,(E) and $,,(E) as elements. For the multi- channel scattering, eqs. [52] and [56] become
Since T is obtained from To by an unitary trans- formation,
r711 T = s . T , . s ~
Equation [70] can be written as
We deduce that
if S is a unitary matrix. The S matrix has been assumed to be unitary as usual. It is possible to show that Q is a Hermitian operator if S is unitary. In fact,
1741 ast t Qt = ( i s =)
2~ MOT~ON AND DELAY TIME 287
If S is unitary, i.e. SSt = I, then
Since Q is Hermitian, it could be diagonalized. The elements of this diagonalized Q matrix are lifetime (or delay time) eigenvalues of the scatter- ing process. The corresponding lifetime eigen- states will be denoted by $i(E) such that
It should be noted that the lifetime eigenstate $i(E) cannot be an eigenstate of the operator To1 = i a/aE since To' and Q do not commute. i a/aE and Q commute when
i.e. only if
This condition seems to be absurd, since it would mean that the phase shift qi(E) depends linearly on the energy E. It should also be noted that the number of lifetime eigenvalues depends on the number of the opened channels in the scattering.
From the expression of the operator Q = - i(as/aE)Sf, one may also write
By a suitable choice of the boundary conditions for the S matrix ( S = I for E + oo) one obtains
This boundary condition for S is reasonable, since it means that as the energy becomes very great, the particle will pass very quickly by the potential range and has no chance to interact.
Equation [78] is rather promising. It permits us to calculate S(E) in terms of Q(E). If Q(E) is known, S(E) can, in principle, be deduced by integrating eq. [78]. Even if eq. [78] cannot be solved exactly, one can still hope to solve it by the conventional perturbation method.
V. Conclusion and Discussion
In this paper, we have tried to make the operators for the time of motion and the delay time become legitimate in quantum theory. We
Can
. J. P
hys.
Dow
nloa
ded
from
ww
w.n
rcre
sear
chpr
ess.
com
by
MC
GIL
L U
NIV
ER
SIT
Y o
n 11
/19/
14Fo
r pe
rson
al u
se o
nly.
288 CANADIAN JOURNAL OF PHYSICS. VOL. 47, 1969
have shown that the energy representation of the asymptotic outgoing scattering states +(+)(E) should be used to describe the physical system in this case. In the energy representation, the operator for the time of motion is T = i a/aE. It should be stressed once again that the operator Tactually represents the time interval which the particle uses to move a distance x from the scattering center to the outside. The operator for the delay time can be derived by simple reasoning. Its expression in terms of the S matrix is Q = - i(as/a E)S?.
The states +(+)(E) may be seen as physical states in Heisenberg's picture. They are indepen- dent of the time parameter t, while the corre- sponding operators, for instance the operator for the time of motion, are time dependent. T varies with t according to T(t) = eiHot T(0) e-iHot = t + T(0). The distribution I+(+)(E)12 repre- sents the probability that the particle has energy equal to E. I+ (+ ) (E ) I~ should have a maximum peak at E, which is the most probable value of E. The time distribution I x ( + ) ( T ) ~ ~ has a maximum peak at T, and the position distribution I+(+)(X)I2 at x,. T, and x, are, respectively, the most probable value of time of motion Tand of position x of the particle. To move a distance x,, the particle -. having an energy E,, i.e. velocity v, = J2tn~,/rn, must use a time T, equal to x,/v,. When the interaction is switched on, the energy distribution I + ( + ) ( E ) ~ ~ has a peak at the same value E, of E (+(+)(E) only differs from +(+)(E) by a phase), but the position distribution I+(+)(x)12 now has a peak at x,' less than x,.
The particle was delayed inside the potential range. Therefore, at a certain time t while the free particle has already been at x,, the position of the scattered particle is still x,' less than x,. Similarly, the time distribution IX(T)I2 for this case has a maximum peak at T,' less than T, and such that, classically, T, = T,' plus the delay time.
If one fixes the distance x, from the scattering center (target) to the point of observation (co~mter) and measures the time of motion T of a free particle having sufficiently steady energy, the results are arbitrary, but one may expect that the most probable value of Tis T, = x,/v,. When the interaction is switched on, one has to wait a time equal to T, plus the delay time to have the particle detected at x,.
It is the operator for the delay time which interests us the most. Among the states $(+)(E),
there are a set of orthonormal states which diagonalize the lifetime (or delay time) operator Q. One may always find this set of eigenstates since Q is a Hermitian operator. This set of eigenstates is assumed to be complete in the sense that any outgoing scattering state +(+) (E) can be expanded in terms of them. Let + i ( + ) ( ~ ) be an eigenstate of the lifetime operator Q having q, as its eigenvalue. Then I)~(+)(E) represents the asymptotic behavior of a collision state having a definite lifetime eigenvalue. Any asymptotic out- going state +,(+)(E) describing a collision process h can be considered to be the superposition of the lifetime eigenstates + i ( ' ) (~ ) ,
If only one lifetime eigenstate, say +,(+)(E), has its eigenlifetime qk(E) under the form of a Wigner resonance lifetime, for instance q,(E) = Tk/ [(E- E0)' + (Tk2/4)], the probability that the collision passes through a resonance inter- mediate state of lifetime qk(E) is la,,12, while the probability that the collision is of the simple type is xi., la,i12. The mean life of the resonance state will be given by
Although we have been using the asymptotic out- going states in our discussion, it is believed that there is no loss of generality, since one usually observes the scattering process at the periphery.
The relation between Q and S matrices promises many interesting features. As we have mentioned previously, one may calculate S in terms of Q by using eq. 1781. Q(E) now replaces the role of the potential V(t). In other words, if instead of being given the form of the potential we are given the form of the lifetime operator, we may derive S(E), at least by the conventional perturbation method and, hence, the amplitudes and cross sections can be calculated in terms oS Q(E). Since there have been many theories for the S matrix, especially the analytic S matrix theory and the Regge pole theory, the relation between Q and Scould help us in finding the properties of the Q matrix in terms of the known properties of
Can
. J. P
hys.
Dow
nloa
ded
from
ww
w.n
rcre
sear
chpr
ess.
com
by
MC
GIL
L U
NIV
ER
SIT
Y o
n 11
/19/
14Fo
r pe
rson
al u
se o
nly.
GIEX: OPERATORS FOR TIME OF MOTION AND DELAY TIME 289
the S matrix. These properties will probably be used as guidelines for our choice of the form of Q. The theory of lifetime operators might also initiate a new technique of a~ialysis of tlie datas in high-energy scattering.
Acknowledgments
The author is indebted to tlie National Re- search Council of Canada for Operating Grant A-3962 under which this work was performed. He also wishes to thank Profs. C. Bloch and M. Froissart for their hospitality at the Department of Theoretical Physics, CEN-Saclay, France,
during the summer of 1968, where a part of these ideas was clarified.
GIEN, T. T. 1965a. J. Math. Phys. 6 , 671. -- 19656. Can. J. Phys. 43, 1978. 1966. (Unpublished).
JACOB, M. and WICK, G. C. 1959. Ann. Phys. (N.Y.), 7, 404.
KEMULE, E. C. 1958. Quantum mechanics (Dover Publica- tions, New York), Sect. 30, p. 162.
LIPPMANN, B. A. 1966. Phys Rev. 151, 1023. SMITH, F. T. 1960. Phys. Rev. 118, 349.
1962. J. Chem. Phys. 36, 248. 1963a. J. Chem. Phys. 38, 1304. 19636. Phys. Rev. 130, 394. 1963c. Phys. Rev. 131, 2803.
WIGNER, E. 1955. Phys. Rev. 98, 145.
Can
. J. P
hys.
Dow
nloa
ded
from
ww
w.n
rcre
sear
chpr
ess.
com
by
MC
GIL
L U
NIV
ER
SIT
Y o
n 11
/19/
14Fo
r pe
rson
al u
se o
nly.
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. E. Papp. 1974. Field-theoretical space-time quantisation. International Journal of Theoretical Physics 9, 101-115. [CrossRef]
Can
. J. P
hys.
Dow
nloa
ded
from
ww
w.n
rcre
sear
chpr
ess.
com
by
MC
GIL
L U
NIV
ER
SIT
Y o
n 11
/19/
14Fo
r pe
rson
al u
se o
nly.