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The classical description of scattering from a quantum system
VoIume 81, number I
THE CLASSICAL
CHEMICAL
DESCRIPTION
PHYSICS LEIYERS
OF SCATTERING
K.L. SEBASTIAN * Donnan Luboratories, University of Liverpool,
Liverpool
1
FROM A QK4NTUM
July 1981
SYSTEM
L69 3l3X. UK
Received 18 March 1981; in final form 14 May 1981
Using the infhrence functional method, we arrive at the best possible classical description of a particle (system) interacting with a quantum system at a non-zero temperature. It is found to be a stochastic one. Applications of the method to Problems of scattering from surfaces are pointed out.
In this letter, we consider the following question: what is the best possible classical description of a system (the primary (P) system) interacting with another (the secondary (S) system) whose description has to be quantumstatistical? The question is motivated by the possible applications of such a description in surface physics, though it would certainly have applications in other fields of physics_ Thus, the method would be of use in inelastic/reac-
tive scattering of atoms and molecules from surfaces_ EIectron-hole pair generation and phonon excitation have important roles in such processes and have attracted much attention [ 12]_ In the framework of our approach, one could imagine the translational motion of the projectile as the P-system and the other degrees of freedom (which could be electronic or vibrational) as the S-system. Sometimes, it may be convenient to include a few of the surface degrees of freedom, viz., those which interact strongly with the projectile, In the P-system_ Rigorously, in order that such an approach be valid, the wavelengths for motion in the P-system should be small_ However, it should be possrble to use the classical paths in the same spirit as in the classical S-matrix method [3] and get a semiclassical approximation of much wider applicability. Assuming that both S- and P-systems obey quantum mechanics, we eliminate the S-system coordinates from the problem, using the influence functional approach of Feynman and Hibbs [4]. The influence functional is evaluated approximately, using operator techniques_ To arrive at the best possible classical description, we make use of stationary phase arguments (SPA) similar to those of Pechukas and Davis [S] and Manz [6] _These authers assume the S-system to undergo a specific transition and fmd the best classical trajectory for the P-system, for that transition_ In comparison, we are not interested in the final state of the S-system and our description is one which is averaged overall possible final states of the S-system. While the main ideas of this letter were already formulated, the author became aware of the recent paper by Mohring and Smilansky [7], who have developed ideas similar to our own. However, our final result, eq. (1 I), differs from theirs in that it is a stochastic one and would lead to the correct classical limit [8] if one decided to treat the S-system too classically. We denote the coordinates of the P-system by Q. Assuming the P-system to be at Qi and the S-system to have a density operator p(ti) at a time ti, the probability has happened to the S-system is [4,7]
of fiiding
the P-system
at Qf at the time tf, irrespective
pCQ@f, Qiti) =JJQ~QI~EIexpC~-l(SCQl - S[Bl)IFCQ, al , where F[Q, Q] is the influence functional
[7] _The action functional S[Q]
of what
(1) = .fzfL(Q,
Q, r)dt, L being the
Permanent address: Department of Chemistry, University of C&cut, Kemla 673 635, India.
14
0 009-2614/81/0000-0000/$02.50
0 North-Holland Publishing Company
CHEMICAL PHYSICS LETTERS.
Volume 81, number 1
.
1 July 1981
lagrangian for the P-system_ The path integrals in (1) are over all Q(t) and Q(Z), satisfying Q(ti) = B(ti) = Qi and Q(tf> =-&tf> = Qf- Putting F[Q, QJ = exp(ifi-l ?[Q, 81) ated using SPA [5,6] lead to 6(s[Q] -S[Q] + G[Q, Q]) = 0, which has, in general, complex solutions for Q(t) and Q(t). Thi s is the procedure adopted by Mohring and Smilansky [7]. We adopt a technique, which would keep all the paths real and would lead to the correct classical limit if one decided to treat the S-system too classically. In the usual operator notatron, the influence functional is J’CQ,
rZ1= tr [rr(t,, ti, Q)P(ti) u(ti, tf, @I -
(2)
tr denotes trace over the S-system coordinates. U(+, ti, Q) is the time-development operator for the S-system if the P-system followed a trajectory Q(t). We assume the hamiltonian for this time development to be
WQI =&j(t) +f(Q(OIA ,
(3)
where f is a function depending only upon the coordinates of the P-system and A is a self-adjoint operator on the S-system. Changing to the interaction picture corresponding to Ho(t) gives
F[Q,al =(tTCex~[-iri-~ If~f(~(~))~t)l~~C~~~C--iti~‘If~f@@~)~~Ol~>>.
(3
((M)) stands for tr[&@&)]
_(?)Tis the (anti-) time ordering operator.&) is in the interaction Kubo’s generalised cumulant expansion [9 ] and retaining only the first two cumulants lead to
@[Q, Q] =& with cpl=-
rf s ri
picture. Using
+Gj2,
(5)
Cf(Q(s)) - f (t%r))l {(A^@Wt
(6)
and a2 =$
dt i *i
ds[f(Q(t))
-f@(t))]
[f(Q(s))<
-f(~(s))Hiil(f)ii*(s),>‘l
(7)
,
*i
where A I(t) =A@) - CA(t)))_ Let us define the real functions R(t, s) and I(t, s) by <~‘(t)8’(s)N + iI(t, s). Then R(s, t) = R (t, s) and I(s, t) = -I(t, s)_ JZq. (1 j may be written as
J’(Qftf, Qifi> = JD 1Q10 El (expCi~-’ (S, [Ql -Sm @I + @R[Q, GI )I), 3
=R(t,
s)
(8)
where S, [Q] = S[Q] - .fi: f(Q(t)) u(t)dt and aR [Q, Q] is the real part of @ [Q, Q] _ ( ),, denotes average with respect to the real gausslan stochastic process u(t) having mean zero and correlation frrnction (u(t)u(s)), = R(t, s). Fq_ (8) can be proved using the equality, = exp [- _f$ dt Jfi ds R(t, s)k(t)k(s)]
(9)
satisfied by real k() *_ Notice that we have made the phase of the integrand ln eq_ (8) real, at the cost of making the action stochastic_ SPA with respect to Q(t) now leads to the noncausal equation
ti
+ u(t) + [ j
ds If(Q(s))
+f (i%s))l
+ s’ cisCf> - f@@Nl] fi-’ W> 4) = 0 t
(101
5 < )” could, in may cases, be expressed in terms of a functional integral, see ref_ [4 1. 15
Volume 8 1, number 1
CHEMICAL PHYSlcs
A similar equation results from the variation of &I, Q and 5 in eq. (10). Physically, the most interesting when one gets the catsal equation
LET-XERS
1 Jury 1981
which we shall call (lOa). Xt can be obtained by interchanging solution to (10) and (lOa) is obtained by putting &) = Q(t),
We interpret (11) as the best possible classical descnption for the P-system. It has: (1) an average force, (2) a random force (noise) and (3) a systematic, damping force acting on the P-system. These are given by the three terms on the rhs of eq, (I. I). The naise and the damping force are deter~ned by the quantum correlation function (@(r)a ‘(s))), this being due to the ftuctuation-dissipation theorem [S,lO] _If Ho(f) is time-independent, then v(t) is stationary_ Our consrderations have been fairly general, except for the fact that the interaction between the two systems
is assumed to have a separable form [see eq. (3)]. The analysis can be extended to cases where the potential is a Iinear combination of separabte terms- Even with the simple separable potential, the method has interesting applications in surface physics. (1) For phonon problems, our approach goes beyond that of Adelman f83 and treats t,‘le solid quantum mechanicalIy_ It should be noted that if the S-system consisted of harmonic oscillators and if A were the Iinear combination of the harmonic oscillator coordinates, then ch[Q, G ] = dit2 exactly. If one takes the classical limit for the S-system too, then the quantum correlation functions would go over to the corresponding classical quanti$ies and eq. (11) would reduce to the equation given by Ade~rn~ [8] _(21 The con-a~abatic effects of the electronic system on the inelastic scattering/trapping of an atom on a metal can now be taken into account by a stochastic approach, This should be of interest in view of the recent suggestion of TuUy [I l] as to the possibility of such an approach. Ho(t) may be chosen as the hamiltonian for the non-interacting system and A may represent the interaction between an orbita! on the atom and a localised orbirdl on the surface. Also, by suitabfy choosing Ho(r), rt should be possrble to make the approximations as good as one desires, For example, Ha(t) may represent the hamiltonian for the electronic part, if the atom foilowed a given trajectory.
The author hatefully a~~owIedg~~ stim~lat~g ~SC~~S~ORS with Professor TB. Crimley and thanks the S.R.C. (UK) for financial support. References El) G-P. Britio and T.B. G&uley, Surface Sci. 89 (1979) 226; 3-W. Gadzukand ii. M&u, Phys. Rev. B22 (1980) 2603. [%I F-0. Goodman and Ii-Y. bacon, Dynamics of gas surface scattering
I6
THE CLASSICAL
CHEMICAL
DESCRIPTION
PHYSICS LEIYERS
OF SCATTERING
K.L. SEBASTIAN * Donnan Luboratories, University of Liverpool,
Liverpool
1
FROM A QK4NTUM
July 1981
SYSTEM
L69 3l3X. UK
Received 18 March 1981; in final form 14 May 1981
Using the infhrence functional method, we arrive at the best possible classical description of a particle (system) interacting with a quantum system at a non-zero temperature. It is found to be a stochastic one. Applications of the method to Problems of scattering from surfaces are pointed out.
In this letter, we consider the following question: what is the best possible classical description of a system (the primary (P) system) interacting with another (the secondary (S) system) whose description has to be quantumstatistical? The question is motivated by the possible applications of such a description in surface physics, though it would certainly have applications in other fields of physics_ Thus, the method would be of use in inelastic/reac-
tive scattering of atoms and molecules from surfaces_ EIectron-hole pair generation and phonon excitation have important roles in such processes and have attracted much attention [ 12]_ In the framework of our approach, one could imagine the translational motion of the projectile as the P-system and the other degrees of freedom (which could be electronic or vibrational) as the S-system. Sometimes, it may be convenient to include a few of the surface degrees of freedom, viz., those which interact strongly with the projectile, In the P-system_ Rigorously, in order that such an approach be valid, the wavelengths for motion in the P-system should be small_ However, it should be possrble to use the classical paths in the same spirit as in the classical S-matrix method [3] and get a semiclassical approximation of much wider applicability. Assuming that both S- and P-systems obey quantum mechanics, we eliminate the S-system coordinates from the problem, using the influence functional approach of Feynman and Hibbs [4]. The influence functional is evaluated approximately, using operator techniques_ To arrive at the best possible classical description, we make use of stationary phase arguments (SPA) similar to those of Pechukas and Davis [S] and Manz [6] _These authers assume the S-system to undergo a specific transition and fmd the best classical trajectory for the P-system, for that transition_ In comparison, we are not interested in the final state of the S-system and our description is one which is averaged overall possible final states of the S-system. While the main ideas of this letter were already formulated, the author became aware of the recent paper by Mohring and Smilansky [7], who have developed ideas similar to our own. However, our final result, eq. (1 I), differs from theirs in that it is a stochastic one and would lead to the correct classical limit [8] if one decided to treat the S-system too classically. We denote the coordinates of the P-system by Q. Assuming the P-system to be at Qi and the S-system to have a density operator p(ti) at a time ti, the probability has happened to the S-system is [4,7]
of fiiding
the P-system
at Qf at the time tf, irrespective
pCQ@f, Qiti) =JJQ~QI~EIexpC~-l(SCQl - S[Bl)IFCQ, al , where F[Q, Q] is the influence functional
[7] _The action functional S[Q]
of what
(1) = .fzfL(Q,
Q, r)dt, L being the
Permanent address: Department of Chemistry, University of C&cut, Kemla 673 635, India.
14
0 009-2614/81/0000-0000/$02.50
0 North-Holland Publishing Company
CHEMICAL PHYSICS LETTERS.
Volume 81, number 1
.
1 July 1981
lagrangian for the P-system_ The path integrals in (1) are over all Q(t) and Q(Z), satisfying Q(ti) = B(ti) = Qi and Q(tf> =-&tf> = Qf- Putting F[Q, QJ = exp(ifi-l ?[Q, 81) ated using SPA [5,6] lead to 6(s[Q] -S[Q] + G[Q, Q]) = 0, which has, in general, complex solutions for Q(t) and Q(t). Thi s is the procedure adopted by Mohring and Smilansky [7]. We adopt a technique, which would keep all the paths real and would lead to the correct classical limit if one decided to treat the S-system too classically. In the usual operator notatron, the influence functional is J’CQ,
rZ1= tr [rr(t,, ti, Q)P(ti) u(ti, tf, @I -
(2)
tr denotes trace over the S-system coordinates. U(+, ti, Q) is the time-development operator for the S-system if the P-system followed a trajectory Q(t). We assume the hamiltonian for this time development to be
WQI =&j(t) +f(Q(OIA ,
(3)
where f is a function depending only upon the coordinates of the P-system and A is a self-adjoint operator on the S-system. Changing to the interaction picture corresponding to Ho(t) gives
F[Q,al =(tTCex~[-iri-~ If~f(~(~))~t)l~~C~~~C--iti~‘If~f@@~)~~Ol~>>.
(3
((M)) stands for tr[&@&)]
_(?)Tis the (anti-) time ordering operator.&) is in the interaction Kubo’s generalised cumulant expansion [9 ] and retaining only the first two cumulants lead to
@[Q, Q] =& with cpl=-
rf s ri
picture. Using
+Gj2,
(5)
Cf(Q(s)) - f (t%r))l {(A^@Wt
(6)
and a2 =$
dt i *i
ds[f(Q(t))
-f@(t))]
[f(Q(s))<
-f(~(s))Hiil(f)ii*(s),>‘l
(7)
,
*i
where A I(t) =A@) - CA(t)))_ Let us define the real functions R(t, s) and I(t, s) by <~‘(t)8’(s)N + iI(t, s). Then R(s, t) = R (t, s) and I(s, t) = -I(t, s)_ JZq. (1 j may be written as
J’(Qftf, Qifi> = JD 1Q10 El (expCi~-’ (S, [Ql -Sm @I + @R[Q, GI )I), 3
=R(t,
s)
(8)
where S, [Q] = S[Q] - .fi: f(Q(t)) u(t)dt and aR [Q, Q] is the real part of @ [Q, Q] _ ( ),, denotes average with respect to the real gausslan stochastic process u(t) having mean zero and correlation frrnction (u(t)u(s)), = R(t, s). Fq_ (8) can be proved using the equality
(9)
satisfied by real k() *_ Notice that we have made the phase of the integrand ln eq_ (8) real, at the cost of making the action stochastic_ SPA with respect to Q(t) now leads to the noncausal equation
ti
+ u(t) + [ j
ds If(Q(s))
+f (i%s))l
+ s’ cisCf
(101
5 < )” could, in may cases, be expressed in terms of a functional integral, see ref_ [4 1. 15
Volume 8 1, number 1
CHEMICAL PHYSlcs
A similar equation results from the variation of &I, Q and 5 in eq. (10). Physically, the most interesting when one gets the catsal equation
LET-XERS
1 Jury 1981
which we shall call (lOa). Xt can be obtained by interchanging solution to (10) and (lOa) is obtained by putting &) = Q(t),
We interpret (11) as the best possible classical descnption for the P-system. It has: (1) an average force, (2) a random force (noise) and (3) a systematic, damping force acting on the P-system. These are given by the three terms on the rhs of eq, (I. I). The naise and the damping force are deter~ned by the quantum correlation function (@(r)a ‘(s))), this being due to the ftuctuation-dissipation theorem [S,lO] _If Ho(f) is time-independent, then v(t) is stationary_ Our consrderations have been fairly general, except for the fact that the interaction between the two systems
is assumed to have a separable form [see eq. (3)]. The analysis can be extended to cases where the potential is a Iinear combination of separabte terms- Even with the simple separable potential, the method has interesting applications in surface physics. (1) For phonon problems, our approach goes beyond that of Adelman f83 and treats t,‘le solid quantum mechanicalIy_ It should be noted that if the S-system consisted of harmonic oscillators and if A were the Iinear combination of the harmonic oscillator coordinates, then ch[Q, G ] = dit2 exactly. If one takes the classical limit for the S-system too, then the quantum correlation functions would go over to the corresponding classical quanti$ies and eq. (11) would reduce to the equation given by Ade~rn~ [8] _(21 The con-a~abatic effects of the electronic system on the inelastic scattering/trapping of an atom on a metal can now be taken into account by a stochastic approach, This should be of interest in view of the recent suggestion of TuUy [I l] as to the possibility of such an approach. Ho(t) may be chosen as the hamiltonian for the non-interacting system and A may represent the interaction between an orbita! on the atom and a localised orbirdl on the surface. Also, by suitabfy choosing Ho(r), rt should be possrble to make the approximations as good as one desires, For example, Ha(t) may represent the hamiltonian for the electronic part, if the atom foilowed a given trajectory.
The author hatefully a~~owIedg~~ stim~lat~g ~SC~~S~ORS with Professor TB. Crimley and thanks the S.R.C. (UK) for financial support. References El) G-P. Britio and T.B. G&uley, Surface Sci. 89 (1979) 226; 3-W. Gadzukand ii. M&u, Phys. Rev. B22 (1980) 2603. [%I F-0. Goodman and Ii-Y. bacon, Dynamics of gas surface scattering
I6