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Eikonal approximation in the configuration space
Volume 54A, number 1
PHYSICS LETTERS
11 August 1975
EIKONAL APPROXIMATION IN THE CONFIGURATION SPACE ~ A. BECHLER Institute of Theoretical Physics, University of Warsaw, Poland
Received 3 June
1975
Conditions of applicability of the eikonal approximation for the nonrelativistic one-particle propagator are investigated. It is shown that the direction of eikonalization cannot be chosen arbitrarily.
Eikonal approximation [1] is in the recent years one of the most intensively explored approximation methods in nonrelativistic quantum mechanics and in the high energy physics [2, 5, 6]. In this note we are going to consider this approximation applied to the time-dependent one-particle propagator G(x, t;y, t’) in the potential theory. Let us take into account the second Born term B2 which can be written as B2
=
3 exp (iky) ~V(k)f-~~ (2ir)3 exp {ip(x—y)}
—
(2ir)
where G
f
2ir exp {—iw(t—t’)}G0(w, p—k),
~
(1)
0 is the Fourier transform of the free retarded propagator, and Vis the Fourier transform of the potential. Peforming w-integration we get B2
=
—iO(t— t’)
f
-~--~-
exp
f-~-~ exp {ip(x—y)}
(iky) I~(k)
(2ir)~
(2~r)~ 2/2m) (t— t’)) + G
(2) 2/2m, p) exp ({(p—k)2/2m}(t-- t’))],
X [G~(p~/2rn,p—k) exp (—i(p 0((p—k) The p-integration in the first term can be written in the parametric form
—if
f~ exp [—i(p2/2m)(t~t’) (2ir)3
dX exp {—i(k2/2m —ie)X}
+
ip(x—y)
±i(pk/m)X].
(3)
We now assume that x—y and t— t’ are large, i.e., either the final or the initial point of the propagation, or both, is in the asymptotic region. We further assume that (x—y)/(t— t’) K/rn is also large. As usually in the eikonal method we assume that the particle does not come close to the potential center (small angle scattering), and therefore only small k’s contribute significantly to (1). We therefore neglect k2 in comparison with pk in (3). Then we get for the first term in (2) -0
(t- t’)
f~
exp {ip(x-y)
-
i(p2/2rn)(t- t’)}
f dXV(y
+
(p/rn)X).
(4)
(2ir)3 The exponent in (4) can be —
written
i((t— t’)/2m)(p—JC)2 +
as
irn(x—y)2/2(t— t’)
(5)
and for large r— t’ we
see that only p’s close to K contribute significantly to the integral. We therefore see that only large p contribute so that neglection of K2 in comparison with pk is reasonable. Shifting now the integration ~r
Supported
in part
by NSF, grant No. GF 36217. 3
Volume 54A, number 1
variable: p B2
=
PHYSICS LETTERS
K + q (q small) we get together with the second term in (2) 2/2rn)(t—t’)l ~3) [x—y (K/rn)(t— t’)] —
=
0(t—
11 August 1975
t’)
exp [i(K
f
dXV(y ±(K/m)X).
(6)
The nth order term can, under above assumptions, be approximated in the same way. After some well known combinatorial rearrangements the k~-integrationsfactorize and we get B~= i(—l )n/fl! 0(t— t’) exp [i(K2/2m)(t— t’)j
~
[x—y (K/m)(t— t’)] —
[~1~
dXV(y + (K/m)X)]
~.
Complete Born series for the propagator can now be summed up yielding 2/2m)(t— t’)] ~ t; y, t’)
=
iO(t
-~
t’)
exp [i(K
[x—y
—
(K/m)(t— t’)] exp [_i
f
dX V(y +
(K/rn) X)].
(8)
Under what conditions K/rn is large? First possibility is that y is large and x is finite. Then K is parallel to the initial monientump 1. Fory finite andx large Kis parallel to the final momentump~.In the scattering problems, however, bothx andy are asymptotic. Then one can choose K=~(p~+p~~.
(9)
This choice corresponds to x = (pf/rn)t andy = (p~/m)t,t— t’ = 2t. Eikonal phase is now accumulated along a straight line path parallel to K and passing through the interaction region [1]. This choice of K allows for a description of scattering in a region wider than forward direction [3, 4]. —
References [1] R.J. Glauber, in Lectures in theoretical physics, eds. W.E. Brittin and L.G. Dunham (Interscience Publishers, Inc., New York, 1959), Vol. 1, p. 315. [21 R. Blankenbecler and R.L. Sugar, Phys. Rev. 183 (1969) 1387. [31 S.J. Wallace, Phys. Rev. Lett. 27 (1971) 622. [41 F. Kujawski, Phys. Rev. D4 (1971) 2573. [51 A. Swift, Phys. Rev. D9 (1974) 1740. [61 H.M. Fried and Y. Hahn, Phys. Rev. D10 (1974) 1908.
4
PHYSICS LETTERS
11 August 1975
EIKONAL APPROXIMATION IN THE CONFIGURATION SPACE ~ A. BECHLER Institute of Theoretical Physics, University of Warsaw, Poland
Received 3 June
1975
Conditions of applicability of the eikonal approximation for the nonrelativistic one-particle propagator are investigated. It is shown that the direction of eikonalization cannot be chosen arbitrarily.
Eikonal approximation [1] is in the recent years one of the most intensively explored approximation methods in nonrelativistic quantum mechanics and in the high energy physics [2, 5, 6]. In this note we are going to consider this approximation applied to the time-dependent one-particle propagator G(x, t;y, t’) in the potential theory. Let us take into account the second Born term B2 which can be written as B2
=
3 exp (iky) ~V(k)f-~~ (2ir)3 exp {ip(x—y)}
—
(2ir)
where G
f
2ir exp {—iw(t—t’)}G0(w, p—k),
~
(1)
0 is the Fourier transform of the free retarded propagator, and Vis the Fourier transform of the potential. Peforming w-integration we get B2
=
—iO(t— t’)
f
-~--~-
exp
f-~-~ exp {ip(x—y)}
(iky) I~(k)
(2ir)~
(2~r)~ 2/2m) (t— t’)) + G
(2) 2/2m, p) exp ({(p—k)2/2m}(t-- t’))],
X [G~(p~/2rn,p—k) exp (—i(p 0((p—k) The p-integration in the first term can be written in the parametric form
—if
f~ exp [—i(p2/2m)(t~t’) (2ir)3
dX exp {—i(k2/2m —ie)X}
+
ip(x—y)
±i(pk/m)X].
(3)
We now assume that x—y and t— t’ are large, i.e., either the final or the initial point of the propagation, or both, is in the asymptotic region. We further assume that (x—y)/(t— t’) K/rn is also large. As usually in the eikonal method we assume that the particle does not come close to the potential center (small angle scattering), and therefore only small k’s contribute significantly to (1). We therefore neglect k2 in comparison with pk in (3). Then we get for the first term in (2) -0
(t- t’)
f~
exp {ip(x-y)
-
i(p2/2rn)(t- t’)}
f dXV(y
+
(p/rn)X).
(4)
(2ir)3 The exponent in (4) can be —
written
i((t— t’)/2m)(p—JC)2 +
as
irn(x—y)2/2(t— t’)
(5)
and for large r— t’ we
see that only p’s close to K contribute significantly to the integral. We therefore see that only large p contribute so that neglection of K2 in comparison with pk is reasonable. Shifting now the integration ~r
Supported
in part
by NSF, grant No. GF 36217. 3
Volume 54A, number 1
variable: p B2
=
PHYSICS LETTERS
K + q (q small) we get together with the second term in (2) 2/2rn)(t—t’)l ~3) [x—y (K/rn)(t— t’)] —
=
0(t—
11 August 1975
t’)
exp [i(K
f
dXV(y ±(K/m)X).
(6)
The nth order term can, under above assumptions, be approximated in the same way. After some well known combinatorial rearrangements the k~-integrationsfactorize and we get B~= i(—l )n/fl! 0(t— t’) exp [i(K2/2m)(t— t’)j
~
[x—y (K/m)(t— t’)] —
[~1~
dXV(y + (K/m)X)]
~.
Complete Born series for the propagator can now be summed up yielding 2/2m)(t— t’)] ~ t; y, t’)
=
iO(t
-~
t’)
exp [i(K
[x—y
—
(K/m)(t— t’)] exp [_i
f
dX V(y +
(K/rn) X)].
(8)
Under what conditions K/rn is large? First possibility is that y is large and x is finite. Then K is parallel to the initial monientump 1. Fory finite andx large Kis parallel to the final momentump~.In the scattering problems, however, bothx andy are asymptotic. Then one can choose K=~(p~+p~~.
(9)
This choice corresponds to x = (pf/rn)t andy = (p~/m)t,t— t’ = 2t. Eikonal phase is now accumulated along a straight line path parallel to K and passing through the interaction region [1]. This choice of K allows for a description of scattering in a region wider than forward direction [3, 4]. —
References [1] R.J. Glauber, in Lectures in theoretical physics, eds. W.E. Brittin and L.G. Dunham (Interscience Publishers, Inc., New York, 1959), Vol. 1, p. 315. [21 R. Blankenbecler and R.L. Sugar, Phys. Rev. 183 (1969) 1387. [31 S.J. Wallace, Phys. Rev. Lett. 27 (1971) 622. [41 F. Kujawski, Phys. Rev. D4 (1971) 2573. [51 A. Swift, Phys. Rev. D9 (1974) 1740. [61 H.M. Fried and Y. Hahn, Phys. Rev. D10 (1974) 1908.
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