Scattering is an important component of any quantum mechanics course. However, the scattering amplitude in the case of a general potential is often calculated using the simple Born approximation, which does not embed general properties such as unitarity or analyticity. We show that a relatively simple extension, the eikonal approximation, offers a significant improvement and demonstrate this in the case of the electromagnetic and gravitational interactions.
REFERENCES
1.
See, for example,
E.
Merzbacher
, Quantum Mechanics
(
Wiley
,
New York
, 1998
), Chaps. 7 (WKB) and 13 (Scattering).2.
Perhaps the best discussion of the eikonal method is that of
R. J.
Glauber
, “
High energy collision theory
,” in Lectures in Theoretical Physics
, edited by
W. E.
Brittin
and
L. G.
Dunham
(
Interscience Pub
.,
New York
, 1959
), Vol.
1
, p. 315
;an excellent discussion and list of references is also given in
R. G.
Newton
, Scattering Theory of Waves and Particles
(
Springer-Verlag
,
New York
, 1982
);a field theoretic derivation is given by
M.
Levy
and
J.
Sucher
, “
Eikonal approximation in quantum field theory
,” Phys. Rev.
186
, 1656
–1669
(1969
);a useful introductory presentation is posted on the internet by
K. V.
Shajesh
, “
Eikonal approximation
,” at ⟨http://nhn.ou.edu/~shajesh/eikonal/sp.pdf.⟩3.
4.
L. D.
Landau
and
E. M.
Lifshitz
, Quantum Mechanics: Nonrelativistic Theory
(
Pergamon
,
New York
, 1977
), Chap. XVII.5.
J. J.
Sakurai
and
J.
Napolitano
, Modern Quantum Mechanics
(
Addison-Wesley
,
San Francisco
, 2011
), Chap. 6.6.
7.
H. M.
van Horn
and
E. E.
Salpeter
, “
WKB approximation in three dimensions
,” Phys. Rev.
157
, 751
–757
(1967
).8.
J. B.
Hartle
, An Introduction to Einsteon's General Relativity
(
Pearson
,
New York
, 2003
), Chap. 10.9.
D. N.
Kabat
and
M.
Ortiz
, “
Eikonal quantum gravity and Planckian scattering
,” Nucl. Phys. B
388
, 570
–592
(1987
).10.
The
expansion is extensively discussed in a recent paper by
W.-M.
Chen
et al, “
Gravitational Faraday effect from on-shell amplitudes
,” J. High Energy Phys.
2022
, 58
.11.
M.
Abramovitz
and
I. A.
Stegun
, Handbook of Mathematical Functions
(
Nat. Bur. Standards
,
Washington, D.C
., 1968
).12.
I. S.
Gradshteyn
and
I. M.
Rhyzik
, Table of Integrals Series and Products
(
Academic Press
,
New York
, 1965
).13.
G.
Hooft
, “
Graviton dominance in ultra-high-energy scattering
,” Phys. Lett. B
198
, 61
–63
(1987
), see also Ref. 12.14.
See, e.g.,
H.
Goldstein
, Classical Mechanics
(
Addison-Wesley
,
Reading, MA
, 1956
), Chap. 3.15.
E.
Golowich
,
P.
Gribosky
, and
P.
Pal
, “
Gravitational scattering of quantum particles
,” Am. J. Phys.
58
, 688
–691
(1990
).16.
H.
Abele
and
H.
Leeb
, “
Gravitation and quantum interference experiments with neutrons
,” New J. Phys.
14
, 055010
(2012
);V. V.
Nevishevsky
et al, “
Quantum states of neutrons in the earth's gravitational field
,” Nature
415
, 297
–299
(2002
).
[PubMed]
17.
See, e.g.,
R.
Ferraro
, “
The equivalence principle and the bending of light
,” Am. J. Phys.
71
, 168
–170
(2002
);L.
Lerner
, “
A simple calculation of the deflection of light in a schwarzschild gravitational field
,” Am. J. Phys.
65
, 1194
–1196
(1997
).18.
19.
See, e.g.,
J. D.
Bjorken
and
S. D.
Drell
, Relativistic Quantum Fields
(
McGraw-Hill
,
New York
, 1964
);B. G.
Chen
et al, Quantum Field Theory Lectures of Sidney Coleman
(
World Scientific
,
Singapore
, 2019
).20.
Y. F.
Bautista
et al, “
From scattering in black hole backgrounds to higher spin amplitudes: Part I
,” J. High Energy Phys.
2023
, 136
;M.
Huber
et al, “
Eikonal phase matrix, deflection angle and time delay in effective field theories of gravity
,” Phys. Rev D
102
, 046014
(2020
).21.
R.
Akboury
,
A.
Saotome
, and
G.
Sterman
, “
High energy scattering and perturbative quantum gravity at next to leading power
,” Phys. Rev. D
103
, 064036
(2021
);S. G.
Naculich
and
H. J.
Schnitzer
, “
Eikonal methods applied to gravitational scattering amplitudes
,” J. High Energy Phys.
2011
, 87
.22.
N. E. J.
Bjerrum-Bohr
et al, “
Light-like scattering in quantum gravity
,” J. High Energy Phys.
2016
, 117
;H.
Chi
, “
Graviton bending in quantum gravity from one loop amplitudes
,” ibid.
99
, 126008
(2019
);D.
Bai
and
Y.
Huang
, “
More on the bending of light in quantum gravity
,” Phys Rev. D
95
, 064045
(2017
).23.
J.
Bodenner
and
C. M.
Will
, “
Deflection of light to second order: A tool for illustrating principles of general relativity
,” Am. J. Phys.
71
, 770
–773
(2003
);E.
Fischbach
and
B. S.
Freedman
, “
Second order contribution to the gravitational deflection of light
,” Phys. Rev. D
22
, 2950
–2952
(1980
).24.
W. R.
Hamilton
, “
Theory of systems of rays
,” Trans. R. Irish Acad.
15
, 69
–174
(1828
). The term “eikonal” was coined by Heinrich Bruns in his 1895 manuscript “Das Eikonal” published in Leipzig by S. Hirzel, an English translation of which by D. H. Delphenich can be found at ⟨http://neo-classical-physics.info/uploads/3/0/6/5/3065888/bruns_-_the_eikonal.pdf⟩.© 2023 Author(s). Published under an exclusive license by American Association of Physics Teachers.
2023
Author(s)
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