This article presents the Zero Range Potential (ZRP) approximation as an alternative to treat quantum scattering problems. The main focus is to provide an undergraduate-level derivation of the ZRP boundary condition that describes the interaction between an electron and a potential, using the spherical-potential-well analytical solutions. The validity of the approximation is discussed qualitatively and quantitatively. Finally, the ZRP approximation is applied to the scattering of an electron by an atom, comparing the results obtained with modern experimental measurements.
References
1.
D. J.
Griffiths
and D. F.
Darrel
, Introduction to Quantum Mechanics
(Cambridge U.P
., Cambridge
, 2018
).2.
J. J.
Sakurai
and S. F.
Tuan
, Modern Quantum Mechanics
, revised edition (Addison-Wesley, Reading
, Massachusetts
, 1994
).3.
L. D.
Landau
and E. M.
Lifshitz
, Quantum Mechanics: Non-Relativistic Theory
(Elsevier
, Oxford
, 2013
), Vol. 3
.4.
5.
B. H.
Bransden
and C. J.
Joachain
, Physics of Atoms and Molecules
(Pearson Education, Prentice Hall
, 2003
).6.
N. F
Mott
and H. S. W.
Massey
, The Theory of Atomic Collisions
(Clarendon Press
, Oxford
, 1949
).7.
8.
H. A.
Bethe
, “Theory of the effective range in nuclear scattering
,” Phys. Rev.
76
, 38
–50
(1949
).9.
T. F.
O'Malley
, L.
Rosenberg
, and L.
Spruch
, “Low-energy scattering of a charged particle by a neutral polarizable system
,” Phys. Rev.
125
, 1300
–1310
(1962
).10.
Y. N.
Demkov
and V. N.
Ostrovskii
, Zero-Range Potentials and their Applications in Atomic Physics
(Springer Science & Business Media
, New York
, 1988
).11.
G. F.
Drukarev
and I. Y.
Yurova
, “Multiple-scattering approach to the vibrational-rotational excitation of molecules by slow electrons
,” J. Phys. B: At. Mol. Phys.
10
, 3551
–3558
(1977
).12.
G. F.
Gribakin
, “Enhancement of positron annihilation on molecules due to vibrational Feshbach resonances
,” Nucl. Inst. Meth. Phys. Res. B.
192
, 26
–39
(2002
).13.
L.
Pricoupenko
, “Pseudopotential in resonant regimes
,” Phys. Rev. A
73
, 012701
–012713
(2006
).14.
R.
Szmytkowski
, “Zero-range potentials for Dirac particles: Scattering and related continuum problems
,” Phys. Rev. A
76
, 052708
(2005
).15.
R.
Szmytkowski
, “Reply to “Comment on ‘Zero-range potentials for Dirac particles: Scattering and related continuum problems’”
,” Phys. Rev. A
73
, 026702
–026707
(2006
).16.
J. M.
Blatt
and V. F.
Weisskopf
, Theoretical Nuclear Physics
(John Weiley & Sons
, New York
, 1952
).17.
E.
Fermi
and L.
Marshall
, “Interference phenomena of slow neutrons
,” Phys. Rev.
71
, 666
–677
(1947
).18.
Z.
Ahmed
, “Studying the scattering length by varying the depth of the potential well
,” Am. J. Phys.
78
, 418
–421
(2010
).19.
G.
Drukarev
, “The zero-range potential model and its application in atomic and molecular physics
,” Adv. Quantum Chem.
11
, 251
–274
(1978
).20.
S. H.
Dong
, X. W.
Hou
, and Z. Q.
Ma
, “Levinson's theorem for the Schrödinger equation in two dimensions
,” Phys. Rev. A
58
, 2790
–2796
(1998
).21.
S. H.
Dong
and Z. Q.
Ma
, “Levinson's theorem for the Schrödinger equation in one dimension
,” Int. J. Theo. Phys.
39
, 469
–481
(2000
).22.
S. H.
Dong
, X. W.
Hou
, and Z. Q.
Ma
, “Nonrelativistic Levinson's theorem in D dimensions
,” Phys. Rev. A
65
, 042717
–042723
(2002
).23.
M.
Kurokawa
, M.
Kitajima
, K.
Toyoshima
, T.
Kishino
, T.
Odagiri
, H.
Kato
, M.
Hoshino
, H.
Tanaka
, and K.
Ito
, “High-resolution total-cross-section measurements for electron scattering from Ar, Kr, and Xe employing a threshold-photoelectron source
,” Phys. Rev. A
84
, 062717
–062730
(2011
).24.
T.
Ohmura
, Y.
Hara
, and T.
Yamanouchi
, “Low energy electron-hydrogen scattering
,” Prog. Theor. Phys.
20
, 82
–88
(1958
).25.
L. B
Madsen
, “Effective range theory
,” Am. J. Phys.
70
, 811
–814
(2002
).26.
T. M.
Miller
and B.
Bederson
, “Atomic and molecular polarizabilities-a review of recent advances
,” Adv. At. Mol. Phys.
13
, 1
–55
(1978
).27.
E. P.
Seidel
and F.
Arretche
, “Rovibrational excitation of rare-gas dimers by electron impact
,” Phys. Rev. A
98
, 052707
–052722
(2018
).28.
J.
Vogt
and S.
Alvarez
, “van der Waals radii of noble gases
,” Inorg. Chem.
53
, 9260
–9266
(2014
).© 2019 American Association of Physics Teachers.
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