The effective range formula for scattering by a short range potential is derived for arbitrary angular momentum of the scattered particle. The derivation is a generalization of a method due to Bethe, and makes no reference to the analytical properties of Jost functions. The derivation is a useful supplement to the textbook coverage of the subject.

1.
H. A.
Bethe
, “
Theory of the effective range in nuclear scattering
,”
Phys. Rev.
76
,
38
50
(
1949
).
2.
P. J. Siemens and A. S. Jensen, Elements of Nuclei, Many-Body Physics with the Strong Interaction (Addison–Wesley, New York, 1987), pp. 14–15.
3.
C. J. Joachain, Quantum Collision Theory (North-Holland/Elsevier, Amsterdam, 1975), pp. 286–292.
4.
L. I. Schiff, Quantum Mechanics (McGraw–Hill, Singapore, 1968), pp. 459–461.
5.
A. Messiah, Quantum Mechanics (North-Holland, Amsterdam, 1991), Vol. I, pp. 406–409.
6.
P. G. Burke and C. J. Joachain, Theory of Electron-Atom Collisions: Part 1, Potential Scattering (Plenum, New York, 1994), pp. 116–134.
7.
R. G. Newton, Scattering of Waves and Particles (Springer-Verlag, New York, 1982), pp. 308–313.
8.
N. F. Mott and H. S. W. Massey, The Theory of Atomic Collisions (Oxford U.P., London, 1965), pp. 48–51.
9.
O.
Dumbrajs
,
R.
Koch
,
H.
Pilkuhn
,
G. C.
Oades
,
H.
Behrens
,
J. J.
de Swart
, and
P.
Kroll
,
Phys. Daten
4–3
,
277
355
(
1983
).
10.
A. Bohr and B. R. Mottelson, Nuclear Structure (Benjamin, New York, 1969), Vol. I, pp. 241–242.
11.
J.
Schwinger
, “
A variational principle for scattering problems
,”
Phys. Rev.
72
,
742
(
1947
).
12.
M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972), p. 437.
13.
L. C.
Biedenharn
and
J. M.
Blatt
, “
Neutron-proton scattering with spin-orbit coupling. II. Variational formulation and effective range theory
,”
Phys. Rev.
93
,
1387
1394
(
1954
).
This content is only available via PDF.
AAPT members receive access to the American Journal of Physics and The Physics Teacher as a member benefit. To learn more about this member benefit and becoming an AAPT member, visit the Joining AAPT page.