The procedure commonly used in textbooks for determining the eigenvalues and eigenstates for a particle in an attractive Coulomb potential is not symmetric in the way the boundary conditions at r = 0 and r are considered. We highlight this fact by solving a model for the Coulomb potential with a cutoff (representing the finite extent of the nucleus). In the limit that the cutoff is reduced to zero, we recover the standard result, albeit in a non-standard way. This example is used to emphasize that a more consistent approach to solving the Coulomb problem in quantum mechanics requires an examination of the non-standard solution. The end result is, of course, the same.

1.
E.
Schrödinger
, “
Quantisierung als Eigenwertproblem 1
,”
Ann. Phys.
79
,
361
376
(
1926
).
Translation is available in
E.
Schrödinger
,
Collected Papers on Wave Mechanics
(
Blackie & Son Limited
,
London
,
1928
). This paper is titled in English as, “Quantisation as a problem of proper values (part I).”
2.
D. J.
Griffiths
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Introduction to Quantum Mechanics
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,
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,
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).
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,
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,
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and
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,
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,
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Frank W. J.
Olver
,
Daniel W.
Lozier
,
Ronald F.
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, and
Charles W.
Clark
,
NIST Handbook of Mathematical Functions
(
Cambridge U.P.
,
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,
2010
), p.
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.
10.
M.
Abramowitz
and
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(
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,
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,
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).
11.

The Laguerre polynomials are usually defined in physics texts (Ref. 2) in a manner that often differs from alternate definitions by a factorial factor. For example, LArfn12+1=(1/(n+)!)GriffLn12+1, so, in our case, LArfn11=1/n!GriffLn11, where LGriffn11 denotes the notation used by Griffiths (Ref. 2) and LArfn11 denotes the notation used in G. Arfken, Mathematical Methods for Physicists, 3rd ed. (Academic Press, Toronto, 1985), or, for example, in Ref. 10.

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