A simple algebraic method, which is as easy to use as the angular momentum algebra, is demonstrated as a pedagogical way to solve certain central force problems exactly. Solutions for the hydrogen atom and the three-dimensional isotropic harmonic oscillator are presented together with a discussion of the limits of the method.

Topics
Central-force, Harmonic oscillator, Algebraic methods, Chemical elements
1.
We are entirely indebted to a paper by
J.
Cizek
and
J.
Paldus
, “
An algebraic approach to bound states of simple one-electron systems
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2.
Although we name the algebraic features that arise in the present work by their technical names, this is done to acquaint the reader with terminology and does not require previous exposure to these terms. For an introduction to Lie algebras that would be suitable for the application to physics problems, we suggest H. J. Lipkin, Lie Groups for Pedestrians (North Holland, Amsterdam, 1965).
3.
This realization was made by Cizek and Paldus, Ref. 1.
4.
A subtle issue arises here in the definition of a Hermitian inner product, for example, the formation of 〈χ|χ〉 from |χ〉=T+|Qq〉 and 〈χ|=〈Qq|T−. The Hermitian inner product for any operator O that can be expressed in terms of V1, V2, and V3 [Eqs. (121314)], is given by 〈ψ|O|χ〉=∫ψ*(ra−2O)χ dΩ, where the integration extends over the whole three-dimensional (physical) space. Although the point is one of considerable sophistication, in practical terms it means that ra−2O, where O is any function of V1, V2, and V3, is self-adjoint. The point is unfamiliar in much of standard quantum mechanics, where the operators describing the system are Hermitian.
5.
An irrep (or irreducible representation) is, in the context of this present discussion, a set of eigenvectors that are related to each other by the action of the ladder operators T±. For angular momentum, they are the set of states with a common value of l and different values of ml; physically, this is the set of angular momentum states that differ only in their directional components. The length of the angular momentum vector is unchanging, or irreducible, for the set.
6.
P. M.
Davidson
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Eigenfunctions for calculating electronic vibrational intensities
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See also
D. J.
Rowe
and
C.
Bahri
, “
Rotational–vibrational spectra of diatomic molecules and nuclei with Davidson interactions
,”
J. Phys. A
31
,
4947
–
4961
(
1998
).
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7.
The Dirac–Coulomb problem is also exactly solvable by this method.
8.
B. G. Adams, Algebraic Approach to Simple Quantum Systems (Springer-Verlag, Berlin, 1994). This monograph elaborates considerably on the present method.
9.
O. L. de Lange and R. E. Raab, Operator Methods in Quantum Mechanics (Clarendon, Oxford, 1991).
10.
A. Frank and P. Van Isacker, Algebraic Methods in Molecular and Nuclear Structure Physics (Wiley, New York, 1994).
11.
L. Harris and A. L. Loeb, Introduction to Wave Mechanics (McGraw-Hill, New York, 1963).
12.
H. C. Ohanian, Principles of Quantum Mechanics (Prentice–Hall, Englewood Cliffs, NJ, 1990).
13.
F. Schwabl, Quantum Mechanics (Springer-Verlag, Berlin, 1992).
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© 2002 American Association of Physics Teachers.
2002
American Association of Physics Teachers
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