We present a numerical matrix method to find quantum stationary states and energies for three-dimensional central forces. The method can be used both for familiar, exactly solvable potentials and for those that are not solvable by analytical methods. As examples, we include the Coulomb potential, the finite spherical well, and the Yukawa potential. This method requires much less mathematical expertise than traditional analytical methods, although it does require some familiarity with numerical matrix diagonalization software.

1.
See any undergraduate quantum mechanics textbook, for example,
D. J.
Griffiths
,
Introduction to Quantum Mechanics
, 2nd ed. (
Pearson/Prentice-Hall
,
Upper Saddle River, NJ
,
2005
).
2.
W. S.
Stacey
and
F.
Marsiglio
, “
Why is the ground state electron configuration for Lithium 1s22s?
,”
Europhys. Lett.
100
,
43002
(
2012
).
3.
F.
Marsiglio
, “
The harmonic oscillator in quantum mechanics: A third way
,”
Am. J. Phys.
77
,
253
258
(
2009
).
4.
See supplementary material available at http://dx.doi.org/10.1119/1.4793594 for a working code in Fortran.
5.
The rigorous argument is a little more involved. See
R.
Shankar
,
Principles of Quantum Mechanics
(
Plenum
,
New York
,
1980
).
6.
We use the terminology “square well” because we are applying this to the radial equation, which is a one-dimensional equation. Of course, in the context of the physical problem, this corresponds to an infinite spherical well.
7.
The astute reader will notice that these wave functions have a cusp at the origin, and that a cusp is essentially impossible to replicate with a finite number of Fourier components. Therefore, there will always be a disagreement between the numerical and analytical results near the origin. For nmax=800, for example, deviations from the analytical results become noticeable below r/a00.1.
8.
F. J.
Rogers
,
H. C.
Graboske
 Jr.
, and
D. J.
Harwood
, “
Bound eigenstates of the static screened Coulomb potential
,”
Phys. Rev. A
1
,
1577
1586
(
1970
).
9.
J. V.
Kinderman
, “
A computing laboratory for introductory quantum mechanics
,”
Am. J. Phys.
58
,
568
573
(
1990
).
10.
C.
Stubbins
, “
Bound states of the Hulthén and Yukawa potentials
,”
Phys. Rev. A
48
,
220
227
(
1993
).
11.
O. A.
Gomes
,
H.
Chacham
, and
J. R.
Mohallem
, “
Variational calculations for the bound-unbound transition of the Yukawa potential
,”
Phys. Rev. A
50
,
228
231
(
1994
).
12.
X.
Luo
,
Y.
Li
, and
H.
Kröger
, “
Bound states and critical behaviour of the Yukawa Potential
,”
Sci. China, Ser. G
35
,
60
71
(
2005
); also available as e-print arXiv:hep-ph/0407258.
13.
E. R.
Vrscay
, “
Hydrogen atom with a Yukawa potential: Perturbation theory and continued-fractions-Padé approximants at large order
,”
Phys. Rev. A
33
,
1433
1436
(
1986
).
14.
A. D.
Alhaidari
,
H.
Bahlouli
, and
M. S.
Abdelmonem
, “
Taming the Yukawa potential singularity: Improved evaluation of bound states and resonance energies
,”
J. Phys. A
41
,
032001
1
(
2008
).
15.
H.
Bahlouli
,
M. S.
Abdelmonem
, and
S. M.
Al-Marzoug
, “
Analytical treatment of the oscillating Yukawa potential
,”
Chem. Phys.
393
,
153
156
(
2012
).
16.
E. J.
Heller
and
H. A.
Yamani
, “
New L2 approach to quantum scattering: Theory
,”
Phys. Rev. A
9
,
1201
1208
(
1974
).
17.
W. H.
Press
,
B. P.
Flannery
,
S. A.
Teukolsky
, and
W. T.
Vetterling
,
Numerical Recipes: The Art of Scientific Computing
(
Cambridge U.P.
,
Cambridge
,
1986
).

Supplementary Material

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.