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Solving for three-dimensional central potentials using numerical matrix methods
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10.1119/1.4793594
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Affiliations:
1 Department of Physics, University of Alberta, Edmonton, Alberta T6G 2E1, Canada
2 Department of Physics, University of Alberta, Edmonton, Alberta T6G 2E1, Canada and Physics Division, School of Science and Technology, University of Camerino, I-62032 Camerino (MC), Italy
Am. J. Phys. 81, 343 (2013)
/content/aapt/journal/ajp/81/5/10.1119/1.4793594
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## Figures

Fig. 1.

(Color online) A plot of the various potentials to be used in this paper, along with the infinite-square-well “embedding” potential extending between  = 0 and  = . In this figure, the Bohr radius and the spherical well radius are both set equal to /10. Also shown is an arbitrarily complicated potential well, to illustrate the point that such a potential poses no further difficulty (compared to the others) using the method described here.

Fig. 2.

(Color online) Energy levels for the Coulomb potential vs. the quantum number , for . Exact analytical results are plotted as squares; numerical results obtained with an embedding infinite square well with width , where is the Bohr radius, are plotted as cross-hairs. The numerical results accurately reproduce the first four bound-state energies. Eventually, the numerical energies become positive (unbound) due to the embedding potential, and for large quantum number they grow in proportion to (shown with a dotted curve), as expected for an infinite-square-well of width . A more accurate result for large , which can be derived from perturbation theory [see Eq. ], is shown as a solid curve.

Fig. 3.

(Color online) Numerical results (plotted as symbols) for the radial wave functions ( = 1 and  = 2, both with ). Analytical results are drawn as solid curves.

Fig. 4.

(Color online) A plot of , the critical value of below which a bound state no longer exists, vs. /, where is the radius of the spherical potential and is the width of the embedding infinite-square-well potential. As the influence of the embedding potential is eliminated; the dotted line shows that the extrapolated value agrees with the well-known analytical result. (The value of was increased to 400 to obtain these converged results.)

Fig. 5.

(Color online) Energy levels for the Yukawa potential (with  = 1) as a function of the screening parameter μ. The solid (red), dotted (blue), and dashed (pink) curves show the 1 , 2 , and 3  levels, respectively. Also shown (with symbols) are results from Ref. . Note the existence of critical values of μ above which the energies become positive.

Fig. 6.

(Color online) The ground state wave function for various values of the Yukawa screening parameter μ with the potential strength varied to keep the ground state energy fixed, . As expected, the wave function becomes increasingly localized around the origin as μ increases.

Fig. 7.

(Color online) The weighted ground-state wave function for various values of the screening parameter μ, with held constant at unity. As μ increases the binding energy decreases, so the wave function becomes more extended in space. Also shown is () for when ; this wave function is on the verge of being delocalized over the entire space available, i.e., .

Fig. 8.

(Color online) The critical screening parameter above which a bound state no longer exists in the Yukawa potential, plotted vs. , where is the Bohr radius and is the width of the embedding infinite-square-well potential. As in the case of the spherical well, as the influence of the embedding potential is eliminated; the solid line ( ) shows that the extrapolated value for the critical screening parameter is , in agreement with previously published results.

## Tables

Table I.

Results for the Coulomb potential with .

Table II.

Results for the Coulomb potential with .

Table III.

Results for the Coulomb potential with .

Table IV.

Ground state energies for the Yukawa potential with  = 1, in units of , obtained via the method described in the text. The results of Ref. are shown for comparison.

/content/aapt/journal/ajp/81/5/10.1119/1.4793594
2013-04-16
2014-04-19

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