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Solving for three-dimensional central potentials using numerical matrix methods
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10.1119/1.4793594
/content/aapt/journal/ajp/81/5/10.1119/1.4793594
http://aip.metastore.ingenta.com/content/aapt/journal/ajp/81/5/10.1119/1.4793594

Figures

Image of Fig. 1.
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.

Image of Fig. 2.
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. (17) ], is shown as a solid curve.

Image of Fig. 3.
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.

Image of Fig. 4.
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.)

Image of Fig. 5.
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.

Image of Fig. 6.
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.

Image of Fig. 7.
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., .

Image of Fig. 8.
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

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Generic image for table
Generic image for table
Table I.

Results for the Coulomb potential with .

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Table II.

Results for the Coulomb potential with .

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Table III.

Results for the Coulomb potential with .

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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.

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/content/aapt/journal/ajp/81/5/10.1119/1.4793594
2013-04-16
2014-04-25
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Solving for three-dimensional central potentials using numerical matrix methods
http://aip.metastore.ingenta.com/content/aapt/journal/ajp/81/5/10.1119/1.4793594
10.1119/1.4793594
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