^{1}and F. Marsiglio

^{2}

### Abstract

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.

This work was supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC), and by the Teaching and Learning Enhancement Fund (TLEF) and a McCalla Fellowship at the University of Alberta.

I. INTRODUCTION

II. FORMULATION OF THE MATRIX MECHANICS PROBLEM

III. THE COULOMB POTENTIAL

IV. THE FINITE SPHERICAL WELL

V. THE YUKAWA POTENTIAL

VI. SUMMARY

### Key Topics

- Bound states
- 21.0
- Wave functions
- 16.0
- Eigenvalues
- 6.0
- Computer software
- 5.0
- Matrix methods
- 5.0

## Figures

(Color online) A plot of the various potentials to be used in this paper, along with the infinite-square-well “embedding” potential extending between r = 0 and r = a. In this figure, the Bohr radius a 0 and the spherical well radius b are both set equal to a/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.

(Color online) A plot of the various potentials to be used in this paper, along with the infinite-square-well “embedding” potential extending between r = 0 and r = a. In this figure, the Bohr radius a 0 and the spherical well radius b are both set equal to a/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.

(Color online) Energy levels for the Coulomb potential vs. the quantum number n, for . Exact analytical results are plotted as squares; numerical results obtained with an embedding infinite square well with width , where a 0 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 n they grow in proportion to n 2 (shown with a dotted curve), as expected for an infinite-square-well of width a. A more accurate result for large n, which can be derived from perturbation theory [see Eq. (17) ], is shown as a solid curve.

(Color online) Energy levels for the Coulomb potential vs. the quantum number n, for . Exact analytical results are plotted as squares; numerical results obtained with an embedding infinite square well with width , where a 0 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 n they grow in proportion to n 2 (shown with a dotted curve), as expected for an infinite-square-well of width a. A more accurate result for large n, which can be derived from perturbation theory [see Eq. (17) ], is shown as a solid curve.

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

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

(Color online) A plot of , the critical value of V 0 below which a bound state no longer exists, vs. b/a, where b is the radius of the spherical potential and a 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.)

(Color online) A plot of , the critical value of V 0 below which a bound state no longer exists, vs. b/a, where b is the radius of the spherical potential and a 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.)

(Color online) Energy levels for the Yukawa potential (with A = 1) as a function of the screening parameter μ. The solid (red), dotted (blue), and dashed (pink) curves show the 1 s, 2 s, and 3 s levels, respectively. Also shown (with symbols) are results from Ref. 13 . Note the existence of critical values of μ above which the energies become positive.

(Color online) Energy levels for the Yukawa potential (with A = 1) as a function of the screening parameter μ. The solid (red), dotted (blue), and dashed (pink) curves show the 1 s, 2 s, and 3 s levels, respectively. Also shown (with symbols) are results from Ref. 13 . Note the existence of critical values of μ above which the energies become positive.

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

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

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

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

(Color online) The critical screening parameter above which a bound state no longer exists in the Yukawa potential, plotted vs. , where a 0 is the Bohr radius and a 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.

(Color online) The critical screening parameter above which a bound state no longer exists in the Yukawa potential, plotted vs. , where a 0 is the Bohr radius and a 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

Results for the Coulomb potential with .

Results for the Coulomb potential with .

Results for the Coulomb potential with .

Results for the Coulomb potential with .

Results for the Coulomb potential with .

Results for the Coulomb potential with .

Ground state energies for the Yukawa potential with A = 1, in units of E 0, obtained via the method described in the text. The results of Ref. 13 are shown for comparison.

Ground state energies for the Yukawa potential with A = 1, in units of E 0, obtained via the method described in the text. The results of Ref. 13 are shown for comparison.

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