^{1,a)}and C. R. Proetto

^{2,b)}

### Abstract

The exact solution to the problem of a hydrogen atom confined in a spherical well (CHA) is discussed; the standard results for the unconfined hydrogen atom (UHA) are recovered as the sphere size becomes large compared to the Bohr radius. The solutions are characterized by a set of three quantum numbers N (= 1, 2, 3,…), L (= 0, 1, 2,…), and M (= − L, − L + 1,…, L − 1, L), and the energy eigenvalues, in contrast to the situation in the UHA, depend on both N and L. All members of a given family n = N + L, however, evolve asymptotically toward the same energy level in the large-sphere limit, recovering the typical n ^{2} degeneracy of the UHA. Besides numerically exact solutions for arbitrary sphere sizes, rigorous analytical approximations are provided for the physically relevant strong- and weak-confinement regimes. A conjecture concerning the ordering of the energy levels is rigorously confirmed. The validity of the virial theorem, Kato's cusp condition, and the role played by the density as an alternative basic variable for the case of the CHA are discussed.

One of us (J.M.F.) is indebted to CONICET of Argentina for financial support at the starting stage of this project. C.R.P. is a fellow of CONICET. The authors thank J. Luzuriaga and V. H. Ponce for a careful reading of the manuscript.

I. INTRODUCTION

II. EXACT SOLUTION IN TERMS OF KUMMER FUNCTIONS

III. NUMERICAL AND ANALYTICAL RESULTS

IV. CONCLUSIONS

### Key Topics

- Eigenvalues
- 10.0
- Semiconductors
- 8.0
- Quantum optics
- 7.0
- Exact solutions
- 6.0
- Boundary value problems
- 5.0

## Figures

Eigenenergies of the compressed hydrogen atom as a function of the atom size, for the four lowest-lying families corresponding to N + L = 1, 2, 3, 4. To leading order, the asymptotic limits for are for , and for (see text). The asymptotic values for are −1/2, −1/8, −1/18, and −1/32.

Eigenenergies of the compressed hydrogen atom as a function of the atom size, for the four lowest-lying families corresponding to N + L = 1, 2, 3, 4. To leading order, the asymptotic limits for are for , and for (see text). The asymptotic values for are −1/2, −1/8, −1/18, and −1/32.

Comparison of the exact eigenvalues of Fig. 1 (lines) with those obtained from the strong- and weak-confinement approximations (discrete points). The inset (corresponding to the rectangle marked at the lower left corner of the figure) shows the accuracy of the weak-confinement approximation given by Eq. (13) , as applied to the ground state. The shaded upper and lower areas represent the regions of applicability of the SCA and the WCA, respectively.

Comparison of the exact eigenvalues of Fig. 1 (lines) with those obtained from the strong- and weak-confinement approximations (discrete points). The inset (corresponding to the rectangle marked at the lower left corner of the figure) shows the accuracy of the weak-confinement approximation given by Eq. (13) , as applied to the ground state. The shaded upper and lower areas represent the regions of applicability of the SCA and the WCA, respectively.

Special sphere sizes (discrete points) for which certain CHA and UHA solutions coincide. Only the negative part of Fig. 1 is shown, following the same line convention. For example, the three black dots at correspond to the UHA solutions with n = 3, l = 0 (first node, at ), n = 3, l = 1 (unique node, at ), and n = 3, l = 0 (second node, at ).

Special sphere sizes (discrete points) for which certain CHA and UHA solutions coincide. Only the negative part of Fig. 1 is shown, following the same line convention. For example, the three black dots at correspond to the UHA solutions with n = 3, l = 0 (first node, at ), n = 3, l = 1 (unique node, at ), and n = 3, l = 0 (second node, at ).

## Tables

Numerical values of , for the six lowest-lying electronic levels of the CHA, in Hartree units. For each λ, the first row (numbers in bold) corresponds to the exact (numerical) solution of Eq. (4) ( ), or Eq. (6) ( ). The second and third rows correspond to the SCA as given by Eq. (12) , and as given by the zero-order approximation , respectively. For the ground state (1,0), the results of the WCA as given by Eq. (16) have also been included (numbers in italics).

Numerical values of , for the six lowest-lying electronic levels of the CHA, in Hartree units. For each λ, the first row (numbers in bold) corresponds to the exact (numerical) solution of Eq. (4) ( ), or Eq. (6) ( ). The second and third rows correspond to the SCA as given by Eq. (12) , and as given by the zero-order approximation , respectively. For the ground state (1,0), the results of the WCA as given by Eq. (16) have also been included (numbers in italics).

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