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Exact solution of the compressed hydrogen atom
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Image of Fig. 1.
Fig. 1.

Eigenenergies of the compressed hydrogen atom as a function of the atom size, for the four lowest-lying families corresponding to  +  = 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.

Image of Fig. 2.
Fig. 2.

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.

Image of Fig. 3.
Fig. 3.

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  = 3,  = 0 (first node, at ),  = 3,  = 1 (unique node, at ), and  = 3,  = 0 (second node, at ).


Generic image for table
Table I.

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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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Exact solution of the compressed hydrogen atom