Eigenstates for a Hydrogen Atom Confined to an Infinite Spherical Potential Well

Apart from its intrinsic interest, the problem of a confined hydrogen atom was motivated by an attempt to model the effects of high pressure on a physical system. It is also relevant to quantum dots and white dwarfs. Among the many approaches to the problem have been numerical solutions, perturbation calculations and variational methods [1–3].
The exact solution for a free hydrogen atom can be represented by a wavefunction of the form
where is the reduced radial function, which can be expressed in terms of a Whittaker function:
This is a solution of the radial Schrödinger equation (in atomic units):
In this Demonstration, we consider an approximation for the hydrogen atom confined to an infinite spherical potential well of radius in the form
where is a parameter that you can vary. The radial functions obey the boundary condition . We consider the states with principal quantum numbers to .
The left-hand plots in the graphic show the radial functions (red curves) for selected values of and . Values of up to 30 hartrees are considered. The free hydrogen atom radial functions (black curves) are also plotted for reference.
The right-hand plots show the energies , in hartrees, as a function of . As becomes large, the free hydrogen atom energies are approached. For sufficiently small values of , the energies become positive, as the effect of confinement overcomes the Coulomb attractive energy.



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[1] C. Laughlin, B. L. Burrows and M. Cohen, "A Hydrogen-Like Atom Confined within an Impenetrable Spherical Box," Journal of Physics B: Atomic, Molecular and Optical Physics, 35(3), 2002 pp. 701–715. doi:10.1088/0953-4075/35/3/320.
[2] ‪S. H. Patil and Y. P. Varshni‬, "Properties of Confined Hydrogen and Helium Atoms," Advances in Quantum Chemistry, Vol. 57 (J. Sabin and E. Brandas, eds.), Amsterdam: Elsevier Academic Press, 2009 pp. 1–24 and references cited therein. doi:10.1016/S0065-3276(09)00605-4.
[3] S. H. Patil, "Wavefunctions for the Confined Hydrogen Atom Based on Coalescence and Inflexion Properties," Journal of Physics B: Atomic, Molecular and Optical Physics, 35(2), 2002 pp. 255–266. stacks.iop.org/0953-4075/35/i=2/a=305.
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