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
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
is a parameter that you can vary. The radial functions obey the boundary condition
. We consider the states with principal quantum numbers
The left-hand plots in the graphic show the radial functions (red curves) for selected values of
. 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
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.