Spherically confined isotropic harmonic oscillator

In this section we shall state some exact results for the spherically confined isotropic harmonic oscillator inside impenetrable walls. The eigenspectral regularities and the characterization of energy states in terms of the electron density and its derivatives at the equilibrium position will be considered. 

In this section, we present some general results on the information theoretical uncertainty-like measures applicable to the standard model systems of hydrogen-like atoms and the isotropic harmonic oscillator. The characteristic features of the spherically confined systems will be highlighted. 

Solid state physicists are familiar with the free- and nearly free-electron models of simple metals [9]. The essence of those models is the fact that the effective potential seen by the conduction electrons in metals like Na, K, etc., is nearly constant through the volume of the metal. This is so because (a) the ion cores occupy only a small fraction of the atomic volume, and (b) the effective ionic potential is weak. Under these circumstances, a constant potential in the interior of the metal is a good approximation—even better if the metal is liquid. However, electrons cannot escape from the metal spontaneously in fact, the energy needed to extract one electron through the surface is called the work function. This means that the potential rises abruptly at the surface of the metal. If the piece of metal has microscopic dimensions and we assume for simplicity its form to be spherical - like a classical liquid drop, then the effective potential confining the valence electrons will be spherically symmetric, with a form intermediate between an isotropic harmonic oscillator and a square well [10]. These simple model potentials can already give an idea of the reason for the magic numbers the formation of electronic shells. 

A different approach was proposed by Aquino [13], where the Hamiltonian matrix is diagonalized in the basis set of an isotropic, spherically confined harmonic oscillator (SICHO), where the CHA Hamiltonian is written as... 

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