ABSTRACT:The model of the confined hydrogen atom (CHA) was developed by Michels et al. [1] in the mid-1930s to study matter subject to extreme pressure. However, since the eigenvalues cannot be obtained analytically, even the most accurate calculations have yielded little more than 10 figure accuracy. In this work, we show that it is possible to obtain the CHA eigenvalues with extremely high accuracy (up to 100 decimal digits) and we do that using two completely different methods. The first is based on formal solution of the confluent hypergeometric function while the second uses a series method. We also compare radial expectation values obtained by both methods and conclude that the wave functions obtained by these two different approaches are of high quality. In addition, we compute the hyperfine splitting constant, magnetic screening constant, polarizability in the Kirkwood approximation, and pressure as a function of the box radius.
ABSTRACT:Using the mathematical properties of the confluent hypergeometric functions, the conditions for the incidental, simultaneous, and interdimensional degeneracy of the confined D-dimensional (D Ͼ 1) harmonic oscillator energy levels are derived, assuming that the isotropic confinement is defined by an infinite potential well and a finite radius R c . Very accurate energy eigenvalues are obtained numerically by finding the roots of the confluent hypergeometric functions that confirm the degeneracy conditions.
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