Harmonics hydrogen-like atoms

Transitions between states are subject to certain restrictions called selection rules. The conservation of angular momentum and the parity of the spherical harmonics limit transitions for hydrogen-like atoms to those for which A/ = 1 and for which Am = 0, 1. Thus, an observed spectral line vq in the absence of the magnetic field, given by equation (6.83), is split into three lines with wave numbers vq + (/ bB/he), vq, and vq — (HbB/he). 

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. 

The energy levels of the hydrogen-like atom are degenerate because more than one state corresponds to a specific value of n (the principal quantum number). These degenerate states are characterized by the quantum numbers I and m, which characterize the spherical harmonics of the wave function. For each value of n there are n values of l(l = 0,.,., n — 1) and for each value of / there are 2/ -h 1 values of m(m = — / -h 1,..., 0,..., / — 1, /) giving... 

The radial factors are a set of functions with two quantum numbers n, the principal quantum number, and /, the same quantum number as in the spherical harmonic functions. The hydrogen-like atom has a single electron and Z protons in the nucleus. The energy eigenvalues of the hydrogen-like atom depend only on the principal quantum number ... 

In variance with the hydrogen-like and Slater functions the potential employed to formally construct the gaussian basis states has nothing to do with the real potential acting upon an electron in an atom. On the other hand the solutions of this (actually three-dimensional harmonic oscillator problem) form a complete discrete basis in the space of orbitals in contrast to the hydrogen-like orbitals. 

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