We first present the exact solutions of the single ring-shaped Coulomb
potential and then realize the visualizations of the space probability
distribution for a moving particle within the framework of this potential. We
illustrate the two-(contour) and three-dimensional (isosurface) visualizations
for those specifically given quantum numbers (n, l, m) essentially related to
those so-called quasi quantum numbers (n',l',m') through changing the single
ring-shaped Coulomb potential parameter b. We find that the space probability
distributions (isosurface) of a moving particle for the special case and the
usual case are spherical and circular ring-shaped, respectively by considering
all variables in spherical coordinates. We also study the features of the
relative probability values P of the space probability distributions. As an
illustration, by studying the special case of the quantum numbers (n, l, m)=(6,
5, 1) we notice that the space probability distribution for a moving particle
will move towards two poles of axis z as the relative probability value P
increases. Moreover, we discuss the series expansion of the deformed spherical
harmonics through the orthogonal and complete spherical harmonics and find that
the principal component decreases gradually and other components will increase
as the potential parameter b increases.Comment: 28 pages, 6 tables, 1 figure
