Microscopic calculations often consider particles placed in single-
particle energy levels subject to two-body interactions. In order
to reproduce collective phenomena one relies on the computer power
to study the system in huge model spaces. Ultimately, such
simulations will describe collectivity adequately, but the
understanding of the phenomena and their symmetry roots are rarely
advanced. It is the goal of this work to illustrate that if one
uses basis with built in collectivity one could describe the
collective phenomena better and would also advance their
understanding. The text contains: (a) the nuclear shell model in
spherical and Elliot's SU(3) basis; (b) the harmonic oscillator in
a one-dimensional box as a toy model of a two-mode system; (c)
generalized eigenvalue problem and the geometrical visualization of
the oblique shell- model basis; (d) illustrative systems such as
24Mg and 44Ti in oblique basis; (e) Study of the SU(3) symmetry and
measuring the its breaking for pf-shell nuclei. This text could be
of value to professors and advanced students who are pursuing
research in unconventional computational methods for quantum
many-body systems.
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