  The purely algebraic technique associated with the creation and annihilation
operators to resolve the radial equation of Hydrogen-like atoms (HLA) for
generating the bound energy spectrum and the corresponding wave functions is
suitable for many calculations in quantum physics. However, the analytical
approach with series is extensively used based on wave mechanics theory in most
of quantum textbooks. Indeed, much More complete than the old solution of
Schr\"odinger's time-independent differential equation (TISE), one can simply
earn all quantum information of a system by using the operational method. In
addition to earlier two models, including the quantum harmonic oscillator and
the total angular momentum, it can undoubtedly be a third fundamental model to
solve Schr\"odinger's eigenvalue equation of the HLA systems in Hilbert space
similar to the harmonic oscillator. We will illustrate how systematically
making an appropriate groundwork to discover the coherent states can lead to
providing the energy quantization and normalized radial wave functions attached
to the matrix representation without additional assumptions.

Quantum Physics