We work in the physical model of hydrogen-like atoms and use the wave functions provided by solutions of the Schrödinger equation with Coulomb potential. For the hydrogen spectra, we develop a theoretical-spectroscopy method to identify the significant quantum states with the highest probability of transition. We exemplify this method for the α, β, γ Lyman transitions, namely for a fixed principal quantum number n less than 5, and find (2 1 0), (3 2 0), (4 3 0) as significant quantum states. Besides this, we obtain other various results: the numerical tables for the radial distributions and the electron density functions, and some observables like θ nodal angles, nodal surfaces, along with the ordered sequences of the average-radii of the sub-shells for a fixed n, such as: 18a0 =〈r〉4f < 〈r〉4d < 〈r〉4p <〈r〉4s = 24a0 (for the shell n = 4 and a0 - the first Bohr radius). Afterwards, we build the scientific 3D-visualization of the orbitals shapes and describe them via the nodal values for θ angles.