We discuss in detail a well known method for obtaining the frequencies of the normal modes of coupled harmonic oscillators that is based on the simultaneous diagonalization of two symmetric matrices. We apply it to some simple illustrative examples recently chosen for the presentation of an alternative approach based on a set of constants of the motion. We show that the traditional method is simpler, even in the case of equal frequencies. We also discuss a problem that commonly appears in elementary courses on quantum mechanics that also requires the diagonalization of two symmetric matrices.

We argue that several potentials proposed recently for the analysis of the vibrational-rotational spectra of diatomic molecules and their thermodynamic properties exhibit a flaw. One can easily show that the parameters $D_e $ and $r_e$ in those potentials are not the dissociation energy and equilibrium bond length, respectively, as the proposers believe. We show how to overcome the mistake in a simple and quite general way.

We overview the main equations of the Rayleigh-Ritz variational method and discuss their connection with the problem of simultaneous diagonalization of two Hermitian matrices.

A Q-switched Nd:YAG laser was focused on Pb-Ti alloy samples in several Laser Induced Breakdown experiments in order to measure the Stark parameters of several spectral lines (58) of single ionized Titanium including the 3504.89 Å, and the 3510.83 Ålines (where we aported new experimental and theoretical values).Diagnostics of the laser induced plasmas (electron density and electron temperature) was performed using Balmer’s H alpha line (6562.7 Å).Temperatures were obtained by the Boltzmann Plot technique with spectral lines of Pb I (after correction for its evident self-absorption).Subsequently, calculations by the Griem approach of the Stark broadening parameters for several spectral lines were performed using the Gaunt factors proposed by Van Regemorter and those proposed by Douglas H. Sampson. In the latter cases the values obtained were very close to the experimental values. It allows us to assume that the calculations made for the spectral lines of Ti II, without experimental information, are more accurate using the Gaunt factors proposed by Sampson.