Phase-space matrix representation of differential equations for obtaining the energy spectrum of model quantum systems

Employing the phase-space representation of second order ordinary differential equations we developed a method to find the eigenvalues and eigenfunctions of the 1-dimensional time independent Schr\"odinger equation for quantum model systems. The method presented simplifies some approaches shown in textbooks, based on asymptotic analyses of the time-independent Schr\"odinger equation, and power series methods with recurrence relations. In addition, the method presented here facilitates the understanding of the relationship between the ordinary differential equations of the mathematical physics and the time independent Schr\"odinger equation of physical models as the harmonic oscillator, the rigid rotor, the Hydrogen atom, and the Morse oscillator.