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

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