Single-particle energy states for a neutron and a proton are obtained by solving the time-independent Schrödinger equation for the mean-field Woods–Saxon potential along with the spin-orbit term. The wavefunctions are expanded as a linear combination of simple sine-wave basis states, which are eigenfunctions of the infinite spherical-well potential. The requisite algorithm based on matrix diagonalization is implemented in Free Open Source Software (FOSS) Scilab. Initial values for the simulation were taken from model parameters given in the book on Nuclear Structure by Bohr and Mottelson, which were then optimized to obtain the best convergence with the available experimental energy values of various nuclei with magic proton and neutron numbers. The level scheme, as well as the energy values for doubly magic nuclei 82208Pb and 2040Ca, which are obtained using our simulation, is presented in this paper. Finally, energy level diagrams for neutrons and protons with respect to mass number A were arrived at, based on those obtained for various magic nuclei. The evaluation method, which is based on the sine-wave basis, is akin to Fourier analysis. When done with the aid of FOSS Scilab, this technique becomes easily accessible to students at the under-graduate (UG) level and may be studied through small projects.