Energy eigenvalues of the Morse oscillator using matrix mechanics based on the harmonic basis states

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This study aims to obtain an approximate solution to the Schrödinger equation of Morse oscillator using matrix mechanics based on harmonic oscillator basis states. Energies of the oscillator were obtained from the solution and compared with analytic solutions and with those obtained using other methods. The convergence of the solution as a function of the number of basis states and energy levels was also investigated. Within the approximation used here, the Morse oscillator was expanded in polynomials containing 13 terms, treated as a perturbation to the harmonic oscillator Hamiltonian. The Schrödinger equation was then projected into a finite subspace and solved using the standard matrix method. Once the Hamiltonian matrix elements were obtained using a Mathematica code, the Hamiltonian was diagonalized to obtain the energy eigenvalues of the oscillator. The results indicated that with 200 basis states, 16 energy levels of the oscillator were reasonably accurate with errors ranging from -3.7x10−6 % to 3.95 %. The first three energy levels of the oscillator were also in good agreement with the variational matrix method reported in the literature. It was also observed that the accuracy improved as the number of base states increased and that the results were generally less accurate for higher levels.