Milne's differential equation and numerical solutions of the Schrodinger equation. I. Bound-state energies for single- and double-minimum potentials

Milne's approach to the numerical solution of the Schrodinger equation via a non-linear differential equation an the quantisation of a quantum action is investigated in detail. An accurate and efficient computational method is presented which allows a rapid second-order convergence onto a desired eigenenergy E n . Numerical sample calculations demonstrate the efficiency of the method, which has special advantages for accurate calculations of high quantum states. The present method can be easily extended to the calculation of quasi-bound levels at resonance (complex-values) energies.