Spectral expansions of a known or unknown smooth function converge very rapidly, and are suited for accurate solutions of differential or integral equations. Based on such expansions, a numerical algorithm for solving the Schr?dinger equation has recently been developed. First a simple numerical example is described in order to illustrate the accuracy properties of spectral expansions, and then the associated numerical algorithm for solving the Schr?dinger equation is described in simple terms. The method proceeds by transforming the differential equation into an equivalent Lippmann-Schwinger integral equation, and then solving the latter by dividing the radial range into partitions. In each partition the (unknown) solution is expanded into a set of Chebyshev polynomials, and the coefficients of the expansion are calculated. A stringent accuracy test of resonant scattering involving barrier penetration for a Morse potential is provided.