A spectral-domain method is described for solving Schrodinger's equation based on the multidomain pseudospectral method and boundary patching. The computational domain is first divided into nonoverlapping subdomains. Using the Chebyshev polynomials to represent the unknown wave function in each subdomain, the spatial derivatives are calculated with a spectral accuracy at the Chebyshev collocation points. Boundary conditions at the subdomain interfaces are then enforced to ensure the global accuracy. Numerical results demonstrate that this spectral-domain method has an exponential accuracy and is flexible, and thus is an attractive method for large-scale problems. With only about four cells per wavelength, the results have an error less than 1% in our typical examples. For a typical quantum well, the method is about 51 and 295 times faster than the second-order finite-difference method for 1% and 0.1% accuracy, respectively. The spectral grid method has also been validated by results obtained by the finite-element method, semianalytical (Airy function) method, and the Numerov's method.