A new, explicit, efficient, and meshfree multi-domain spectral method (MDSM) is presented to calculate the eigenstates of arbitrary semiconductor quantum wells (QW). The method is functional expansion based where the quantized energy states are approximated using efficient basis sets and the boundaries are considered using Tau approach. For bounded domains, Chebyshev polynomials and power series are used as the basis sets, while non-orthogonal predefined exponential basis sets are used for the half bounded domains. Many QW structures were studied and the results are compared with published results for validation. The comparisons exhibit the accuracy and efficiency of the presented method. For QWs that can be analyzed analytically, the relative errors in the calculated quantized energy levels are in the order of 10- 12 with very small number of bases in few ms. © 2010 Elsevier B.V. All rights reserved.