A general method for finding the energies and wave functions for aniso
tropic masses and potentials has been studied. This method uses basis
wave functions expanded in spherical harmonics. The coupled differenti
al equations for the radial components of each basis function are inte
grated numerically. The energy eigenfunctions and their radial derivat
ives are then matched at a spherical boundary. Special treatment is ne
eded to ensure linear independence of the basis functions at the bound
ary. The applicability of this method and the speed of convergence are
tested on anisotropic harmonic oscillators. The method is then applie
d to the Coulomb potential with anisotropic masses. With a basis of fi
ve or fewer spherical harmonics, our method produces energies which co
nverge to values lower than those previously reported. We have also ob
tained energy levels for the Coulomb potential with threefold mass ani
sotropy. This method should be applicable to other anisotropic problem
s with a single potential minimum. In particular, it should facilitate
the employment of full-potential Green-function band theory.