Grover’s algorithm for quantum searching of a database is generalized to deal with arbitrary initial amplitude distributions. First order linear difference equations are found for the time evolution of the amplitudes of the r marked and N - r unmarked states. These equations are solved exactly. An expression for the optimal measurement time $ T \sim O\left( {\sqrt {N/r} } \right) $ T \sim O\left( {\sqrt {N/r} } \right) is derived which is shown to depend only on the initial average amplitudes of the marked and unmarked states. A bound on the probability of measuring a marked state is derived, which depends only on the standard deviation of the initial amplitude distributions of the marked or unmarked states.