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The quantum resonances (QRs) of the kicked particle are studied in a mostgeneral framework by also considering {\em arbitrary} periodic kickingpotentials. It is shown that QR can arise, in general, for {\em any rational}value of the Bloch quasimomentum. This is illustrated in the case of the mainQRs for arbitrary potentials. In this case, which is shown to be preciselydescribed by the linear kicked rotor, exact formulas are derived for thediffusion coefficients determining the asymptotic evolution of the averagekinetic energy of either an incoherent mixture of plane waves or a general wavepacket. The momentum probability distribution is exactly calculated and studiedfor a two-harmonic potential. It clearly exhibits new resonant values of thequasimomentum and it is robust under small deviations from QR.Comment: Revised Version, to appear in Physical Review

General quantum resonances of the kicked particle