The propagation of quantum massive particles in the general plane wave spacetime and external, non-plane, electromagnetic waves is studied. The asymptotic conditions, the "in" ("out") states and the cross sections are analysed. It is observed that despite of the singularities encountered the global form of these states can be obtained: at the singular points the Dirac delta-like behavior emerges and there is a discrete change of phase of the wave function after passing through each singular point. The relations between these phase corrections and local charts are discussed. Some examples of waves of infinite range (including the circularly polarized ones) are presented for which the explicit form of solutions can be obtained. All these results concern both the scalar as well as spin one-half quantum particle; in latter case the change of the spin polarization after the general sandwich wave has passed is studied.