The paperpresentsaquantummechanicaltreatmentforanalyzingtheSmith–Purcell radiation generatedbychargedparticlespassingoveraperiodicconductingstructure.Inourtheoreticalmodel, the electronsinteractwithasurfaceharmonicwaveexcitednearthediffractiongratingwhenthe electron velocityisalmostequaltothephasevelocityofthesurfacewave.Then,thesurfaceharmonic waveiselectromagneticallycoupledtoaradiationmode.Thedynamicsofelectronsareanalyzed quantummechanicallywheretheelectronisrepresentedasatravelingelectronwavewitha finite spreading length.Theconversionofthesurfacewaveintoapropagatingmodeisanalyzedusingthe classical Maxwell'sequations.Inthesmall-signalgainregime,closed-formexpressionsforthe contributions ofthestimulatedandspontaneousemissionstotheevolutionofthesurfacewaveare derived.Theinclusionofthespreadinglengthoftheelectronwavetotheemissionspectrallineis investigated.Finally,wecompareourresultsbasedonthequantummechanicaldescriptionofelectron and thosebasedontheclassicalapproachwhereagoodagreementisconfirmed.