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.