Quantum mechanical formalism of particle beam optics

A general procedure for construction of the formalismof quantum beam optics for any particle is reviewed. The quantum formalismof spin-1/2 particle beamoptics is presented starting {\em ab initio} with the Dirac equation. As an example of application the case of normal magnetic quadrupole lens is discussed. In the classical limitthe quantum formalismleads to the well-known Lie algebraic formalismof classical particle beamoptics.