Optical Focusing in Constant Radius Accelerators

Abstract

This corresponds to the classical virial theorem, since in the presence of a magnetic field the right-hand side of (1) is replaced by (a -it) in the quantum case.Perhaps the most useful form of the virial theorem is the case corresponding to no magnetic field, -<r.FMa.p>-<E-e*-|S>.It is clear that (5) and (6) do not apply for continuum states. For example, for a free particle (a-p)=(p 2 /E). Alternatively, for a static potential cj>, we consider the expectation value for any stationary state of the anticommutator of £ and H to obtain <0(£-«0)>«l,(that is, </3a-p) = 0) which gives the well-known result 0)=t/E for a free particle. However, for a bound state in the Coulomb field, it follows from (6) thatso that in this case,•e<00> = E»--l.The nonrelativistic limit e{4>)--2{T), T the nonrelativistic kinetic energy operator, follows from (7b); and more generally from (6) the usual limiting form e(r-V<t>) = 2{T) results.