When an electron beam is split with a beam splitting apparatus, half the electrons are transmitted and half are reflected. This means that the energy of electric potential of the beam is now reduced, which means that there should be an equal and opposite blue shift in the kinetic energy (and momentum) of the beam. Proof: Energy of electric potential of an electron beam with N electrons and with average spacing of the electrons, z: E = (1/4*pi*epsilon)*(e^2)*f(N,z) where e = electric charge and f(N,z) = (1/z)*SUM where SUM is a summation depending on N, which at high N I have calculated to be roughly SUM=10N So that f(N,z) = 10N/Z Now, E = (1/4*pi*epsilon)*(e^2)*(10N/z) This means that the average potential energy of a single electron in the beam is (1/4*pi*epsilon)*(e^2)*(10/z) = 10^(-9)eVm/z If z, the average electron spacing = 1cm = 10^(-2)m then the average potential energy of the electron is 10^(-7)eV. This means that when you split up the beam you will have two distributions with N/2 electrons each and average spacing 2z, then you get the average potential energy per electron as 2*(1/4*pi*epsilon)*(e^2)*(N/4z) = 1/2*10^(-7)eV The energy of electric potential of the electron is halved. And it is clearly measurable because it is on the order of micro electron volts. So does this blue shift the kinetic energy (p^2/2m) and momentum? Has this already been measured and analysed somewhere in the literature of electronics?
