Schrodinger wave eqn for a beam of monoenergeticc electrons

[sagarbhathwar]

The problem is to apply Schrödinger wave equation to a beam of mono energetic electrons and show that the probability of finding electron at each point on the beam is constant


(d2ψ/dx2) + (8∏^2m/h^2)(E-V)ψ = 0
I have been taught to apply this to a single particle for various cases(potential wells, potential barriers and potential steps). But I don't know how to apply this for a beam of electrons.

I would appreciate any help. Thank you

P.S. - I have no idea how to go about approaching the problem. So, I am unable to show any work. I apologize.

 

[member 553083]

The Schrödinger equation can be applied to a beam of mono-energetic electrons by considering the wave function of the beam as an envelope of the individual wave functions of the electrons. The equation for the wave function of the beam can be written as:(d2ψ/dx2) + (8∏^2m/h^2)(E-V)ψ = 0where m is the mass of the electron, h is Planck's constant and V is the potential energy of the beam. This equation can be solved to find the wave function of the beam. Integrating this wave function over the length of the beam gives the probability of finding an electron at each point on the beam, which is found to be constant.