https://authors.library.caltech.edu/3520/1/FEYpr49b.pdf
We are trying to solve the mystery described in the two previous blog posts.
In the above paper, Richard P. Feynman calculates the Schrödinger equation behavior of a particle under a weak (electric) potential V.
If V is very weak, one can first solve the free particle equation, and then treat the slight change caused by V as a perturbation. Or is this really true?
If we have a beam of electrons, our experience is that an electric field deflects all electrons. A cathode ray tube works this way.
It is not that most electrons would pass the field undeflected, and there would be a small beam of "scattered" electrons.
The Schrödinger equation is a linear differential equation. If we solve the equation numerically, using small time steps Δt, can we solve the equation approximately by collecting a perturbation term which we add to the original free field solution? It cannot really be called a "perturbation" if it is able to deflect the whole beam of electrons.
The question: can the Schrödinger equation or any linear wave equation describe the bending of waves under a potential?
In optics, one calculates the refraction at the surface of glass by fitting waves that propagate in air to waves that propagate in glass. The wave equation is not used to describe the behavior at the surface.
We need to check the literature. Clearly, the perturbation method of Feynman cannot be used if we are dealing with the bending of a whole electron beam. But is the real problem in the linearity of the Schrödinger equation?
Bend a beam solution?
Suppose that we have a solution which describes the behavior of an electron beam in a V = 0 potential.
Let us then simply "bend" the beam in spatial dimensions. The Schrödinger equation is no longer satisfied because the nabla squared term will change its value in the bent solution. We can make the Schrödinger equation to hold again if we introduce a complex valued potential V which, when multiplied by Ψ, restores the equality in the equation. Here we assume that the wave function is not zero anywhere.
If we are bending a plane wave, then we can choose V to be "almost" real-valued.
If our wave function would be real-valued, then we would face the problem how to restore the equality when Ψ(t, x) = 0?
The Feynman approximation of summing the "scattered" waves is bad
Feynman assumes that the bulk of the wave function remains as it was when V = 0, and sums the "perturbation" terms caused by a weak potential V != 0.
He then assumes that the perturbation wave propagates as a separate, "scattered" wave.
This approximation makes sense if we want to approximate the wave function for a very short spatial distance. But if we let the beam continue far away, then the approximation is very bad: it does not model the bending of the beam in any way!
Why does Feynman obtain a correct estimate for Coulomb scattering, if the approximation method is this bad? We have to find out.
Green's functions
The above Wikipedia article defines a Green's function as a solution to a Dirac delta source in a linear differential equation:
L G = δ(t = t_0, x = x_0).
One can then find a solution for an initial value problem of the source in the equation
L g = f(t, x)
by building the source function f from Dirac deltas and summing various G to obtain g.
The Wikipedia article claims that a relativistic propagator gives the probability amplitude for a particle to travel from x to y. Is that really so?
In Feynman diagrams, a new particle is born from a "disturbance" to the free wave equation of that particle. A disturbance is natural to describe as a source to the free wave equation. In that context, a "propagator" of that source is the Green's function.
Is it correct to say that the propagator gives the probability amplitude of a "particle" to travel from x to y? Probably no. Rather, it tells the probability amplitude that a source produced a particle at x and that particle is seen at y.
A correction to Feynman's "The Theory of Positrons" paper?
The approximation method of Feynman, which we described above, is bad. One can try to fix it by treating the (small) term VΨ as a source in the Schrödinger equation. The source creates new waves which we can calculate using the mathematical Green's function.
(UPDATE Nov 27, 2020: for the Schrödinger equation, the Green's function is a sharp wave packet and its diffusion in time. It makes no difference if we treat VΨ as a source, or as a "new wave"! Thus, our "fix" really changes nothing.)
However, this does not make the electron beam to bend. This not a good approximation either, in that case.
This new approximation method might work if the potential V is only significant in an area which is smaller than the wavelength of an electron. That may be the reason why Feynman was able to get the correct Coulomb scattering formula!
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