Mathematical treatment of electron double slit experiment Can someone please provide me with the mathematical treatment of the double slit experiments with electrons?
The diffraction pattern seems to resemble that generated by photons (light) counterpart, but I don't know if the exact mathematical expression of the patterns are identical. I am not satisfied with the usual diffraction of light theory argument applied to electrons either because firing electrons upon a single slit does not produce interference pattern. Whereas single slit illuminated by light will produce maxima-minima.
 A: You can find an extensive treatment of the double-slit experiment with electrons in Feynman Path Integral approach to electron diffraction for one and two slits, analytical results (Beau, 2012).
The paper discusses both Fraunhofer and Fresnel regimes. These regimes do hold for electrons.
Interestingly it does not use the standard semi-classical trajectories superposition picture but instead superposes all possible paths, thus bypassing the whole wave/particle duality conundrum.
A: Diffraction has two regimes: Fraunhofer diffraction and Fresnel diffraction.
The Young's slit experiment is governed by Fraunhofer diffraction , where the diffraction pattern is simply the Fourier Transform of the aperture function; for a double-slit aperture the aperture function is a double 'top-hat' function; and the Fourier Transform of this is a cosine function, which is symmetric about the origin, if this is central between the two slits.
A: That electrons diffract according to the de Broglie wavelength was confirmed back in 1925, the Davisson-Germer experiment. 

Davisson attended the Oxford meeting of the British Association for the Advancement of Science in the summer of 1926. At this meeting, he learned of the recent advances in quantum mechanics. To Davisson's surprise, Max Born gave a lecture that used diffraction curves from Davisson's 1923 research which he had published in Science that year, using the data as confirmation of the de Broglie hypothesis.

.....

As Max von Laue proved in 1912 the periodic crystal structure serves as a type of three-dimensional diffraction grating. The angles of maximum reflection are given by Bragg's condition for constructive interference from an array, Bragg's law

There is no open question on the validity of the de Broglie wave framework. Thus all the optics mathematics for photons can be used for electron beams, single slit, double, and diffraction gratings mathematics.
We are now in 2016 and applying the mathematics of optics to electron beams  has been verified innumerable times, there exist electron microscopes after all. 
