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.
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$\begingroup$ What makes you think electrons don't exhibit single slit diffraction too? $\endgroup$– knzhouCommented May 1, 2016 at 3:35
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1$\begingroup$ From this link youtube.com/watch?v=DfPeprQ7oGc. But I guess I could have misheard the speech since it was noisy around when I watch the video. Anyway, even if single slit electron diffraction does exist, will the mechanism be exactly the same with light? If yes, how one derives it, that's my concern. $\endgroup$– nougakoCommented May 1, 2016 at 3:46
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1$\begingroup$ Depends on the level of the derivation. Using nonrelativistic QM, it basically comes out the same with the electromagnetic field replaced with the wavefunction. $\endgroup$– knzhouCommented May 1, 2016 at 3:48
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$\begingroup$ Would you please write it as an answer? $\endgroup$– nougakoCommented May 1, 2016 at 3:51
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$\begingroup$ On which physics is that analogy between wavefunction and EM field based? In the far field diffraction pattern, the pattern is the Fourier transform of slit's transmission function. Now, if I were to simply replace the slit's transmission function with the electron's wavefunction, as you suggested, I will be violating the requirement that a wavefunction should be continuous. I don't think there is such a thing as top-hat wavefunction, is there? $\endgroup$– nougakoCommented May 1, 2016 at 3:56
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3 Answers
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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.
answered May 2, 2016 at 8:26
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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.
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$\begingroup$ Thanks for the response. Unfortunately, what you describe is the diffraction of light, not of electrons. Moreover, the Fourier transform of a double top-hat function is not a purely cosine function. It's modulated by a sinc function arising from the single slit diffraction. $\endgroup$– nougakoCommented May 1, 2016 at 2:59
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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.
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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.
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$\begingroup$ I am perfectly aware that the de Broglie wavelength in electron diffraction acts exaclty like the wavelength of light. But again, what concerns me is that, if the very exact math in light diffraction also applies for electrons, how can one derives from the Schroedinger equation. Ok let's illustrate it with some equations, the diffraction pattern of light due to a single slit is $$\tilde{E}(k_x,k_y) = FT[rect(x/a.y/b)] $$ where $a$ and $b$ are the slit sizes. If I apply this for $e^-$ diffraction, how can I derive the above equation from Schroedinger equation of the system? $\endgroup$– nougakoCommented May 1, 2016 at 4:15
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$\begingroup$ Have you derived the above equation from Maxwells equations simply? equivalently there will be a derivation but I do not know if somebody has gone to the trouble since it needs QED, not Schrodinger. Since the de Broglie relation has been verified everybody uses the optics equations. Possibly designers of electron microscopes may have gone the full road. $\endgroup$– anna vCommented May 1, 2016 at 4:26
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$\begingroup$ Yes I know how the FT relation in the far field came about from Maxwell's equation. $\endgroup$– nougakoCommented May 1, 2016 at 4:38