Quantum mechanics Interference While learning wave-particle duality of an electron I found the wave with which we associate the electron is the probability wave. Now my questions are that if that is so


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*How do electron show interference because the probability wave is just a mathematical interpretation and not an actual one ("Like in case of lights")?

*I even read that if we take a single electron for a double silt experiment then it can be present at same time at two silts ( Now some people may say because of wave nature of electron but after all-electron is a single particle how is it possible ) explain this?

*How many waves does one electron produce?

*What actually is matter wave?


Some people try to relate Young's experiment with interference of electron. Please tell me till what extent this is correct
 A: Your questions are very pertinent- nobody can give a definitive answer to all of them.
Quantum mechanics, which is currently our best framework for describing nature on a small scale, assumes that every particle has an associated "wave function". There is much debate about exactly what the wave function represents physically, so no-one can give you an authoritative explanation.
When the two-slits experiment is performed with electrons, it produces an interference pattern, even if the electrons pass through the apparatus one at a time. One conclusion that can be drawn from this is that the wave-function associated with an electron interferes with itself. Whether the electron itself goes through one slit or both cannot be determined from experiment. The two slit experiment has been performed with other particles, and with quite large molecules. To date, experiments have suggested that quantum interference effects occur even with systems consisting of around 2000 atoms.
Quantum theory assumes that an individual electron has one associated matter wave- it evolves in accordance with the time dependent Schrodinger equation.
No-one can say definitively what a matter wave 'actually is'- all we know is that modelling particles as matter waves leads to predictions that agree quite precisely with experimental results.
A: Indeed probabilities cannot interfere. Probability is always positive and real valued. The behavior of electrons in QM is described by wave functions. These are complex valued and can be reflected and diffracted at boundaries such as a double slit system. The result of an electron wave function passing though a double slit is an interference pattern  as also occurs when light passes through a slit system. 
One property that you can find from the wave function is the charge density $e|\psi|^2$. If you drop the factor $e$ you obtain the famous probability density.
A: Ashutosh Panda


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*What we know for sure is that the probability of finding the electron on the screen depends on the geometry of the barrier. This is to be expected because the electron interacts with the barrier. If the barrier has one hole you get a certain pattern, if it has two holes you get a different pattern. The pattern you get with two holes resembles the one you get with water waves, but it does not logically follow that in the case of electrons we have also waves that interfere. Quantum mechanics does correctly predict the pattern but it does not give us a detailed description of the interaction.

*There is no evidence that the electron is present at both slits at the same time. So, there is nothing to explain.

*An electron does not produce any wave. You can explain this experiment by assuming that the electron behaves as a wave (the wavelength is given by the de Broglie relation, lambda=h/p), but this is not the only  available explanation.

*Matter wave is a different name used for particles to point out their wave-like behavior.
In conclusion, the math used to describe classical waves can be applied to quantum particles but you cannot deduce from here that particles are waves.
A: Interference is the most poorly explained concept in physics, it is based on older classical thinking which is OK for many problems.  The modern method is the probability wave but it is complex to apply the correct boundary conditions for the double slit.  One way to understand it is to see that the wave function requires that the photon or electron must travel n multiples of its wavelength, any path that that is too long or too short is not highly probable.
Dark areas after the double slit are where almost no photons land! Bright areas get almost all photons.
The electron does not go thru both slits, just one, but it had a chance to go either way.
Because matter must interact with matter thru the EM field, and the EM field only works with wavelengths, the result is wavelike observations for small particles. 
