In a double slit interference experiment, the wave function at the screen with both slits open is not exactly equal to the sum of the wave functions with the slits individually open one at a time. The three scenarios represent three different boundary conditions and as such, the superposition principle should not be applicable. However, most well-known text books in quantum mechanics implicitly and/or explicitly use this assumption that is only approximately true. In this talk, I will discuss recent results  in which we have used the Feynman Path Integral formalism to quantify contributions from non-classical paths in quantum interference experiments that provide a measurable deviation from a naive application of the superposition principle. A direct experimental demonstration for the existence of these non-classical paths is difficult to present. We find that contributions from such paths can be significant and we propose simple three-slit interference experiments  to directly confirm their existence. I will also describe more recent work  in which we have gained an analytical handle on the problem. I will end with showing some results from ongoing experiments in my lab which are aimed at testing these theories.
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 A.Sinha, Aravind H.V., US, On the Superposition principle in interference experiments. Scientific Reports (2015).