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When a mechanical wave goes from one material to an other, some fraction of it returns back.

Same thing with light (massless), but what happens with an electron? When the "wave function" changes material (?), does some of the "electron" return back? I think that this is not the case, but with quantum mechanics you never know... and my knowledge of it stops at Quantum Harmonic Oscillators.

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Short answer: Yes

Slightly longer answer: If you scatter the wavefunction of a propagating electron from a potential (surface of a material for example), it generally splits into two parts - a transmitted part and a reflected part. As the names indicate, the reflected part represents a 'reflected' electron, the transmitted part a transmitted one. However, you must not make the mistake of understanding the particles wavefunction as representing the 'position' of the particle. In this case, the absolute value of the wavefunction squared $S = \left| \Psi(x) \right|^2$ represents the probability of finding the particle in a measurement after the particle has scattered.

So, the wavefunction contains transmission and reflection, but a measured particle can only experience one of both.

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