conservation laws - Diffracted electron - where does it gain additional momentum?

When an electron is diffracted, the momentum after the diffraction has different direction than before.

Where does the electron gain this momentum?

This is related to this question, but it's different enough to be posted separately: Does the diffracted electron radiate photons?

Answer

It all depends on the size of the slit with respect to the energy of the electron.

If the electron has very high energy and the slit is large, one can talk of momentum conservation in a classical way, as : "if a ball changes direction hitting an edge where does it get the momentum?" it gets it by hitting the edge and transferring a bit of its energy to the bulk of the edge.

The electron though is a quantum mechanical entity, and with a slit size commensurate to the magnitude of $\hbar$, the Heisenberg uncertainty principle allows an indeterminacy in the momentum if the position is known:$\Delta x \times \Delta p \gt \hbar$.

Alternatively one can think of solving the quantum mechanical problem "electron of momentum p + slit" and see that the solution is a probability distribution, that will give a probability for the electron to come out at an angle without violating classical conservation laws.