Conservation of momentum in single slit experiment

Question

I am reading about the uncertainty principle and its stated that we can not simultaneously determine the position and momentum of a photon. But my question is, if you know the initial momentum of the photon, and assuming that a is very small, than we can say that the central maxima would be very wide. The individual photon would go into the slit with momentum $p$, but they leave with a momentum in either the $+y$ or $-y$ direction. Can someone explain why that is the case?

conservation laws heisenberg uncertainty principle momentum

rob · Answer

The transverse momentum is exchanged with the slit, but you'll never measure it.
If interacting with the slit gives the photon a transverse momentum $p_y$, the material which makes up the slit must have acquired a transverse momentum $-p_y$.  Pick any realistic construction for the slit and compute the velocity corresponding to this momentum, then compare that corporate velocity change to the random atom velocities typical of thermal noise at your favorite temperature.  The noise is bigger.
Remember that the primary function of the slit is to absorb most of the photons, each of which is carrying a much  larger momentum $p_x$. If completely absorbing the photons doesn't make the slit move downstream, the transverse momentum is also negligible.

Conservation of momentum in single slit experiment

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