In the derivation of Snell's Law from momentum conservation, the tangential component (parallel to the plane which separates the two media) of a photon's momentum is not changed, but what is the physical explanation or insight to conclude it?
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It is a manifestation of Noether's theorem.
In short, if you have translational invariance in particular direction you have a conservation of momentum in this direction.
Use of Hamilton equations lets you prove it very easily. Suppose there is a translational invariance in $i$-th direction. Hamiltonian ${\mathcal H}$ does not depend on this coordinate and partial derivative with respect to it is zero. Then:
$$ \dot{p}_i=-\frac{\partial {\mathcal H}}{\partial q_i}=0 $$
Thus $p_i=$const.
In your case Hamiltonian has translational invariance parallel to the boundary (it doesn't matter where to refract along the boundary, since medium is uniform).
stinglikeabeer
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Momentum is always considered as only the perpendicular component having an effect. Same when billiard balls collide.
The reason is understood from simple classical mechanics:
During a collision, a normal force is created, always being perpendicular to the surface (to the contact point). So only perpendicular acceleration can happen, meaning only the perpendicular component of $v$ can be altered. And thus only the perpendicular component of momentum, $p_y=mv_y$, is affected.
No parallel forces means no change in parallel motion.
Steeven
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