[Physics] Snell’s Law and momentum conservation

conservation-lawshamiltonian-formalismmomentumopticsrefraction

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?

Best Answer

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).