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I was thinking whether there is any analogous Snell's Law for classical particles. for example I have a problem where a particle (in 2D) crosses a boundary separating potential $U_1$ and $U_2$. It enters the boundary with angle $\theta_1$ with the normal to the boundary. I wanted to find the $\theta_2$ that the particle makes with the boundary when it has entered the new region. If there isnt any Snell's law for classical particles, How to solve this problem ?? Are the energy and momentum conservation equations enough?enter image description here

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  • $\begingroup$ Is this helpful? $\endgroup$
    – J.G.
    Nov 2, 2022 at 15:39

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Newton has done this for you!
He used his ideas about Mechanics and assumed that the component of velocity of the corpuscles (the name he gave to the particles of light) perpendicular to the interface, as they moved from one medium into another medium, changes due to the change in the potential, whereas the components of velocity parallel to the interface did not change as there were no forces in that direction, so $v_1\sin \theta_1= v_2\sin \theta_2$.

So the refractive index $=\frac{\sin \theta_2}{\sin \theta_1} = \frac{v_1}{v_2}$ which from your diagram means that the speed of light is less in medium $2$ than in medium$1$ which was later shown experimentally to be incorrect.

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  • $\begingroup$ We need to explain what causes the instantaneous change of momentum in the direction orthogonal to the discontinuity in the potential. And this is the impulsive force, that can be derived as the space derivative of the jump in the potential. This should complete the picture with the dynamics $\endgroup$
    – basics
    Nov 2, 2022 at 15:40

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