What are examples of Solutions of Newton's Laws that have analogs to Solutions of the Wave Equation? The idea is to give examples of processes that deal with properties of a particle that have clear wave analogues.
 A: The two traditional properties which were considered analogous were refraction/sliding-down-a-potential-step and reflection. But all properties are analogous, because Newton's laws emerge as the small $\hbar$ limit of the Schrodinger equation.


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*Refraction: suppose a marble is rolling along a surface and encounters a smooth ramp which takes it down to a lower level. What happens to the particle? You can see that it is refracted--- it moves at a different angle. If the ramp is translation invariant in y, and the particle comes in with $p_x$,$p_y$, then the outgoing momentum is $p_x+\Delta p$,$p_y$, and the matching conditions for refraction are satisfied if the wavenumbers $k_x$ and $k_y$ are proportional to the momentum.

*Reflection: if a particle hits a wall, the angle of incidence is equal to the angle of reflection. If you boost to a moving frame, you can find the condition for the angle of reflection when the wall is moving. The condition again matches momenta as though they were wavenumbers.

*Conservation laws: if a particle is entering a force field, this is like light propagating in a material with a smoothly varying index of refraction. Again, the deflection is consistent if the momentum is the wavenumber. The conservation of energy becomes the conservation of frequency in a time-invariant medium.


These coincidences led Hamilton to suspect that particle motion should be thought of as a small-wavelength approximation to a wave equation.
