Connection between phase, wave number and momentum

[cin-bura]

hello!

Does anybody know the physical justification of the de broglie relation p=\hbar k? or (i guess equivalently) for p=-i\hbar ∂/∂x ?

i came across a more general "law" in laser physics, where momentum is seen as the gradient of the phase of a scalar light field (used in eg the paraxial wave equation).

what is the exact connection between the phase (or actually it's gradient) and momentum, and where does it come from physically?

thank you!

 

[PhilDSP]

Hello cin-bura,

Do you have a reference for the statement that "momentum is seen as the gradient of the phase" ?

Long ago Lord Raleigh found that one way to characterize waves is the phase velocity. That is the velocity of the plane in which the phase is constant. That's more of a mathematical extrapolation than a physical observable because it's only in special situations that that plane is aligned with the actual wave front. But from the phase velocity many facts about the wave can be determined which de Broglie used to great advantage.

For the rationale and derivation you can see de Broglie's thesis or Noble prize lecture:

http://dieumsnh.qfb.umich.mx/archivoshistoricosMQ/ModernaHist/De_Broglie_Kracklauer.pdf

http://nobelprize.org/nobel_prizes/p...ie-lecture.pdf