Quantum Momentum, Wavelength, Impulse and Special Relativity?

Addendum Sept. 8, 2022: The conservation of photon momentum parallel to the moving mirror holds i.e. p(incident) sin(AA) = p(reflected) sin(BB). (This is the same approach that may be used to derive Snell's law, however, the angle BB is unknown.) If p=hbar/wavelength, then pi/pr = wavelength r / wavelength i and the momentum result matches the wavefront approach described in the note. An independent relation between cos(BB) and cos(AA) may be obtained using Fermat's principle i.e.

cos(BB) ={ -2v/c+ (1+vv/cc)cos(AA) } / {1-2(v/c)cos(AA)+vv/cc} from Ref(2) (For example, one may take the parallel momentum expression above, write p=hbar/wavelength and insert the wavelength relationship given in Ref 2. Using the cos(BB) - cos(AA) given directly above one may see that the two approaches are compatible.

The concept of impulse exists in Newtonian mechanics, but wavelength is not associated with momentum in such a case Even for light (which has a wavelength) reflection from a stationary mirror or refraction into a stationary medium, the idea of momentum without wavelength may be applied (1). In quantum mechanics, however, momentum is directly linked to wavelength through p=hbar/ wavelength, but what does this relationship mean for interactions? Wave mechanics were known long before quantum mechanics and special relativity and often involve following two wavefronts as they interact to see if their spacing remains the same. The interaction itself, however, seems to be very much like a Newtonian impulse one. Thus there are two types of “mechanics” occurring, one linked to impulse and the other to wave features i.e. delays which disrupt the spacing of wave fronts.

    In particular we try to associate this idea with the reflection of light from a moving mirror. In (2) the author considers such a scenario with two wavefronts, each undergoing identical impulse-like interactions, but calculates how the movement of the mirror during the time the two wavefronts react affects the distance between the two reflected wavefronts. The author argues that this approach allows one to obtain the same results as special relativity. In this note we argue that such an analysis may give insight to quantum mechanical interactions i.e. help clarify how momentum and wavelength are linked in a practical problem. As a result, we argue that quantum mechanics of a free particle and special relativity are closely linked and that quantum mechanics like wavefront mechanics combines Newtonian interactions with an uncertainty in the particle position i.e. two extreme points separated by a wavelength.