Quantum Mechanics and the Doppler Effect

 For sound and light, changes in energy due to moving frames are manifested in terms of a change in frequency as energy itself is identified with frequency. On the other hand, the energy of photons and phonons is given by E=pc which follows from the more general Einstein energy momentum equation pp + mo mo = EE ((1)). A solution for mo=0 E=pc is to set E=id/dt and p=-id/dx with both operating on exp(-ipx+iEt). This begs the question as to the meaning of this exponential function. It matches the solution of the sound wave equation: d/dt d/dt (partial) density +ccd/dx d/dx density = 0, but E=pc is a different equation. Thus, we argue that exp(-ipx+iEt) has a different meaning than density(x)=Probability(x). As deBroglie showed in the 1920s, one may suggest that matter particles (e.g. electrons) also have a frequency (and wavelength) and satisfy ((1)) in analogy to phonons and photons. Using E=id/dt (partial) and p=-id/dx in ((1)) yields the Klein-Gordon quantum equation from which the Schrodinger equation follows in the low kinetic energy limit. One may note that exp(-ipx+iEt) may interfere if added to exp(-ipx+iEt + phase shift) as shown in electron interference experiments. Thus, having px-Et as a Lorentz invariant is essential. There is not only a “doppler” shift, but an interference result with no particles at a point x and t and this maps into the same physical result for x’,t’ in a boosted frame. Thus, the idea of exp(ipx - iEt) seems to be closely linked to Lorentz invariance. 

  A bigger issue occurs if one considers the Dirac form of the energy-momentum equation:   (E-V)(E-V) = pp + mo mo ((2)). It is possible to use the simple approximation of V(xo) = constant at each xo and consider an exp(i p(xo) xo -iEt). Given the idea of superposition, however, one may consider V(x)=Sum over k Vk exp(ikx). This then leads to a solution of ((2)) with E=id/dt, p=-id/dx, of W(x,t)=exp(iEt) Sum over p a(p)exp(ipx). At this point, exp(ipx) is a relative probability P(p/x). 

   We try to investigate these ideas further in this note.