Impulses, Schrodinger Equation and Newtonian Gravity

  In a previous note (1), we argued that one may consider a potential V(x) as being an average of impulse hits i.e. V(x)= Sum over k V(k,x) with the rule that -d/dx V(k,x) = k V(k,x) i.e. k is the amount  of momentum imparted to a quantum particle and V(k,x) is a conditional probability P(k/x). Thus, one works entirely with momentum at the level of impulse, so p=m1v1 = m2v2 are treated in the same manner. One develops a formalism based on momentum and invariance i.e. V(k,x)=V(k)exp(ikx) with the quantum particle also taking on an exp(ipx) conditional probability (or rather a distribution Sum over p a(p)exp(ipx)) so that exp(ikx)exp(ipx)=exp(i (p+k) x) i.e. an impulse hit based solely on momentum. Mass only enters when one imposes conservation of average energy at each point i.e. 

   1/2m [Sum over p a(p)pp/2m exp(ipx)] / [Sum over p a(p)exp(ipx)] + V(x) = E  ((1))

(This means, however, that mass is present in a(p).)

  The above approach seems to apply to nuclear and electromagnetic forces (potential), but a problem arises with the gravitational potential because it does not seem to be impulse based. In other words,  V=GMm/r where m is the mass of the quantum particle. Thus:

dp = impulse = Integral F dt = m Integral -GM/r dt  ((2))

  In other words if dt is constant, the impulse depends on m, or said in another manner, the acceleration and not the momentum change of a particle is the same. Quantum mechanics, 

however, seems to be based on the idea of momentum i.e. exp(ipx) describes a free particle with momentum p. There is no discernment of mass in exp(ipx), nor is the wavelength which is proportional to 1/p. For the nuclear and electromagnetic potentials, one may use an impulse picture with unusual complex probabilities (because the particle momentum is not clearly defined as it changes continually due to impulse hits and there is no time reversal balance). For gravity, one has a contradiction because dp is proportional to m and this is not constant for p=m1v1=m2v2. 

  This might suggest treating gravity in a different manner from nuclear or electromagnetic potentials as in the Schrodinger-Newton approach (3). On the other hand, one might treat mass as a constant of the system and still try to treat dp as constant in an impulse situation, as is done in some cases in the literature.

     We try to investigate these ideas in more detail in this note.