Classical Stochasticity in Quantum Mechanics and Multiple Solutions to Pressure Balance

We have argued in previous notes that quantum formalism emerges from the one dimensional photon reflection-refraction scenario which is readily observable in the classical world. What about the situation for a particle with rest mass? In previous notes, we have considered the case of a constant E,p and V1 which changes at xo to E,p2,V2 (all again constants with E remaining unchanged). This problem is then associated with an incident and reflected particle for x<xo and a particle with p2 for x>-xo.

   In a classical world, however, how does one obtain reflection and refraction together? If there is a strong wall, a particle may reflect with -p and the same kinetic energyE, but it does not pass through the wall. If there is a ramp at xo leading to a horizontal floor a little higher than that for x<xo, then one has a conserved total E, but a new p2 and V2. The ramp, however, is not a sudden change at xo, but occurs over a range of x. As a result, it seems the quantum scenario, rather than leading to the classical case in a high energy limit (as in bound states), is associated with fluctuations and non-sharp boundaries. It is fine to have a sharp wall at x=xo, but in quantum mechanics, one has atoms with electrons moving in space according to probability distributions. Furthermore, there is movement due to temperature. One does not have a sharp edge or ramp scenario alone, but fluctuations which could lead to both. This suggests stochasticity at a point of discontinuity, but also stochasticity in the regions V1 and V2. In other words, V1, V2 and the sharp discontinuity points are all approximations (averages). 

   If one accepts the fluctuations at xo (ramp and sharp wall), then classically one might impose a pressure balance equation at an idealized xo, i.e.:  AAp -BBp=p2CC where AA represents a positive number etc. AA represents the incident particle, BB the reflected and CC the particle which moves up the ramp. Nonrelativistically and in the photon case, the pressure balance equation is also equivalent to a conservation of particle number or energy equation. Thus one has two unknowns (if  A is known), i.e. B and C, and one equation. Here, however, we wish to stress a different, but related issue. The pressure balance equation allows for multiple solutions because it only exists at xo. For example, BB=0 is a solution as is BB=.5, CC=.5. If p2 is almost equivalent to p, one might expect that BB=0. In other words, there is a specific solution which goes beyond pressure balance, i.e. here seems to be a p (momentum) biased solution. This forces one to go beyond classical mechanical ideas, even if one introduces a fluctuating wall-ramp scenario at xo. The fluctuating scenario suggests fluctuations in V1 and V2 as well, suggesting one may have some kind of dynamical equilibrium in these regions. Would this be enough to create a momentum bias, linked say to a continuity of a function of p,x and its first derivative at xo? This extra momentum bias seems to be the extra feature which emerges in a quantum treatment of the problem, which only occurs because one introduces fluctuations at the xo point. We suggest that a dynamical equilibrium approach solves the problem if one maximizes Shannon’s entropy subject to a the constraint of a constant average classical action written as:

-P(t) iEt +  ipx P(x) with a Lagrange multiplier of 1.

   “i” is introduced to a have a periodic/dynamic probability, i.e. one that does not grow.

  Thus we argue that quantum mechanics for a particle with rest mass emerges because a classical pressure balance equation for a fluctuating wall-ramp picture at a V1-V2 junction allows for multiple solutions. Ifr V1 is almost equal to V2, one has a biased or selective solution which should force BB to be 0. This may be done through continuity, but there must be a physical basis for the function used and we suggest that this is maximization of Shannon’s entropy subject to an average classical action  introduced to make the probability periodic due to the dynamical nature of the problem. Thus there is not just stochasticity at the discontinuity point xo, but throughout x.