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Speculation on Quantum Free Particle Probability and Spatial Equilibrium Francesco R. Ruggeri Hanwell, N.B. Oct. 31, 2025 We argue that the toss of a coin or a die may be seen as an equilibrium in time. In particular, for a coin there is an intrinsic spatial feature (two sides of the coin) which a priori defines probability for heads/tails and after a very large number of tosses, one essentially has 50% of the results as heads and 50% as tails. In such a case, time runs in one direction. One may run a movie backwards, but then time still runs in one direction, i.e. backwards. Spatial equilibrium, however, involves motion in space in both directions, as in the case of pressure in an ideal gas with no potential V(x). Thus, we argue that for a spatial equilibrium, one must consider both directions of space. This leads to an interesting situation if one has a directional problem associated with a spatial equilibrium equation which should not be directional based. In particular, we consider the example of one dimensional reflection-refraction at an n1-n2 index of refraction junction. At first this has the appearance of an equilibrium in time through: 1= P(reflect) + P(refract). Just like the coin toss, after many photons N impinge at the junction, there will be N P(reflect) which reflect and N P(refract) which refract. In a previous note, we saw that one may write this probability conservation equation in terms of dynamic variables, i.e. AA/c - BB/c = CC/c2 —> AAp - BBp = CC p2 ((1)), using E=pc with E being the same for the incident, reflected and refracted photons. Here AA,BB,CC are the fluxes of the three. We also noted that written in this way, one has pressure balance. We argue here that a pressure balance equation represents a spatial equilibrium and only makes sense in terms of motion in both x directions. A steady stream case or a single photon scenario, however, only involves one direction of motion. We thus argue that one must introduce equations which explicitly show directional motion and that these must create the pressure equilibrium equation which has not sense of direction. To do so, we consider AA, the incident flux and note that flux is the number of particle per sec. As a number it has no direction, although one may associate it with a velocity vector. We suggest that for a pressure spatial equilibrium case, it must be associated with both directions of motion (i.e. x) and that one should write a probability in terms of p which governs interactions. This suggests: AA = Aexp(ipC) Aexp(-ipC). (Here C is a constant to account for units.) In other words, if one performs a probabilistic equilibrium calculation one must have a proper equilibrium with motion to the left and right described. A physical interaction of a single photon, however, only involves one direction and so is linked with one piece A exp(ip) we argue. Furthermore, as a probability this should be continuous in x and so instead of exp(ipC), suggest A exp(ipx) as the proper probability. We then see that changing the direction of the x-axis leaves exp(ipx) unchanged. As a result, one tries to establish an equation which involves the incident photon moving to the right or left, but this must ultimately be associated with a pressure equilibrium equation ((1)) which must involve both directions of the incident flow, i.e. an equation in exp(ipx) and its complex conjugate. We suggest that this is the reason that a so-called square root Aexp(ipx) of flux emerges from AA, i.e. free particle quantum mechanical behaviour.
Equilibrium in Time versus in Space We suggest that the usual toss of a coin or die is linked with an equilibrium in time. The coin and die have physical spatial properties which define the probability a priori, but any toss is completely uncertain. After N (large number of tosses), 50% of the results for a coin should be heads and the rest tails. In other words, one has an equilibrium in time. As more tosses are performed, for N very large, one still has very nearly 50% heads and 50% tails. We argue that time runs in one direction. One may run a movie backwards, but time still runs in one direction. This is not the case for a spatial equilibrium. If one considers a pressure equilibrium. In such a scenario, one must consider particle motion in both x directions, not just one. This is the physical equilibrium linked with pressure. We suggest that this turns out to be a key idea in the analysis of one-dimensional reflection-refraction at an n1-n2 index of refraction junction. One Dimensional Reflection-Refraction In the case of 1D reflection-refraction at an n1-n2 index of refraction junction, one has the usual time equilibrium equation: 1= P(reflect) + P(refract) ((2)) Just as in the coin toss situation, for N very large, N P(reflect) photons reflect and N P(refract) refract. There is no notion of a spatial equilibrium at this point. We also note that in a physical case, there is a specific direction of motion, but there is no indication of such a direction in ((2)). In previous notes, we pointed out that ((1)) may be written in terms of dynamic variables: AA/c = BB/c + CC/c2 → AAp - BBp = CCp2 ((3)) using E=pc Here E is the same for the incident, reflected and refracted photons and AA, BB, CC represent the fluxes for the three. We stress, as noted in before, that ((3)) is a pressure balance equation, i.e. a spatial equilibrium which physically should be linked to motion in both directions along the x-axis. ((3)), however, is only linked with one direction of motion, whether one thinks in terms of a steady state situation or a single photon. We thus argue that one consider ((3)) as an equation representing spatial equilibrium and motion in both x directions and find other equations which are linked to direction. This is the main point we make in this note. To achieve this goal, we consider: AA = Aexp(iCp) * Aexp(-iCp) ((4)) (Here C is a constant to deal with units) ((4)) shows that a flux may be linked to motion in both directions as it must, and we use the variable p, momentum, because it is the variable of impulse hits.
With ((4)), one may now describe reflection-refraction in one direction (which occurs physically) by using one piece of ((4)), i.e. A exp(iCp). This “probability”, however, should be continuous in space, so we suggest using instead of exp(ip): exp(ipx) ((5)) (This has the added feature that exp(-iEt+ipx) is Lorentz invariant.) We then write the one directional equation: Aexp(ipx) + Bexp(-ipx) = C exp(i p2x) at x1, the n1-n2 junction ((6)) Similarly, one may have a continuous first derivative, or consider continuity of momentum as well: Ap exp(ipx) -p B exp(-ipx) = Cp2 exp(ip2 x) at x1 ((7)) If x=0, one may multiply the two equations to obtain the pressure equilibrium ((3)). We note that ((6)) and ((7)) hold for an incident and reflected photon on the left hand side of x1 and for p2 on the right or the incident and reflected on the right and p2 on the left. Thus, the notion of the equation holding for motion in both directions as an overall spatial equilibrium holds if one considers an average of an n1-n2 motion to the right with an n2-n1 with motion starting on the right and moving to the left. The pressure equation does not distinguish between these, but a physical scenario does. Thus, the physical scenario needs to use a different set of equations than the overall pressure balance one, we argue. We also note that if one does not chose x1 =0, then one may multiply ((6)) by the complex conjugate of ((7)), i.e. p AAexp(ipx)exp(-ipx) -p BB exp(-ipx)exp(ipx) -pAB exp(i2px1) + pAB exp(-i2px1) = p2 CC exp(Ip2x) exp(-ip2x) ((8)) In order to obtain the pressure balance equation ((3)), one needs to average over x1. In such a case, one the complex conjugate of ((7)) represents motion in the opposite direction from ((6)). We suggest that given a one directional physical problem (reflection-refraction), casting this as a spatial equilibrium one, i.e. one with pressure balance, requires having a non-directional equation ((3)) linked directly to the conservation of probability 1= P(reflect)+P(refract). This, in turn, means that there must exist equations which describe the one directional motion. These, however, must be invariant under p→-p and x→-x and must combine to remove the sense of direction in order to obtain a pressure balance equation which classically allows motion in both directions. We argue that this is how the notion of a “square root” of flux AA arises, i.e. Aexp(ipx)
Aexp(-ipx). In other words, this is how the notion of a free particle wavefunction appears, we speculate. Conclusion We argue that classical spatial equilibrium involves motion in both directions, as in a pressure balance. We then note that for a one directional problem, i.e. reflection-refraction at an n1-n2 junction, probability conservation: 1 = P(reflect) + P(refract) may be recast as a pressure balance equation: AAp - BBp = CCp2 where AA, BB, CC are the fluxes of the incident, reflected and refracted photons. This leads to a problem. Pressure balance should be based on an equilibrium involving motion in both x-directions, but n1-n2 reflection clearly involves motion in one direction. We thus suggest that there should be equations which describe one dimensional motion, but combine to yield the pressure balance equation which involves both directions, or no sense of direction. We argue that one may write AA = Aexp(ipC) Aexp(-ipC) to describe both directions. The variable p is used as it describes impulse hits. Then exp(ipC) (C=constant for units) then represents one directional motion. Such a function, however, should be continuous in x and so we propose Aexp(ipx) and try to create a one directional equation using it, i.e. Aexp(ipx) + Bexp(-ipx) = Cexp(ip2x) at x=0 and continuity of the derivative d/dx at x=0. These equations hold for both the incident and reflected photon moving from the left to right in n1-n2 and for them moving in n1 to n2 from the right to left. A given equation, however, only describes one kind of direction, but combining the two continuity equations leads to a single equation with no direction present, as it represenst both, i.e. the pressure balance equation. We thus speculate that this is how the quantum free particle exp(ipx) emerges from the notion of spatial equilibrium. In other words, there are two spatial directions and both are involved in a spatial equilibrium. Thus, if one has a one directional problem (reflection-refraction) linked to a spatial equilibrium equation (pressure balance), one must find directional equations which combine to create this non-directional pressure balance one.
