Speculation on Quantum Free Particle Probability and Spatial Equilibrium Part 4

Speculation on Quantum Free Particle Probability and Spatial Equilibrium Part 4 Francesco R. Ruggeri Hanwell, N.B. Nov.10, 2025 In previous notes, we argued that one may introduce a probability into Newtonian elastic scattering if one assumes that for an incident (e1,e2) energies and (p1,p2) momenta vectors, any (ei,ej), (pi,pj) which satisfy conservation of momentum and energy have equal probability. We then found that for a single particle, the relevant probability is exp(-iEt+ip dot r). This is a unit modulus complex number which is Lorentz invariant, and exp(i p dot r) is time reversal invariant, i.e. p→-p and r→-r yield the same value. The question then becomes: How does one use exp(-iEt+ipx) or for time-independent problems, exp(ipx)? This is not a usual probability linked to quantity, e.g. N P(reflect) = the number of particles which reflect. exp(ipx) is a probability to have a certain momentum, but there is no reason it cannot carry a weight, i.e. a math factor. The question is: What does such a factor mean physically? Secondly, exp(ipx) depends on x (space), but exp(ipx) is not the probability to be at x, but it is somewhat related in that for any x and a given p, the modulus has the same value, suggesting equal probability to be at any x. This mapping, however, really only holds for a single exp(ipx). It seems that one can say two things about exp(ipx). First, if it is a probability for a momentum p, then one should be able to write p exp(ipx) or function(p)exp(ipx). Secondly, one should be able to write AND (multiply) and OR (add) statements. The idea of an OR or addition of exp(ipx)s is important, because one may be able to write an equation for constant kinetic energy: -1/2m grad dot grad W(x) = numberW. The general solution should be a W(x) which contains a sum of all exp(ipx) type solutions which have this kinetic energy. (In the relativistic case, one may consider d/dt d/dt W - cc grad dot grad W = -momocccc W.) The question we consider in this note is: Given that one may write W and grad W, what can only really do with these? They do not directly represent physical observables. Furthermore, if one wishes to impose continuity on W and gradW, which is reasonable mathematically, one should have a physical argument to justify it, and we suggest that this is a little complicated. Ultimately, we try to argue that one may consider continuity of W as well grad W in certain cases based on the association with a classical quantity probability equation. We also suggest one may reate W* grad W which may be generalized to W* grad W + (grad W*) W, as discussed in Part 3 and provides a link to classical probabilities. exp(i p dot r) Probability We have argued in previous notes that one may introduce a probability into Newtonian elastic scattering if one assumes that a given initial (e1,e2) set of energies and (p1,p2), initial momentum vectors can result, with equal probability, in any (ei,ej) (pi,pj) (vectors) set such that energy and momentum is conserved. A probability function is needed and in Parts 1 and 2, we discuss how we choose: exp(-iEt+ i p dot r) ((1)) This is a unit modulus complex number because all free particles carry the same real value weight. This seems to be linked to the idea that for a fixed p, any x has the same modulus and so ((1)) is consistent with: P(x)dx =dx/L ((2)) where L is an arbitrary length One may multiply ((1)) by its complex conjugate to obtain ((2)) (L=1), but this is a math process. It begs the question: What is the physics? If one considers exp(ipx), one must ask: What does this probability really represent? It is not a “quantity” probability which can be multiplied by the usual “number of particles”. Nevertheless, it can carry a real value weight, which begs the question: What does such a weight mean physically? We suggest that one may say at least two things about exp(ipx): (A) One should be able to add exp(ipx)’s (with weights) in OR situations as exp(ipx) is a probability. This is important, because if one solves: -1/2m grad dot grad W = kinetic energy W ((3a)) nonrelativistically or -EEgrad dot grad W = -momocccc W ((3b)) relativistically Then one should obtain a solution which is a sum of all possible exp(ipx) type probabilities which are consistent with the energy value. (B) One should be able to write p exp(ipx) (or function(p) exp(ipx)) as a weight for p, because exp(ipx) represents a probability to have a certain momentum p. The next question is: What boundary conditions may one associate with W? If one treats W(x) as a math function, one may introduce continuity, continuity of grad W and other boundary conditions, as is typically done, but how does one justify these by thinking of exp(ipx) strictly as a probability and not a math function? Attempt to Link exp(ipx) to Boundary Conditions, Continuity etc We start with condition (B) as having a non-math foundation, namely exp(ipx) is a probability to have p so one should be able to multiply p by exp(ipx). If this is the case, one may also write: P dot p exp( i p dot r) ((4a)) and {Sum over p a(p)pp/2m exp(ip dot r)} ((4b)) ((4b)) is an unusual sum because there is interference which occurs. Nevertheless, this is what follows if one takes exp(ip dot r) seriously. One still does not have any kind of boundary or constraint equation. One might suggest that ((4b)) leads to: {Sum over p a(p)pp/2m exp(ip dot r)} / ( Sum over p a(p)exp(ip dot r) + V(r) = E ((5)) ((5)) is the Schrodinger equation. There are, however, physical problems which have only one or say two p values and one does not know V(r) explicitly. These must be considered as well. We note that exp(ipx) is a directional probability as exp(ipx) and exp(-ipx) have different values. The notion of p exp(ipx) should be linked with impulse, but given the notion of probability, one cannot help note that in classical equilibria, impulse is linked with the idea of pressure, which holds in all directions in a point in space. If one wishes to develop classical type probability equations linked to classical probabilities (unlike the exp(ipx) directional probability), then it seems that one should extend exp(ipx) to map into pressure. As discussed in previous notes, this may be done for a one-dimensional reflection-refraction problem. Before considering this problem, however, one should consider the form of pressure classically. In general one has: p proportional to impulse = Force and p =mv proportional to flux, i.e. # particles per second ((6)) One may write: pressure is proportional to pp ((7)) This involves a quadratic form of p. It is, however, equally possible to write: Pressure is proportional to AA p ((8)) Here AA is the flux (which includes speed intrinsically). The form ((8)) is relevant to photons, while ((7)) is not because photons do not follow p=mv. One may consider the notion of p exp(ipx) in a photon problem such as a n n1-n2 index of refraction junction reflection-refraction. In such a case: 1= P(reflect)+P(refract) → AA/c -BB/c =CC/c2 → pAA-pBB = p2 CC ((9)) ((9)) has the form of a pressure equation which is a balance at some junction point which one may always call x=0. In such a case, it seems that there is justification (beyond mathematical continuity) to argue that there should be a balance: f(A) p exp(ipx) + f(B) (-p)exp(-ipx) = f(C) p2 exp(ip2x) at x=0 ((10)) f(A) is an arbitrary function of A, the square root of flux. One may ask: Can f(A) = AA = flux? We argue that it cannot because exp(ipx) is not a usual probability which may be multiplied by a quantity, such as the number of particles etc. This suggests a second continuity equation, we argue, namely: g(A) exp(ipx) + g(B)) exp(-ipx) = p2 g(C) exp(ip2x) at x=0 ((11)) A solution is g(A)=f(A) =A etc and one finds that based on an equation using classical probabilities ((9)), one is led to two continuity equations, one linked to exp(ipx) and the other to p exp(ipx) = -id/dx exp(ipx). Thus, one may use exp(ipx) and its derivative d/dx in equations which would otherwise be ascribed to mathematical continuity. In Part 3, we noted that if x not=0 in the above problem, that one obtains x dependent interference terms, but that these may be eliminated by defining current as: 1/m W* dot (-igrad) W - 1/m (-grad(W*)) dot W ((12)) in the simple case W-> exp(ipx) If W and grad W are continuous at a point x, one may create the combined expression ((12)). We suggest that it is important to be able to use exp(ipx) and its d/dx derivative in continuity/constraint type equations (without having to think about the physical justification), as it is these continuity equations (or constraints) which actually solve the problem. In particular, in n1-n2 reflection-refraction, the two continuity equations in exp(ipx) and d/dx exp(ipx) solve for B and C in terms of A, something cannot be done by simply using ((9)). Thus, ((9)) is physically important and may be measured, but there is an underlying probability (directional probability of p= exp(ipx)) which is influenced by ((9)), in that it may also be used in continuity equations at a junction point. exp(ipx) acts as if it is a probability representing quantity, when formally it doesn’t. We note that in such cases, it carries unusual weights (e.g. square root flux). One may write: Aexp(ipx) + Bexp(-ipx) = Cexp(ip2x) at x=0 ((13a)) and Apexp(px)-pBexp(-ipx) = Cp2exp(ip2x) at x=0 ((13b)) These appear as “quantity” type probabilities, even though they are not. Usually it is argued that ((13a)) and ((13b)) are justified by math continuity, but we argue that it is the link between exp(ipx) in ((13a)) and ((13b)) with ((9)) which ultimately justifies the continuity/constraint equations like ((13)) which are used in quantum mechanical problems. Thus, there seems to be no problem with solving for W in -1/2m grad dot grad W = E W in a scattering problem and then applying “boundary conditions” on W at say a reaction radius for a hard-sphere. For example, one may say that W=0 for r<ro and any solution using exp(ipx)s outside ro. In such a case, the physical solutions outside ro are: exp(ipz) and f(theta) exp(ipr)/r ((14)) Conclusion In conclusion, classical probabilities seem to be quantity probabilities in that one may multiply them by a physical number. Examples include multiplying the probability .5 for a coin toss by N, the number of tosses, or P(reflect), the probability for a photon to reflect by the number of photons. In previous notes, we suggested that there is a probability exp(-iEt+ip dot r) which describes Newtonian elastic scattering. This is not a “quantity” probability. This is why exp(ipx) is a complex number with a unit modulus. exp(ipx) is the probability to have momentum p, but any free particle has the same real-value weight. Hence, one cannot multiply exp(ipx) by the number of particles etc. We suggest, however, that it may carry a math weight and may certainly be multiplied by p or a function of momentum. This leads immediately to the formulation of the time-independent Schrodinger equation, but we are more interested here in establishing continuity-boundary equations for sums of exp(ipx)s. We note that one should in principle be able to add various exp(ipx) probabilities if physical OR situations arise. Then, one may solve -1/2m grad dot grad W = kinetic energy W to find a sum (OR case) set of exp(ipx)s which have the same kinetic energy. Mathematically, one typically applies continuity or boundary conditions to W, but in a physical problem there should be physical justification for such applications. That is the focus of this note. We suggest that given that one may write sums of pexp(ipx)s, in the case of a photon, p flux = pressure. Thus, an equilibrium equation involves an expression such as: 1= P(reflect)+P(refract) → AA p - BBp = pw CC, where AA, BB, CC are fluxes of the incident, reflected and refracted photons. This is a pressure type equilibrium equation and we suggest that a p weight exp(ipx) equation should be linked to it. This suggests that one may have continuity of p weight exp(ipx)s together with a continuity equation in weight exp(ipx)s because the weight of p exp(ipx) cannot be flux due to the fact that exp(ipx) is a not a quantity probability. This is in fact a very good thing because the two equations allow for the solution of B and C in terms of A, whereas the 1= P(reflect)=P(refract) does not. Thus, the classical probability equation seems to physically justify the notion of continuity for W and grad W, which is usually justified by math continuity. We argue that a physical problem requires a physical reason for continuity-boundary condition-constraints for exp(ipx) and their derivatives.