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Free Particle exp(ipx) As A Momentum Conserving Probability Francesco R. Ruggeri Hanwell, N.B. Dec. 6, 2026 We have suggested in previous notes that there is a tendency to consider quantum mechanics as a kind of hybrid probabilistic-deterministic theory because it is supposed to ultimately “merge’ with classical mechanics for high energy levels (tiny wavelengths). Newtonian mechanics is deterministic and so somehow it seems quantum mechanics should also be seen as deterministic, hence the sense of surprise at tunneling when KEave(x)=0, even though vrms(x)=0 does not mean that actual values creating the probability are 0. Here, we wish to argue that exp(-iEt+ipx) (or the exp(ipx) part for spatial interactions) is really a momentum probability scheme which may possibly bypass many aspects of the actual physics to describe a problem in a simpler form in terms of a momentum conserving probability. A key idea is that exp(ipx) is actually the form to use when one has randomness as we will argue for reflection-refraction from an n1-n2 index of refraction junction. We argue that this idea is already known for a particle in an infinite well for which one has sin(pave x), i.e. 1/2i (exp(i pave x) - exp(- i pave x)). In other words, one has the form exp(i pave x), even though it is linked with an average p. Given that one is not considering deterministic physics, there is nothing wrong with exp(i pave x) being composed of Sum over p a(p) exp(ipx), where exp(ipx) holds in all space, not just within the infinite well potential. exp(ipx) is a momentum conserving probability, but on average momentum must also be conserved by a free statistical entity within the well. Similarly, W(x) for a single particle bound state consists of many exp(ipx)s, but represents a fixed E ave and on average this Eave must be conserved in time, hence the time dependent exp(-i Eave t). As a result, one may see that the quantum mechanical exp(-iEt+ipx) is an attempt to describe a system in a simple manner in terms of momentum conserving probability. To see this in more detail, we consider (1) which describes reflection and refraction of a photon at an n1-n2 junction in depth. In a previous note, we have already pointed out that in n2, c/n2 simply means that the photon is interacting on average with atoms/molecules in n2, but has c in between such interactions. It is only on average that one has c/n2 and even E and p2 = p n2. Nevertheless, one may use exp(ip2 x) as a probability which describes average properties of the system, just as it does in the examples in the above paragraph. The question then becomes: Do the microscopic details of the photon physics (the actual interactions) dictate the probabilities to reflect/refract or does the simplified exp(ipx) set linked to momentum conservation do so? Based on (1), it seems that reflection is completely independent physically from “refraction”, the latter requiring re-radiation and affecting all glass molecules. As a result, it appears that it is difficult to suggest that it is the microscopic interactions which dictate probability because reflection and refraction are microscopically so different. On the other hand, reflection-refraction is not a .5-.5 probability random process as experimental results show. The question then becomes: If some kind of systematic probability driver is not supplied by the microscopic interaction, what provides it? We argued in previous notes that one may introduce the probability exp(ipx) into Newtonian scattering to account for an initial p1,p2 (momentum vector set) having equal probability to produce any pi, pj (outcome set) as long as momentum is
conserved. We suggest here that it is exp(ipx) that one must use for random interaction if one insists on conserving momentum. Thus, exp(ipx) bypasses microscopic details of interaction and provides an “ any outcome pair of momenta” is equally likely if momentum is conserved. This is a major simplification of the problem, but the question is: What constraints does it impose on physical results? For a free particle, it suggests that it may interact probabilistically with both slits of 2-slit apparatus if the slits are about hbar/p apart. For randomness, one would expect equal probability for each slit and this is the case as long as one also considers exp(ipx) for each slit to a the same point on a screen. In the case of reflection-refraction at an n1-n2 index of refraction junction, we try to see how the simplification exp(ipx) can be consistent with the microscopic details of the interactions and at the same time dictate the probabilities to reflect and refract. We try to consider why an exp(ipx) scheme predicts how nature behaves while a random one with no momentum conservation considerations does not. In particular, we argue that exp(ipx) represent randomness in a two body scattering problem in which momentum is conserved and so this same exp(ipx) must be carried over into problems which involve interactions instead of simply suggesting that one has .5 probability to reflect and .5 to refract because the two processes are independent and hence random. They are random, but with conserved momentum and hence an exp(ipx) scheme is required, we argue. Reflection and Refraction Details According to (1), a photon passing from a vacuum into glass may reflect off of an outer electron, i.e “does not penetrate inside the glass” and “deflects”. Whatever the actual mechanism causing this specific reflection, it is a one time impulse type situation and seems to be completely independent of the refraction process. Refraction is said to involve a photon interacting with an outer electron of the first glass atom/molecule and leading to re-radiation in a particular direction. Furthermore, the entire glass is involved because this interaction-reradiation process must be repeated in a systematic manner to create the average value of c/n2 for light speed and p2=p*n2 for momentum and even E as an average. Thus, the microscopic mechanisms for reflection and refraction are completely different and it seems, completely unlinked. It is known that there is a probability to reflect and one to refract and they must add to one, but given the independence of the processes one would expect complete randomness and so might expect that the probabilities would be .5 and .5, which they are not experimentally. This begs the question: What is happening? We have argued in previous notes that one may introduce a probability to have a momentum into Newtonian elastic scattering. We argued that given an initial (e1,e2) energy set and (p1,p2) (initial momentum vector set), one may have any (ei,ej) (pi,pj)’s with equal probability as long as energy and momentum are conserved. We argued that the overall probability (product–AND) for ei,ej is P(ei)P(ej) (and a similar result for pi,pj) and insisted on Lorentz invariance, leading to: exp(-iEt+ ipx) ((1))
exp(ipx) is a probability which conserves momentum and allows for equal probabilities for any any pi,pj pair. This equal probability, however, is equivalent to randomness in the sense that no pi,pj pair is favored and no ei,ej either as long as momentum and energy are conserved. The notion of randomness seems to be associated with exp(ipx) in a spatial problem. We argue that one cannot simply state that the probability to reflect is .5 (and to refract .5) due to randomness. Randomness in interactions with momentum conservation suggests that the reflected photon must be linked with a probability of exp(-ipx) and the initial photon, with exp(ipx). What does one do with the average p2= p *n2 of the second (glass) medium? Physically, this is not the same as a photon with p2 moving in a vacuum. P2 is only an average as the photon interacts with an electron leading to reradiation in a certain direction, and this process is repeated. When not interacting, the photon moves with c, but time is lost during interactions and so the average speed is c/n2. Can one assign any properties to an average p2? We note that the refraction process is independent of the reflection one and so one does not really need to follow the photon in time as it is absorbed and reradiated, creating the average p2. Idealizing this photon at the n1-n2 junction, one would argue that one had a photon with p2 and c2 (grossly simplifying the situation as this is not the microscopic picture at all). It does allow, however, for a sense of randomness for the refraction process as well and the notion of conservation of momentum. We argue that one may simplify the problem of n1-n2 reflection-refraction by considering the probabilities exp(ipx), exp(-ipx) and exp(i p2 x). The final step is to link these probabilities. We do so by following previous notes in which we argue that the exp(ipx) approach suggests one has uncertainty intervals in x. This probabilistic scheme means that one does not have a microscopic picture of the incident, reflected and refracted photons, but rather probabilities all three in a tiny dx region about x=0. We then suggest using: Aexp(ipx) + B exp(-ipx) = C exp(i p2 x) at x=0 ((2a)) Apexp(ipx) - Bp exp(-ipx) = Cp2 exp(ip2 x) at x=0 ((2b)) ((2a)) and ((2b)) allow one to solve for AA/c, BB/c and CC/c2. Using AA=1, BB/c and Cc/c2 are the probabilities to reflect and refract. The point we make is that randomness must be applied already to two body scattering and leads to exp(ipx) for interactions and so one must use the exp(ipx) forms and not .5 for reflection and .5 for refraction to describe randomness. Conclusion In conclusion, we argue that given Newtonian two-body scattering, randomness, i.e. equal probability for any outcome to an initial e1,e2 (energy set), p1,p2 (momentum vector set), ei,ej, pi,pj that conserves energy and momentum means having a Lorentz invariant probability exp(-iE+ipx). In a spatial problem one only uses exp(ipx). Thus, exp(ipx) is associated with randomness in problems involving interactions with conservation of momentum. This is the key
idea of this note. We suggest that one cannot use .5, .5 probabilities for randomness in 2-choice situations if one has interactions involving momentum conservation. Rather an exp(ipx) scheme must be used. We consider the problem of reflection-refraction at an n1-n2 index of refraction junction. The microscopic details of the interaction are complicated and are described in (1). The point we make is that reflection is independent from refraction. Thus, one should expect random probabilities for each. This might lead one to expect .5 for reflection and .5 for refraction, but we argue that one must consider the exp(ipx) form for randomness associated with interactions with conservation of p (which always holds). The incident and reflected photons should be associated with exp(ipx) and exp(-ipx). The refracted photon is described on average by c2=c/n2 and p2=n2 * p. We suggest that because the reflection and refraction interactions are completely different, it is possible to idealize the refracted photon by exp(ip2 x) suggesting randomness which conserves momentum. This allows one to take a complicated microscopic problem involving two independent processes and to calculate randomness using exp(ipx)s which conserve momentum. This leads to ((2a)) and ((2b)) and to a solution for the probabilities to reflect and refract, we argue. References 1. Sokolov, G. and Sokolov, V. Refraction and Reflection of Single Photons vs WaveTheory (The General Science Journal) https://www.gsjournal.net/Science-Journals/Research%20Papers-Relativity%20Theory/D ownload/9567
