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Free Particle Quantum Probabilities and Classical Conditional Probability Relations Francesco R. Ruggeri Hanwell, N.B. Oct. 27, 2025 In previous notes, we argued that one may introduce a unit modulus complex probability into Newtonian mechanics to account for the uncertainty in the outcomes of elastic two body scattering. Such scattering must conserve energy and momentum and so for an initial (e1,e2) and (p1,p2) vectors, one may propose that any set (ei,ej) and (pi, pj) has the same product probability as the original. At first, one might think that this leads to exp(iE) and exp(ip), but enforcing Lorentz invariance and time reversal invariance (i.e. p→-p x–.-x) one obtains exp(-iEt+ipx) (or exp(-iEt+i p dot r). The question then becomes: How is this probability used? First, one may note that it is physically relevant because it may be applied to the analysis of 2-slit interference and one dimensional reflection-refraction at an n1-n2 index of refraction junction. On the other hand, if one considers a large number of particles with various momentum which move in a box without interacting, then one has P(x)=1/sqrt(L) and P(p) = 1/B, where L is some length and B, some momentum range. This implies there is no special preference for any momentum. At first it seems that exp(-iEt+i p dot r) has nothing to do with the non-colliding particles in a box, because there are no interactions and exp(-iEt+i p dot r) was specifically constructed with interactions in mind. Nevertheless, one may observe that exp(-ipx)exp(ipx) = 1 which is linked to P(x). We suggest that even though a priori, the non-colliding particles in a box example does not seem to be directly linked to exp(-iEt+ipx), there is another Newtonian case which involves interactions, namely those with a potential V(x). Such interactions must conserve momentum and energy of the particle-V(x) system and so exp(-iEt)exp(ipx) should apply to such an analysis. In Newtonian mechanics, however, one does not use exp(ipx) to calculate motion with a V(x) present. As a result, one must account for this. It seems that if one uses wavelength = hbar/p, classical values of p lead to a wavelength which is so tiny compared to the system length that one may use the Newtonian approach dp/dt = -dV/dx. This does not mean, however, that exp(ipx) is irrelevant. It manifests itself in 2-slit interference and 1D reflection-refraction (n1-n2 junction) and it is possible that there are systems for which the wavelength is of the order of the system length. In such a case, one should be able to analyze the system using exp(ipx)s. This would imply a probabilistic OR situation with free particle probabilities exp(ipx) and various impulse hits leading to: { Sum over i a(p) pp/2m exp(ipx) } / W(x) +V(x) = E, where W(x)=Sum over p a(p)exp(ipx). Given the presence of p and x, one might suggest that P(p/x) = a(p)exp(ipx)/W(x) as done in (1). We note that exp(ipx) is symmetric in p and x and so a conditional probability treatment should deal with a system with uncertainty in both p and x. A given p in exp(ipx) has no uncertainty in p and so it is not until one considers the V(x) interaction that uncertainty in p arises. In (1), using conditional probability relations, we showed that such an assumption leads to P(x) = W*(x)W(x) and P(p) = a*(p)a(p). Here we argue that the classical relations of conditional probability do not appear until one obtains a situation with both uncertainty in p and x as exp(ipx) is symmetric in both of these variables. We also example the case of a(p1)=1, a(p)=0
for p not= p1 (1 dimension). We show that this then leads to P(x)=P(p)=P(p/x)=P(x/p) = 1 which describes the non-colliding particles in a box. Thus, there is a roundabout way to link the exp(ipx) formalism to the noncolliding particles in a box scenario. Classical Non-Colliding Particles in a Box One may first consider a box of length L with a single particle placed at some x. If there is no bias, one would expect a uniform distribution, i.e. P(x) dx = dx/L ((1)) A second scenario to consider is a single particle moving with a constant speed. If one makes a measurement at a given time t, one may ask: What is P(x)? It should again be given by ((1)), but as pointed out in previous notes, there is a fundamental difference between the scenario of ((1)) and that of the moving particle, namely that for ((1)) one may verify the measurement at a later time, but this is not possible in the moving particle case. Phrased differently, classical equilibrium is usually concerned with a system which does not change in time and a single moving particle changes. To remedy this, one may add many such particles, but there is no reason then to restrict p to a single value. One may also have many p values as long as the particles are non-colliding. In such a case, one has: P(x)dx =dx/L and P(p)dp = constant ((2)) Newtonian Energy-Momentum Conserving Probability In previous notes, we argued that one may introduce a probability into Newtonian 2-body elastic scattering for which energy and momentum are conserved. For an initial (e1,e2) and (p1,p2) vector, we suggest that all (ei,ej) and (pi,pj) pairs which conserve energy/momentum have the same product probability as the original. This thinking should then be applied not only to elastic 2-body scattering, but also to V(x) interactions as they also conserve p and e overall. At first one might think that the probabilities are: exp(iE) and exp(i p) ((3)) Here a unit modulus complex number is used because all free particles have the same real value weight. The problem with ((3)) is that changing the direction of the x-axis changes exp(ip) to exp(-ip). Furthermore, given that E and p are present, one wishes to have Lorentz invariance which is linear in E and p. This seems to suggest: exp(-iEt+ i px) (or exp(-iEt+ i p dot r) ((4)) The question then becomes: How is this new probability ((4)) used? One may point out that it may be applied to 2-slit interference and one dimensional reflection-refraction of photons at an n1-n2 index of refraction junction. Usual Newtonian
mechanics cannot be used in calculations in either of these cases. Thus, ((4)) seems to be legitimate, yet Newtonian mechanics seems to match experiment for: dp/dt= - dV/dx and .5mvv + V(x) = E ((5)) We have argued that in principle exp(ipx) should be used to describe such a scenario. We suggest that: Wavelength = hbar/p ((6)) For classical values, the wavelength is so tiny relative to the system length that the usual Newtonian approach applies. In principle, however, one may imagine a system whose length is of the order of a wavelength. In such a case, one may use exp(ipx) as a probability and consider elastic scattering with “points’ in the potential, i.e. V(x) = Sum over k Vk exp(ikx). This suggests: {Sum over i a(p) pp/2m exp(ipx) } / W(x) + V(x) = E ((7)) where W(x) = Sum over p a(p)exp(ipx) Classical Conditional Probability Relations We introduced exp(ipx) as a probability linked with elastic two body scattering. The fact that it contains p and x, two variables, seems to suggest that some kind of conditional probability is present, but it is not so clear what that is. Given the symmetry in p and x in exp(ipx), one would expect uncertainty in both p and x, but the elastic scattering case has a single known p given by exp(ipx). It is only when one considers interactions with a potential V(x), i.e. ((7)), that one has an expression uncertain in x and p. We thus seek conditional probabilities here. In (1), we suggested that: P(p/x) = a(p)exp(ipx) / W(x) ((8)) From this, we showed, using conditional probability relations, that: P(x) = W*(x)W(x) and P(p) = a*(p)a(p) ((9)) Thus, the presence of x and p together in exp(ipx), which arose from an attempt to have exp(-iEt+ipx) be Lorentz invariant and linear in E, p at the same time. There is a further point to note. One cannot simply use exp(i |p|), This would give p and -p the same probability, but would suggest that: P(p)P(-p) = exp(i 2 |p|) ((10)) which cannot be. Given the conditional probability viewpoint in ((7)) ((8)), ((9)), we ask: How does one consider the noncolliding particles in a box problem? If one sets: a(p1)=1 a(p)=0 fo p not= p1, then:
P(p/x) = 1 P(x) =1 P(p) = 1 ((11)) These are the results for the noncolliding particles in a box. There are no collisions there and so there is no need for exp(ipx). In ((7)), there are collisions and so one requires the presence of exp(ipx). Ultimately, however, the exp(ipx) description may be used to explain the noncolliding particles in a box. Conclusion In conclusion, we try to show how the exp(ipx) (from exp(-iEt+ipx)) probability used to describe Newtonian 2-body elastic scattering gives rise to classical conditional probability expressions. We suggest that the presence of x and p in exp(ipx) suggests that such a conditional probability scheme should somehow appear. We argue that given the symmetry between p and x in exp(ipx), one should expect an expression demonstrating uncertainty in both variables. We note that exp(ipx) applies to 2-body scattering, but should equally well apply to collisions with a potential V(x). Newtonian mechanics, however, which does not use exp(ipx), accurately describes V(x) interactions through dp/dt = -dV/dx or .5mvv + V(x)=E. We suggest that this is the case because wavelength = hbar/p is tiny compared to the system length. If one imagines a wavelength of the order of the system length, then ((7)) should apply. This then is an expression with uncertainty in both x and p and we suggest, as in (1), that P(p/x) = a(p)exp(ipx) / W(x). In (1), we showed that P(x) = W*(x)W(x) and P(p) = a*(p)a(p) using classical conditional probability relations. Here we note that one may create this conditional probability framework, but that for a(p1)=1, a(p)=0 for p not= p1, one has: P(x)=1, P(p)=1 and P(p/x)=1. This describes a box of noncolliding particles with a uniform distribution of p values. exp(ipx) does not appear because there are no collisions. Nevertheless, an exp(ipx) framework does reduce to the noncolliding particles in a box picture using conditional probability relations. References 1. Ruggeri, Francesco R. Wave Function Squared as Spatial Probability Density (preprint, zenodo, 2018)
