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Joint Phase Space (x-p) Probability and Free Particle Quantum Mechanics Part 2 Francesco R. Ruggeri Hanwell, N.B. Nov. 22, 2025 In Part 1, we argued that P(x)dx=dx/L (where L is an arbitrary length) applies to a particle at rest. If one views such a particle from a frame moving at constant -v, it is moving. We argued that in such a case, the relevant probability is not simply linked to the position x, but also to the momentum p at x because this delivers an impulse hit. In particular, probability is presumably to be used in a physical problem with interactions and probability and a dynamical particle requires a dynamical probability describing both x,t and E,p. This led to exp(-iEt+px) which actually predicts intervals in space hbar/p and time hbar/E. In other words, a probabilistic view point leads to these results. Here, we ask the question: What if one simply wishes to use deterministic physics to describe a moving particle? After all, Newtonian mechanics is deterministic and one does not really need to focus on P(x)=1/L or its modification for a moving particle. In such a case, a particle moving with constant v (starting at x=0, t=0) is described by x/t=v both in Newtonian mechanics and special relativity. There is no uncertainty in x and t here which begs the question: Why does it appear in the probabilistic formalism of Part 1? In particular, exp(-iEt+ipx) derived in Part 1 implies physical regions: dx = hbar/p and dt=hbar/E and these seem to be absent in x/t=v and this is a problem. We suggest that a probabilistic approach is buried in the Lagrangian formalism which leads to the result d/dt v = 0, i.e describes the particle’s motion in space. This motion is actually x/t=v, but this equation only presents a subset of the description of a particle that is moving with constant speed, even in a deterministic Lagrangian/Action approach. We have already pointed this out in (1). We briefly explain it again here, but the real question is why does one need to consider the Lagrangian in order to describe x/t=v? In other words, the probabilistic approach associated with v=x/t only emerges in the Lagrangian formalism using a relativistic action Lt = -Et+px, with x/t=v. Ignoring the action and Lagrangian leads to an absence of hbar/p and hbar/E and one is only concerned with x/t=v. We note that the Lagrangian is primarily linked to the result: p = dL/dv partial and so argue that one is not simply interested in motion (x/t=v), but also in interaction because p is associated with an impulse hit. Thus, the Lagrangian approach is more than simply a mechanism which describes x,t motion, it is associated with p and E (through Hamiltonian = E = pv-L.) Using the Lagrangian approach suggests that it does not suffice to consider x and t by themselves when p and E are present. One must consider all four variables x,t,E,p which is the same point made in Part 1. The physics of the problem dictates that it is not enough to simply think in terms of x,t,v. The Lagrangian approach, which is constructed to ultimately yield x/t=v, when written in terms of E, p, i.e. Lt = -Et+px demonstrates uncertainty in x, hbar/p and t, hbar/E as shown in (1). This uncertainty must then be written in terms of a probability which is Lorentz invariant and leads to exp(-iEt+ipx). The point is that one would not even consider a probability if one did not consider E,p and x,t together as a complete treatment of a moving particle. In other words, x/t=v is an incomplete description of the state of a particle moving with constant speed, even in a deterministic framework.. At first this might seem surprising, but it is well known that energy and p are linked to interactions. In special relativity, one does not
consider x,t,v alone and then compute p and E from these, but considers four vectors and Lorentz invariant equations. It is possible to create a Lorentz invariant equation involving all four state variables x,t,E,p and it turns out that this is the relativistic Lagrangian which describes motion, so this approach is more general than simply stating x/t=v. Thus, delta x = hbar/p and delta t= hbar/E appear in both the probabilistic approach of Part 1 and the deterministic one of the Lagrangian used here. A Priori Probability Approach of Part 1 In Part 1, we started with an a priori probabilistic description of a particle at rest somewhere in a length L, i.e. P(x)dx = dx/L ((1)) Although ((1)) is well-known, one might argue that in deterministic physics one would simply give the x position of the particle which holds for all t. There would be no need for a probability equation ((1)). In Part 1, our entire argument was based on finding a probability which applied to a description of the particle in ((1)) when viewed from a moving frame. In other words, one probabilistic approach led to another. We argued that for a moving particle, position is not sufficient because the particle interacts by delivering an impulse hit. A complete physical description would be: What is the probability to find the particle at x and have it deliver an impulse hit of p? This suggests a link between x,p, E,t instead of x,t, v simply describing the whole picture. The relation is given by the Lorentz invariant; Invariant = -Et+ px ((2)) We then argued for a Lorentz invariant probability exp(-iEt+ipx) whose modulus preserved P(x)=1/L and P(t)=1/T. As noted above, this analysis is based on an a priori probability scheme ((1)). One might argue that there is no need for ((1)) if one uses deterministic physics. In other words, x/t=v does not seem to imply a probability exp(-iEt+ipx) at all. Moreover, exp(-iEt+ipx) implies physical intervals hbar/p =delta x and hbar/E= delta t and this do not seem to be at all present in the equation x/t=v. Lagrangian Formalism The Lagrangian formalism is an approach which allows one to obtain Newton’s second equation: dp/dt = Force which ultimately becomes a differential equation in x with d/dt’s and allows one to find x(t) ((3))
Specifically: d/dt dL/dv partial - dL/dx = 0 ((4)) with L=T-V usually with T = -mo sqrt(1-vv/cc) in the relativistic case or .5movv in the nonrelativistic case. We point out, however, that there seems to be something more involved in a Lagrangian scheme because: dL/dv partial = p = momentum ((5)) ((5)) is an equation involving momentum which delivers an impulse hit. Furthermore, Newton’s second law dp/dt = Force is all about force as well. Thus, one begins with a formula regarding force and arrives at one which simply describes x(t). This suggests that the result of the force on the particle is a trajectory based on x(t). As a result, a particle which moves with a constant speed should be described by: x/t=v ((6)) There is no probability in ((6)) and no hbar/p and hbar/E, there is no appearance of p and E. It is known, however, that: p = dL/dv partial ((7a)) and E=Hamiltonian = pv - L ((7b)) with L= -mo sqrt(1-vv/cc) ((8)) Thus, the Lagrangian and Hamiltonian are intimately linked with p and E. This leads one to a consideration of special relativity in which one has: Lorentz invariant: -Et+px ((9)) Furthermore, using x/t=v (x=0,t=0 initial points) Lt = -mot sqrt(1-vv) = -Et+px (with c=1, x/t=v) ((10)) Lt is called the action, and it is the variation of the action which yields the differential equation in x and d/dt’s which ultimately gives x(t). It seems one cannot separate E,p from x,t in Lt, the action even though the ultimate solution of the Lagrange equation d/dt dL/dv partial - dL/dx = 0 yields x/t=v which is a simpler solution. We suggest that this differential equation “loses” some information of the problem contained in Lt, as it follows from Lt and L itself contains information about p and E through ((7a)) ((7b)). We argue that Lt is more general than the differential equation containing L and the result x/t=v. In other words, x/t=v is a subset of the information describing the moving particle. A full description must account for E,p,x,t with x/t= v and this description is given by -Et+px= Lt. This Lorentz invariant form which includes p,E,x,t for a moving particle seems to represent a full description, we argue. This idea is similar to the one we made in Part 1 in which we argued that it is not sufficient to
talk about t and x for a moving particle, one must also include E and p because these exist and affect the interactions of the particle. There is no point noting that a particle is at x from an interaction point of view unless one knows p, i.e. the impulse hit one is to receive. If one examines the specifics as done in (1), one sees that for: Lt = -Et+px one has delta x = constant/p and delta t = constant /E ((11)) Even though there is a deterministic trajectory x/t=v, there is also uncertainty in the deterministic action/Lagrangian formalism (which was supposed to be entirely deterministic). To describe this probability, one must make use of ((11)) together with the notion of no special weights for x and t and this yields: exp(-iEt+ipx) ((12)) As the intervals are present and the magnitude shows no preferential weight for x and t. Thus, the result ((12)) which emerged from probabilistic arguments in Part 1, actually emerge from deterministic considerations of Lt (action) because it is not enough to consider only x(t) which follows form d/dt dL/dv partial - dL/dx = 0 which seems to contain a subset of the information present in Lt. Thus, Lt seems to contain more information than x(t) and this information appears to be physically relevant. Furthermore, it gives rise to a probability which is not present in the deterministic solution x/t=v, suggesting that there are problems in nature which require such a probability, such as 1-dimensional reflection-refraction from an n1-n2 index of refraction junction. Conclusion In Part 1, we argued for the existence of a probability exp(-iEt+ipx) for a moving particle based solely on probability arguments. We started with a particle at rest in a region L and asked: What probability pertains to the particle when seen from a frame moving with constant -v? We argued that from a physical point of view, it is not sufficient to know that the particle is at x, one needs to know p as well because this is related to the impulse hit one receives (and a similar argument for E,t). In other words, one has a dynamical probability which may be used in dynamical problems which exhibit probability. Here we note that traditionally one describes a deterministic particle with constant speed by x/t=v. There then is no probability and hence no exp(-iEt+ipx) and no hbar/p, hbar/E. This begs the question: Why are there two schemes? Part 1 is based on probabilistic notions and presumably suggests physical intervals delta x = hbar/p and delta t = hbar/E, but where are these in x/t=v? If they are physical, they should be present, but do not appear. We argue here that this seems to suggest that x/t=v is a subset of the information needed to fully describe a particle moving at constant speed, even in a deterministic approach. We suggest that L, the Lagrangian, yields p = dL/dv partial and Hamiltonian = E = pv -L and so starting with L and Lt (the action) one has much more information than what one obtains from d/dt dL/dv partial - dL/dv=0. which ultimately yields x/t=v. One must consider the full L, Lt information and as shown in (1), Lt = -Et+px for x/t=v in the relativistic (and nonrelativistic
cases). This means that even in the deterministic Lagrangian/Action approach one has the uncertainty or probability hbar/p = delta x and hbar/E = delta t. Thus, the deterministic approach also points to these physical uncertainty regions and exp(-iEt+ipx). References 1. Ruggeri, Francesco R. Classical Free Particle Lagrangian, Special Relativity, Quantum Mechanics (preprint, zenodo, 2022)
