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Quantum Free Particle Probability and the Notion of Outcomes Francesco R. Ruggeri Hanwell, N.B. Dec. 1, 2025 Classically, probability is considered in terms of the number of possible outcomes. For example, a die toss has six possible outcomes and a coin toss, two. Probability calculations make use of probabilities and hence of the total number of possible outcomes. Thus, they are not deterministic calculations which follow “what happens” as in Newtonian mechanics. We suggest that if free particle quantum mechanics is probabilistic, it must also involve the notion of possible outcomes in an experiment and not represent a deterministic approach. We suggest, however, that something interesting applies to quantum mechanics because we have suggested that free particle quantum mechanics arises from the Lorentz invariant A = -Et+px. We note that there exist x,t trajectory points which follow x=vt. We also note, however, that one may introduce the idea of intervals dx, dt (because Newton’s dx→0, dt→0 is an idealization) and that A is unchanged for x,t and x+hbar/p , t+hbar/E. In other words, we suggest that there are intervals with probability about each x,t trajectory point. A trajectory point is associated with deterministic physics, but probability about this point is not. In other words, one has a kind of hybrid probability scheme. Thus, exp(ipx) and exp(-iEt) really represent fluctuation probabilities about a trajectory point x and t. These ranges of possible x and t fluctuations about a trajectory point x,t allow for further probability to occur in certain interactions. We consider two examples. The first is 2-slit interference which involves a deterministic particle moving along a straight line towards a 2-slit mechanism perpendicular to its path. At the 2-slit apparatus, one considers a probabilistic calculation as the uncertainty in x, i.e. dx=hbar/p allows for possible interaction with both slits. This involves considering all possible outcomes (i.e. motion through one slit or the other which is a classical situation), but exp(i p dot r1) and exp(i p dot r2) “sit on top” of these trajectories at the 2-slit apparatus and one has interference of the probabilities linked with each classical possible outcome. Thus, there are two sets of probabilities which are intertwined, the probability range hbar/p for each path and the two possible paths which have equal probability. One must consider the possible outcomes for both sets together. We suggest that a similar analysis applies to the case of 1-D reflection-refraction for x=0 at an n1-n2 index of refraction junction. Thus, we suggest that a deterministic free particle with x=vt may encounter probabilistic interaction situations, such as a two-slit apparatus or an n1-n2 index of refraction junction. These are probabilistic only because there is a probability fluctuation scheme in x associated with each deterministic x point. It is this probability which allows for a second probability to arise due to interaction with the slits or at the 1-D n1-n2 junction. In such a case, one must consider all possible outcomes which means that one does not only consider the exp(ipx) type probabilities, but also those linked with the physical interaction, i.e. two possible paths, one from each slit, or an incident, reflected and refracted photon at x=0 (n1-n2 junction). It is a combination of all possible paths which must be used in calculations. We thus suggest that even though there is a sense of determinism, i..e free particles move as x=vt and these exist in the 2-slit case and the 1-D reflection-refraction case, in a probabilistic calculation one must consider
all probabilistic outcomes consistent with these. Thus, quantum mechanical calculations are true probabilistic calculations in the sense that they involve all possible outcomes in an analysis. Probability and the Notion of Outcomes Probability calculations necessarily involve the notion of possible outcomes because probability is usually defined as: P(i) = i / total outcomes for a uniform distribution ((1)) such as a coin or die toss or some other function in ((1)) for the non-uniform case. Either way, the notion of possible outcomes is key. Quantum Free Particle Probability Due to t and x Uncertainty We suggested in previous notes that Lorentz invariance, together with the desire for dx and dt intervals which go beyond Newton’s idealization of dx→0, dt→0 in an interaction and conservation of momentum and energy lead to the free particle probability: exp(-iEt+ipx) ((2)) Given A = -Et+px (Lorentz invariant) and an x,t on the trajectory x=vt, one may write: x+hbar/p and t+hbar/E ((3)) and retain the same value of A. ((3)) is a particular choice of periodic interval which is associated with the periodic function exp(-iEt+ipx) which is both Lorentz invariant and allows for equal weights for any (ei,ej) (pi,pj) (momentum vector) outcome in a Newtonian elastic 2-body interaction (with e1,e2, p1, p2 as initial values). This, in turn, is linked to conservation of energy and momentum. We wish to consider exp(-iEt+ipx) in more detail because it is associated with physical interval regions dx=hbar/p and dt=hbar/E. Imagine that one has two particles with rest masses m1 and m2 sitting a distance of x1 apart at rest. Classically, these are essentially two independent systems, but exp(-imot) suggest that one may write: exp(-im1t)exp(-im2t) ((4)) ((4)) is the same for a single particle with mass m1+m2, but the two separated particles and m1+m2 are very different physically. In particular, a reaction involving a single m1+m2 interacts with the full mass, while an interaction with a separated m1 and m2, might only interact with one. As a result, we argue that one should not associate
hbar/ (m1+m2)cc ((5)) with both cases, but only with a single m1+m2, or with two particles m1 and m2 which are physically interacting at the same x point due to a force which links them through action-reaction. If one performs a Lorentz boost, then exp(ip1x) and exp(ip2x) only create a dx=hbar / (p1+p2) when they are interacting together at an x as in a 2-body scattering problem. If p1 and p2 are associated with classical trajectory x1=v1t1 and x2=v2t2 such that x1=x2, but times are different (t1 not= t2), then it makes no sense to use hbar/ (p1+p2) as a dx. This suggests that exp(ipx) is an interaction probability and taking a product (AND situation) should be done in context with an interaction. We extend this idea in the next section to additional probabilities which arise from exp(ipx). Two Features of exp(-iEt+ipx) As we have pointed out before, exp(-iEt+ipx) is linked with a classical trajectory x=vt. exp(-iEt+ipx), however, does not need to use x and t values which are on the trajectory. In fact, x,t values in exp(-iEt+ipx) describe fluctuations about particular trajectory points and that is how probability is introduced. We note that in many situations, one has only an x-type of interaction as in the case of 2-slit interference and 1-D reflection-refraction at an n1-n2 index of refraction junction. Thus, one only need use exp(ipx) in these cases, and even for V(x) bound problems, but V(x,t ) cases require both the notion of exp(-iEt) and exp(ipx)s. In other words, exp(ipx) is a probability which sits upon a deterministic free particle path and drives a second probability associated with some type of physical interaction scheme in x. We clarify this with two examples. Given a 2-slit apparatus, Newtonian mechanics suggests that a particle or photon (as a particle) should pass through one slit or the other. hbar/p=dx, linked with exp(ipx) suggests that if the slits are about hbar/p apart, then there is probability to interact with either, i.e. the probabilities within dx mean that one may have a particle pass through one slit or the other. Both must be considered as possible outcomes in a probabilistic calculation based on exp(ipx). This is the second probability in the problem and is linked with two classical trajectory lines which are drawn from the center of each slit to a point on a screen far away, i.e. r1 vector and r2 vector leading to: exp(ip dot r1) + exp(ip dot r2) ((4)) Each x point has its own exp(i p dot r), but there are two of these linked to the probability to pass through one slit or the other. The total set of outcomes is linked with the exp(ip dot r) probabilities and the two possibilities (one slit or the other) and this creates the sum in ((4)) which accounts for all possible outcomes. We argue that this total possible outcomes calculation really pertains to the area near the 2-slits, but then is mathematically extended throughout the x-y plane. A second example is 1-D reflection-refraction at an n1-n2 junction. One may consider the notion of a classical trajectory x=vt for an incident, reflected and refracted photon. These do not occur at the same time, but neither does a heads up and heads down state in a coin toss. One
deals with possible outcomes in a probabilistic calculation. The difference in the quantum mechanical case, is that there are two probabilities, the exp(ipx) fluctuation one and the type of outcome, reflected and refracted. We argued in the previous section that exp(ipx) is an interaction probability and should be considered at the interaction point, i.e. the x=0 n1-n2 junction. There, one has exp(ipx) as possible probabilities for each type, incident, reflected and refracted. This, however, is an OR situation, like the 2-slit interference case, and so one may use continuity arguments at the x=0 point, i.e. continuity of exp(ipx)s and their first derivative: Aexp(ipx) +B exp(-ipx) = Cexp(ip2x) at x=0 ((5a)) Apexp(ipx) -Bp exp(-ipx) = Cp2 exp(ip2x) at x=0 ((5b)) What differs from classical calculations is that one has two sets of probabilities, the exp(ipx) associated with a classical trajectory point and the identity of the photon (incident, reflected and refracted). Both sets are linked with outcomes and so the full set of outcomes must be considered, not simply the incident, reflected and refracted cases. Thus, ((5a)) and ((5b)) apply and it is exp(ipx) which allows one to have a dx region of uncertainty (like in the 2-slit case) for which all three photon states may probabilistically exist. Thus, we argue that ((5a))/((5b)) are very much usual probabilistic calculations with respect to possible outcomes. It is just that there are two sets of interrelated outcomes and classically one only thinks of one (i.e. the incident, reflected, refracted state). Conclusion In conclusion, classical probability is linked with possible outcomes. That is the nature of probability, we argue. We suggest that quantum free mechanics is associated with two sets of probabilities, one which creates the second, and so one must consider all possible outcomes together at the same level. The driver probability level is that of exp(-iEt+ipx). We argue that this applies to an interaction, i.e. one cannot take a p1 at a classical time t1 and p2 at a very different t2 time (with a separation much more than hbar/E) and write exp(ip1x)exp(ip2x). An AND multiply situation is used if the particles are interacting. In many cases, there is only an x interaction and one may drop exp(-iEt). We argue that exp(ipx) (which represents an x range of probabilities within hbar/p) also creates the possibility for other probabilistic results, as in the 2-slit interference case. Without dx=hbar/p, the particle would simply pass through one slit. It may now interact with both slits probabilistically and so this is a second level of probability. exp(ipx) sits on top of a classical trajectory and so one may write exp(ip dot r1) for a classical trajectory from slit 1 to a point on a screen and exp(i p dot r2) for r2 from the second slit to the same screen point. One then uses exp(ip dot r1) + exp(ip dot r2) which physically represents an interaction at the 2-slit apparatus, but may be mathematically extended throughout the x-y plane. Similarly, for 1-D reflection-refraction at an n1-n2 index of refraction junction, incident, reflected and refracted photons are linked with classical trajectory and each x trajectory point is associated with an exp(ipx) range of probabilities within hbar/p. We argued above that one only wants to use exp(ipx) at the interaction point and so within a dx there, one may have exp(ipx),
exp(-ipx) and exp(ip2x) (incident, reflected, refracted photons). This allows for two sets of probability outcomes which must be handled at the same level (together) through continuity of exp(ipx)s and their first derivatives as expressed by ((5a)) and ((5b)). Thus, we argue that free particle quantum mechanics with simple interactions is based on the probabilistic notion of possible outcomes which is central to classical probability.
