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Quantum Wavepackets, Bohemian Potential and Interaction Probability Part 2 Francesco R. Ruggeri Hanwell, N.B. Dec. 22, 2025 In Part 1, we argued that the Heisenberg uncertainty principle (standard deviation p) * (standard deviation x) >= hbar/2 only applies when a measurement (i.e. interaction) has occurred. Otherwise, the notion of standard deviation does not exist. Thus, if one has a Gaussian wavepacket, it is the measuring device which has created it and constrained is standard deviations of momentum and position. A free particle, which has not been measured, is described by exp(-iEt+ipx), as argued in Part 1, and not by a wavepacket. Here we ask: How does one interpret a wavepacket calculation? In (1), we argued that exp(-iEt) is a free particle energy conserving probability and exp(ipx), a momentum conserving one. Then, a sum of exp(ipx - iEt) represents an OR situation in probability theory. There is superposition and so one obtains a weighting of possible outcomes which is the probabilistic interpretation of the scenario. If one considers two slit interference, the viewpoint is the same. exp(ipx) is a probability which allows for momentum hits of p in a range of dx=hbar/p outside the center-of-mass, which follows x=vt. This built-in stochasticity of impulse delivery leads to the notion that probabilistically, a particle may interact with both slits if their separation is about hbar/p. One writes an OR probability statement: W(x)= exp(i p dot r1) + exp(i p dot r2), where r1 is the position vector from the center of slit 1 to a point far away on a screen, and r2, a vector from the center of slit 2 to the same point. The result, W*(x)W(x) is a probabilistic expression which describes the possible outcomes probabilistically. (1). Seen in this way, the Gaussian wavepacket should be interpreted in a similar manner and not as a physical description of the particle which moves freely. A single bound state in a potential V(x) is localized by a potential and one may consider only OR cases of exp(ipx), i.e W(x) = Sum over p a(p)exp(ipx). In the measurement case, there is no quasi-permanent localization and one uses a linear superposition of exp(-iEt+ipx)s. Nevertheless for a Gaussian wavepacket, two probability distributions describing possible outcomes are created. One has W*(x)W(x)= exp(- (x-u)(x-u) / (2 sigma(t) sigma(t))) where sigma is the x standard deviation. In addition, W(x) = Sum over p a(p) exp(ipx), leading to a momentum outcome distribution of a*(p)a(p). As written, sigma(t) for x which means that the Gaussian is expanding. We called this unphysical if one considers the wavefunction as representing a description of a single physical entity (2) at times beyond t=0. Like in the 2-slit case, we argue that it represents an outcome distribution. Thus, one would not consider any more time values than t=0, at which time the measurement is completed. An exp(-iEt+ipx), describing a free particle, is measured by a specific device which creates the Gaussian form and gives it a specific standard deviation in space and momentum. The measurement is then done and W*(x,t=0)W(x, t=0) represents the possible outcome positions of the particle at t=0 (i.e. probabilities for x portions of a phase shift). The momentum distribution yields possible p outcome values. As soon as the measurement is finished, the particle moves as exp(-iEt+ipx), with p being probabilistically linked to a*(p)a(p) (a Gaussian as well), where W(x,t) = Sum over p a(p)exp(-iEt+ipx).If one has a second measuring device which creates a tiny momentum standard deviation, one would be able to carry out experiments which would determine roughly the standard deviation of p created by the first measuring device. Furthermore, one would be able to measure phase shifts as well by comparing a free particle
measured by a device with a tiny p standard deviation and a large x one with free particles which were not measured. Probabilistic View of Free Particle Quantum Mechanics Our arguments above are based on a probabilistic interpretation of exp(-iEt) and exp(ipx). These are strict free particle probabilities, not a dynamical description of the particle in space and time. The dynamics are associated only with interaction, i.e. p may strike in an interval of hbar/p about the center-of-mass which follows x=vt. E (nonrelativistic kinetic energy) is also stochastic in time with respect to an interaction. The free particle probability has a unit real modulus of 1, because every free particle has the same real weight. There is no distribution of energy as in a Maxwell-Boltzmann gas. The complex valued part describes momentum conserving outcomes of Newtonian 2-body elastic scattering. For any given e1,e2 (energies) and p1,p2 (momentum vectors), one might argue that there is equal probability for any (ei,ej) and pi,pj outcomes as long as energy and momentum are conserved. Given independence, one expects energy and momentum probabilities of: exp(iC1 E) and exp(iC2 p) (p along the x-axis) with C1,C2 being constants for unit purposes ((1)) ((1)), however, only holds in one frame because p=m1v1=m2v2 have the same momentum in a given frame, but not in a Lorentz boosted one. Thus, we suggest a probability of: exp(-iEt+ipx) ((2)) The point seems to be that if ((2)) is officially a probability, then the rules of probability apply to it. For example, it may be added in OR situations and multiplied in AND ones. A superposition, therefore, is an OR probability scenario of possible outcomes (1). We suggest that this is the description of a wavepacket calculation. Gaussian Wavepacket A wavepacket is created by an OR situation of exp(-iEt+ipx)s: W(x,t)*W*(x,t) Gaussian = C exp(- (x-u)(x-u)/ (2 sigma(t)sigma(t)) ((3)) A free particle is represented by exp(-iEt+ipx), we argue, and so to obtain ((3)) physically, we suggest that a special measurement must be made on the free particle. In other words, there is an interaction and the particle is localized for a short time and ((3)) is created with the Gaussian form and the specific values of the momentum and spatial standard deviations given by the
measurement. The Heisenberg uncertainty principle applies to ((3)) and not to exp(-iEt+ipx) as argued in Part 1. Given that a measurement may localize a particle for a short time, this may induce a phase shift in space when one interferes with an exp(-iEt+ipx) which was not measured. The x standard deviation yields a probability for a particle spatial x part of the phase shift. The point we make is that unlike a single particle bound state in V(x) which is always localized even though there is tunnelling, as this is a time-independent situation, the measurement is short lived. The Gaussian wavepacket describes possible outcomes at t=0 when the measurement ceases. After this, only a certain exp(-iEt+ipx) outcome is chosen out of the possible results. If one writes W(x,t) = Sum over p a(p) exp(-iEt+ipx) ((4)) Then a(p)*a(p) is the probability to have a certain p outcome. We note that due to the measurement, one may have a spatial phase shift which shows that an interaction occurred. Thus, exp(-iEt+ipx+phase) may describe the free particle after the interaction, i..t for t>0. We argue that one should not use the wavepacket as a description of the particle in time. It is a probabilistic outcome set which occurs at the measurement time and allows one to see weights of possible results. If this is the case, one should be able to perform a second measurement which creates a very small standard deviation of p and one would then be able to sample many particles, measured by the first device and approximately determine the standard deviation of p of the first measuring device. Furthermore, one could have free particles, measured by the second measurement which would have a large standard deviation of x interfere with a particle which were not measured, in order to see if a spatial phase shift is present. Conclusion In conclusion, in Part 1 we argued that a free particle should move as exp(-iEt+ipx) and that the spreading of the x standard deviation in time of a Gaussian packet is unphysical. Here, we reiterate that exp(-iEt+ipx) is a probability and so a sum of exp(-iEt+ipx) with a(p) weights represents an OR situation. In other words, it represents possible outcomes and not a dynamical description of a single particle as implied in (2). We suggest that it is a measurement (i.e. specific interactions) which create the Gaussian wavepacket and its x and p standard deviations. The wavepacket only holds at time t=0 when the measurement is finished. The particle then moves as an exp(-iEt+ipx + phase shift). The Gaussian wavepacket calculation (only evaluated at t=0), then gives the probability for p, i.e. a*(p)a(p), where W(x,t) = Sum over p a(p)exp(-iEt+ipx) and the Gaussian= W*(x,t)W(x,t). We suggest that the Gaussian spread in x at t=0 may indicate a spatial phase shift p delta x, with delta x governed by the probability of the Gaussian and the p value, by the probability a*(p)a(p). We suggest that one may test these ideas experimentally. Given the first measurement and its x and p standard deviations, one may perform a second measurement with a device which creates a very small p standard deviation and a large x one. Then one may approximate the standard deviation of p of the first device by finding the value of p. Furthermore, allowing a particle from the second device to interfere with free particles which were never measured, one
may see a phase shift due to both measurements. To have a phase shift due to one measurement, one may use a measuring device with a tiny p standard deviation, i.e. large x one. One may then allow free particles measured by this device to interfere with free ones which were not measured. References 1.Ruggeri, Francesco R. Quantum Free Particle Probability and the Notion of Outcomes (preprint, zenodo, 2025) 2.Grande, P.L. and Fadanelli, R.C. and Vos, M. A Valsov-Bohm Approach to Quantum mechanics for statistical systems (2025) https://www.semanticscholar.org/reader/0fba7926dc5f1cfc9eedb995eea3a45da1a4b38e
