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Speculation on the Quantum exp(ip dot r) as a Sampling State Francesco R. Ruggeri Hanwell, N.B. Oct. 15, 2025 The notion of probability exists in the classical world. The classical world, however, has well defined states (be it the face of a coin or die) or the momentum or energy of a particle. p Probability arises in terms of weights of one deterministic state versus another. One might consider a two body elastic collision (relativistic or nonrelativistic) with initial e1,e2 and p1, p2 (vectors). If one does not know the outcome states, one might argue that any ei,ej and pi, pj (vectors) states have the same weight as long as they satisfy conservation of energy and momentum. This is classical probability and leads to exp(i E) and exp(i p). A complex number (unit modulus) is used because there is no real valued weight for a free particle. The problem is that exp(ip) and exp(-ip) should have the same value and we have argued in previous notes, that one may remedy this by introducing a second variable, e.g. x, which also changes sign when the x-axis is reversed. We also argued for Lorentz invariance, leading to exp(-iEt+ip dot r). Even though p is a constant vector in exp(i p dot r), r is not. We suggest, however, that this is not merely a classical probability with different values for different r points. We postulate that one has a sampling of r state within a characteristic length of hbar/|p|, i.e. the wavelength. One has a kind of physical state created by a probabilistic sampling nature, but one that is controlled by the form exp(i p dot r) and limited to a wavelength region. This result is consistent with exp(-iEt+i p dot r) which ensures conservation of energy, momentum and probability. We suggest that probability in the quantum realm is somewhat different from that in the classical. Product (AND) and sum (OR) scenarios still hold, but instead of probability being simply a weight for one deterministic state or another, we suggest that one has sampling or resonance states. exp(-iEt+i p dot r) is one example, but we argue that this line of reasoning should be extended to other physical situations, such as 2-slit interference. One cannot reject classical probability notions for a sampling state for exp(ip dot r) and then accept classical probability notions when describing exp(i p dot r1) + exp(i p dot r2). It is the sampling exp(i p dot r) incoming state which interacts with both slits. This means that one has the possibility of two momenta for a single particle, namely p along r1 and p along r2 where r1 and r2 are vectors from the centers of each slit to the same fixed x point on a screen far away. A particle cannot have two p vectors at the same time, but it can sample one and then other. This means that a kind of sampling resonance or entangled state is created. If one places a measuring device behind one slit, one may break this resonance and create a single state exp(i p dot r1) (for example), which is still a sampling state, but only in x and not in p. Alternatively, one may have a second particle collide with one that has passed through a 2-slit apparatus. Classical physics would suggest that exp(i ptest - r) interacting with the particle will reveal either the p along r1 or along r2. One might be tempted to use the overlap probability like a classical probability, but this then destroys the notion of interference. We thus suggest that the particle is in a sampling state which samples p1 along r1 and then p2 along r2 etc. In other words, a p1 linked with slit 1 does not mean the particle passed through slit 1, we suggest.
Free Particle Probability We have suggested before that free particle quantum probability has its roots in usual classical probability in Newtonian mechanics. If one has an elastic two body collision, then for an initial e1,e2 and p1, p2 vector, one might argue that any ei,ej and pi, pj (vectors) have the same product probability if they obey conservation of energy and momentum. This holds relativistically and nonrelativistically and leads to: exp(i E) and exp(i (px)) for a simple example of p vector = (px) (i) where i is a unit vector in the x direction ((1)) exp(i (px)) and exp(-i (px)), however, should have the same value and so we suggested in previous notes introducing x. If the x-axis switches direction, then (px)--> - (px) and x→ -x. In such a case, one may also argue for Lorentz invariance, leading to: exp(-iEt + i p dot r) ((2)) At first this appears as a classical probability. It is complex with a modulus of 1 because a free particle has no real value weight. Nevertheless, one would simply insert t and r points because p and E are fixed. We suggest, however, that based on experiment, something very different is occurring. We argue that one has a sampling probability in ((2)) and that the system is in a kind of resonance in which p is fixed, but different x points are sampled in a wavelength: Wavelength = hbar/ h|p| ((3)) region. This is not the notion of classical physics. Rather, one has a new kind of sampling state with a probability linked to conservation of energy, momentum and as we argue later, probability. In other words, the introduction of t and r lead to a new conservation idea, namely that of probability. We suggest that this resonance/sampling state has physical consequences. Two Slit Interference We argue that the existence of a sampling of r state represented by ((2)) is justified by the fact that 2-slit interference only occurs for slit separation of about a wavelength. A free particle is associated with a wavelength and this suggests that it is sampling this region. Otherwise, it seems that there is no reason to expect a certain length to be associated with a free particle. This sampling idea is non-classical (i.e. one does not simply give weight to different pointsthere is a resonance). If this is in fact the case, this type of thinking must be maintained when trying to analyze what happens as a particle reaches and passes through a 2-slit apparatus. For consistency, one cannot use the notion of resonance for ((2)) and then switch to classical probability when discussing the slits. We argue that the resonance particle interacts with both slits and creates a new resonance-sampling state, but this time in momentum as well as in x. A particle with momentum p received this from an interaction (impulse hit etc). This hit not only created p, but also a
sampling-resonance state in r. Now a new interaction based on 2-slit apparatus occurs and this should create a new resonance-sampling state, but one based on a p (momentum) linked to each slit. Thus, one does not simply use the classical OR math: exp(i p dot r1) + exp(i p dot r2) ((4)) where r1 and r2 are vectors from the centers of each slit to the same point on a screen far away. Rather, one has created a sampling-resonance state with two momenta p along r1 and p along r2. The particle should be sampling these and so at one time it may have p along r1 and at another, p along r2. If one strikes the particle with another one, one may obtain an overlap Born’s rule weight: { Integral dr exp(-i pfinal dot r) V(r) (exp(i p1 dot r) + exp(i p2 dot r) } power 2 ((5)) Here p1 is a vector along the original r1 and p2, along r2. This might suggest that there is no interference resonance. One might choose pfinal = -p1 and argue that the particle passed through slit 1, but we argue that this is misleading. The fact that an interference pattern, if one allows many particles to pass through the 2-slit apparatus, occurs suggests a physical resonance-sampling system. This is not a classical weight for each slit possibility. This suggests that the particle should be sampling the two different p values, just as it samples different r values in exp(i p dot r). This also explains why a measurement just behind one slit breaks the sampling resonance and destroys interference. It is the sample resonance with its two p values which allows one to have the interference pattern. By conservation of momentum, however, a particle striking the one which passed through the slits hits one p or the other as seen by the weights in ((5)). To see that the form ((4)) conserves probability, one may note that multiplying ((4)) by its complex conjugate and integrating yields 1 + 1, i.e. showing the classical choice of equal probabilities for each slit, The actual pattern on the screen, however, is one due to interference, i.e. a sampling of p resonance state, we argue. As a result, probability in quantum mechanics is really associated with sampling resonance states and should not be thought of as simply a weight for different deterministic states. Conclusion In conclusion, we argue that classical probability is simply a weight for different deterministic states. For example, .5 heads + .5 tails is not a resonance. The system is not sampling heads and tails in time and changing from heads to tails. Once flipped, the coin is either heads or tails. One does not know which because one has a lack of information, not a resonance-sampling situation. We argue that the 2-slit experiment indicates that exp(i p dot r), which represents a free particle, is not simply a weight for different r values for a fixed p. If this were the case, there would be no justification for interference. Rather, we argue for r point sampling within a wavelength region hbar/ |p|. The form exp(-iEt+i p dot r) is linked with conservation of energy , momentum and probability as well as being Lorentz invariant. The sampling of r is indicated by the notion of a wavelength. If one only had weights for different r points with no sampling, there
would be no physical notion of a wavelength. The fact that a wavelength is considered physical is that 2-slit interference only occurs if the slit separation is of the order of the wavelength. If exp(i p dot r) represents a sampling resonance in r, we argue that a particle which reaches the slits interacts with both and creates a sampling resonance with both p values, i.e. exp(i p dot r1) + exp(i p dot r2), where r1 and r2 are vectors from the centers of each slit to the same point on a screen. The actual form is required for probability conservation as discussed above and also momentum conservation. It seems that a particle in such a sampling resonance should be switching from p along r1 to p along r2 in time, just as it samples r points. Thus, a measurement showing that the particle has p along r1 far from the 2-slit system does not mean that the particle passed through slit 1 if there is a sampling-resonance in p occurring.
