Speculation on the Quantum exp(ip dot r) as a Sampling State Part 2

Speculation on the Quantum exp(ip dot r) as a Sampling State Part 2 Francesco R. Ruggeri Hanwell, N.B. Oct. 15, 2025 In Part 1, we tried to argue that a quantum mechanical wavefunction describes a sampling-resonance type of state. Here we wish to clarify that this is an ensemble state, but that it still governs a single particle, as several examples will show. We argue that exp(-iEt+ i p dot r) is a mathematical function linked to conservation of momentum, energy and probability, as stated in Part 1. We show from a scattering experiment how important interference is to conserving probability. The main purpose of this note, however, is to question why quantum properties, e.g. interference etc are attributed to a particle when they are only manifest in an interaction. In Part 1, we argued that one may consider a Newtonian problem with e1,e2 and p1,p2 (vectors) and then argue there is equal probability for any ei,ej and pi,pj (vectors) outcome which conserves momentum and energy. This leads to a probability trial of exp(i E) and exp(i (px)) (one dimensional momentum). A complex number is used because there is no real value weight as in an ideal gas. This probability is changed to exp(-iEt+i p dot r) to make it Lorentz invariant and to allow (px)--> - (px) and x→ -x to have the same probability. We argue that exp(-iEt+i p dot r) is a math function linked to probability and conservation of energy, momentum and probability, but that it largely arises due to the nature of the forces in the two body scattering problem. This allows one to postulate equal probabilities for any ei,ej such that ei+ej = e1+e2 and pi,pj such that pi+pj = p1+p2. We suggest that if a wavefunction is a math object (probability), then one may perform basic probability math calculations without necessarily attributing them to physical explanations linked to the particle. The math may be consistent with results of interaction and conservation of energy, momentum and probability. Thus, one may not need to justify interference on physical grounds in terms of the particle somehow having probability attributes which add and cancel. We also discuss the idea that exp(-iEt+i p dot r) describes an ensemble result, but that a single particle result must be consistent with it. The Origin of exp(-iEt+ i p dot r) In Part 1, we argued that exp(-iEt+ i p dot r) arises from elastic two body Newtonian scattering. Two particles with energies e1,e2 and momenta p1, p2 (vectors) collide and one assumes equal weight for any ei,ej, pi, pj outcomes if: E1+e2 = ei+ej and p1+p2 = pi+pj (vectors) ((1)) ((1)) depends on the behaviour of the force in two body elastic scattering. We then argued for a probability of the form: exp(i E) and exp(i (px)) for one dimensional p ((2)) A complex number with unit modulus is used because one does not have real value weights as in an ideal gas. We stress that ((2)) is simply a math function which yields energy and momentum conservation results. In Part 1, we noted that exp(-ip) and exp(ip) should have the same value and don’t and so modified ((2)) to: exp(-iEt + i p dot r) ((3)) ((3)) is Lorentz invariant and also under (px) → - (px) and x→-x. We still stress that ((3)) is a math function which now represents conservation of energy, momentum and probability. To see the conservation of probability in a dramatic sense, we consider the following example. Conservation of Probability We initially argued for conservation of energy and momentum in Newtonian 2-body elastic scattering and introduced a complex function with unit modulus. Next, we multiplied E by t and wrote p dot r in a Lorentz invariant manner. The question now becomes: How does one interpret the presence of x and t? Momentum and energy conservation must still remain. We suggest that there are two effects of x and t, namely: (A) Conservation of probability (B) exp(-iEt+i p dot r) represents a steady state or ensemble result Here we consider a dramatic example of conservation of probability and argue that exp(-iEt+i p dot r) is a math function which satisfies various conservation properties. It does not need to be tied directly to physical behaviour of the particle. In other words, interference of exp(-iEt+i p dot r) with a more general W(x,t) form does not need to be given a physical explanation. One only needs to note that this interference removes probability (steady stream particles) from one region and places them in another so there is particle conservation. A clear example of both (A) and (B) is the case of a plane wave exp(i (px) x) which scatters from a potential V(x) in a time-independent manner. Mathematically, one writes: exp(i (px) x) + f(theta) exp(i kr) / r ((4)) The second term represents the probability of scattering and the first, the incident probability. Now this may be considered as a steady stream or ensemble problem and so exp(i (px) x) actually represents an incoming beam and the second term, all possible scattered particles. The probability exp(i p dot r) is not restricted to a single particle, although steady state-ensemble results affect the single particle case as will be seen in 2-slit interference. The point we wish to make, which is well-known in the literature, is that there is math interference between exp(i (px) x) and the second term which accounts for particles (probability) being removed from the incident beam and placed in the scattered one. This interference occurs for back scattering values of the second term. The usual math of probabilities (OR situation) introduced above takes care of conservation of energy, momentum and probability conservation and one has no need of explaining, in terms of the particle, why interference occurs. The point is that the final result must conserve probability (particle number) and so some particles must be removed from the initial beam and appear as scattered results. The periodic nature of exp(i p dot r) allows for this to occur. For example, if one considers: exp(i p1 x) + exp(i p2 x) ((5)) and multiplies it by its complex conjugate, one obtains: 2+ .5 cos( (p1-p2)x ((6)) This leads to removal of particles in one place and an identical addition in another so that the integral of ((6)) yields 2, i.e the two initial particles. This demonstrates that probability is conserved through the use of cos(px) and sin(px). More Examples Regarding the Ensemble Nature of exp(i p dot r) Originally, we considered a single two body elastic scattering example. This does not seem to be related at all to an ensemble or steady stream of particles. This leads to exp(i E) and exp(i (px)) which were modified to exp(-iEt+i p dot r) to make the expression Lorentz invariant. This still does not directly imply a steady stream or ensemble. We suggest, however, that in a time-independent problem with exp(i p dot r), one may use different r values. This may be interpreted to a steady stream of particles which occupy these different r positions. The periodicity of the probability shows how such particles may interact with a potential subject to conservation of energy, momentum and probability. A clear example of a steady stream-ensemble situation is the reflection-refraction of a photon which reaches an n1-n2 index of refraction junction. If one only considers a single photon, then it is not clear how one could write a time-independent equation with an incident, reflected and refracted photon all in the same equation, i.e. A exp(i p x) + B exp(-i px) = C exp(i p2 x) at x=0 where n1=1 and p=n2p ((7a)) and Ap exp(ipx) -B exp(- p x) = C p2 exp(i p2 x) at x=0 ((7b)) A third example is the two slit example of Part 1. The form: exp(i p dot r1) + exp(i p dot r2) ((8)) with r1 and r2 being vectors from the centres of each slit to the same point on a screen far away, describes a full interference pattern, i.e. one created by an ensemble or steady stream of particles. The point is that a single particle passing through the slits must be consistent with the result of ((8)). We argue that one does not have to account for why interference occurs physically if exp(i p dot r) is a math probability function. The point then seems to be that the interaction at the two slits creates a scenario consistent with ((8)) and so the physics in the interaction with the slits. As in Part 1, this implies that one cannot think of this problem in terms of the usual classical probability scenario of weights for non-inteferring probabilities, we argue. There is, however, a physical feature associated with the particle, namely that it does not interact only at its center-of-mass point as in classical physics. Rather, it may interact with both slits if they are about a wavelength apart. This leads one to think of properties of a particle, but it seems that it is really interaction considerations which are important and that exp(-iEt+i p dot r) provides the mathematics for describing these. Conclusion In conclusion, we suggest that the free particle probability exp(-iEt+ i p dot r), discussed in Part 1, is a math function which allows for the conservation of energy, momentum and probability. This occurs through interference, i.e. the addition and subtraction of sin(px) and cos(px) values and we suggest that this is a math process. We argue that one does not necessarily need to account for it physically in terms of properties of the particle, although exp(i p x) suggest that a particle does not interact at its center-of-mass, but over a range of a wavelength hbar/|p|, allowing it to interact with both slits of 2-slit apparatus if they are about a wavelength apart. The final result of the interaction is one based on exp(ip dot r) math which is consistent with the kind of interaction which occurs which is very different from the classical one. exp(-iEt+ i p dot r) seems to be a book-keeping function which accounts for the various conservation laws which must be upheld and shows that one does not have interaction at the center-of-mass point as in classical physics. We suggest that the wavefunction represents a steady-state or ensemble result, but that a single particle must behave in a consistent manner with such an ensemble. For example, if the 2-slit interaction prevents particles from reaching certain x points on a screen (corresponding to an angle of theta with the y axis with the slits along the x), and the probability is a math calculation using exp(i p dot r)’s representing an ensemble, then this result must still describe the result for a single particle passing through the slits.