We try to argue that the idea of probability associated with motion may have classical beginnings in that the center-of-mass (cm) of a piece of mass moving with constant velocity v is present within a dx at some time, but moves out of dx. Flux is constant, but at one time the particle cm leaves dx and arrives at the adjacent dx. Constant flux suggests one does not need to consider time. In such a case, one might associate a periodic function showing probability being removed from a region dx (at x) (i.e. a negative distribution) followed by it being added (a positive distribution) to a region x+dx. This suggests a negative-positive cycle, but this labels certain x regions as negative and others as positive which is inconsistent with all x regions having the same weight based on a constant v. Given that time is removed, it seems this cycling may have nothing to do with velocity v or perhaps anything else for that matter. As a result, one may postulate an Aexp(ibx), where b is a constant, for motion. We will argue for a phase linear in x. AAexp(-ibx)exp(ipx) yields a constant AA, which is flux, and treats all x points the same. At this point, one cannot go farther unless one considers other issues.We consider two such issues. First we ask what happens if the particle is viewed from a frame moving at constant v? The velocity of the particle will not be the same, but we have argued that exp(ibx’) may be the form, yielding no information about b when (x,t) changes to (x’,t’). If on the other hand, the phase cannot change if the particle is viewed from the moving frame, one requires a Lorentz invariant. This suggests that the phase may actually be -Et+px. (We also show that it is not necessary to use special relativity to obtain the form exp(ipx). Rather one may consider ideas of pressure balance associated with continuity of exp(ibx).) As a second consideration, we examine a one dimensional reflection-refraction problem at a point xo for which a probability equation 1= Probability(reflect)+Probability(refract) exists. If exp(ipx) for constant E describes motion, this motion changes with changing p, yet one wishes probability to be a continuous function at xo. This suggests different weight coefficients for different exp(ipx) which seem to be associated with P(reflect), and P(refract). If x=0, one has an equation in the weights, with A+B (x<xo) matching C (x>xo) at xo. If one stops at this point, exp(ipx) does not seem to have any impact. One may note, however, that in general xo is not zero and that one wishes at the very least to have the derivatives match. This immediately links the A,B,C values to the p (momentum) discontinuity. Thus using special relativity to create an exp(ipx) has practical implications because it allows one to deal with p discontinuities. We also show though that special relativity, i.e the use of -Et+px, may be bypassed and the form Aexp(ipx) obtained as a result of its first derivative yielding p which is needed in a pressure balance equation. This then suggests A is the square root of flux.
We try to argue that the idea of probability associated with motion may have classical beginnings in that the center-of-mass (cm) of a piece of mass moving with constant velocity v is present within a dx at some time, but moves out of dx. Flux is constant, but at one time the particle cm leaves dx and arrives at the adjacent dx. Constant flux suggests one does not need to consider time. In such a case, one might associate a periodic function showing probability being removed from a region dx (at x) (i.e. a negative distribution) followed by it being added (a positive distribution) to a region x+dx. This suggests a negative-positive cycle, but this labels certain x regions as negative and others as positive which is inconsistent with all x regions having the same weight based on a constant v. Given that time is removed, it seems this cycling may have nothing to do with velocity v or perhaps anything else for that matter. As a result, one may postulate an Aexp(ibx), where b is a constant, for motion. We will argue for a phase linear in x. AAexp(-ibx)exp(ipx) yields a constant AA, which is flux, and treats all x points the same. At this point, one cannot go farther unless one considers other issues.We consider two such issues. First we ask what happens if the particle is viewed from a frame moving at constant v? The velocity of the particle will not be the same, but we have argued that exp(ibx’) may be the form, yielding no information about b when (x,t) changes to (x’,t’). If on the other hand, the phase cannot change if the particle is viewed from the moving frame, one requires a Lorentz invariant. This suggests that the phase may actually be -Et+px. (We also show that it is not necessary to use special relativity to obtain the form exp(ipx). Rather one may consider ideas of pressure balance associated with continuity of exp(ibx).) As a second consideration, we examine a one dimensional reflection-refraction problem at a point xo for which a probability equation 1= Probability(reflect)+Probability(refract) exists. If exp(ipx) for constant E describes motion, this motion changes with changing p, yet one wishes probability to be a continuous function at xo. This suggests different weight coefficients for different exp(ipx) which seem to be associated with P(reflect), and P(refract). If x=0, one has an equation in the weights, with A+B (x<xo) matching C (x>xo) at xo. If one stops at this point, exp(ipx) does not seem to have any impact. One may note, however, that in general xo is not zero and that one wishes at the very least to have the derivatives match. This immediately links the A,B,C values to the p (momentum) discontinuity. Thus using special relativity to create an exp(ipx) has practical implications because it allows one to deal with p discontinuities. We also show though that special relativity, i.e the use of -Et+px, may be bypassed and the form Aexp(ipx) obtained as a result of its first derivative yielding p which is needed in a pressure balance equation. This then suggests A is the square root of flux.
We suggest that quantum mechanics is concerned with preserving two conservations, one related to a kind of probability and the other to a mechanical balance.These two conservations exist at the level of a single particle and are interlinked. Specifically, the derivative d/dx, which measures the effect of a change in a kind of probability, also measures the associated resistance or impulse for a free particle. For a free particle, the “kind of probability” is exp(ipx) and d/dx exp(ipx) = ip exp(ipx) showing that there is an impulse related to p linked with the probability. In other words, a free particle moves through space in a probabilistic manner related to the resistance associated with the impulse p. These two conservations, a kind of probability and a kind of resistance, lead to two independent equations when one considers continuity at points which change momentum. We consider again the problem of a photon moving in the x direction which reflects or refracts at an interface along the y axis between n1-n2 (indices of refraction). Classically, probability is conserved i.e. 1= P(reflect) + P(refract) ((1)). The steady state problem, however, has the appearance of an equilibrium and should have a second conservation, namely one in space, i.e. a balance of pressure. This balance, linear in momentum, however, is the identical equation as ((1)). Thus, there are still two unknowns P(reflect), P(refract) and one equation. In the quantum case, double conservation may classically be manifested by the same equation. This seems to suggest, we argue, that ((1)) is a consequence of an equation based on p (which is linked to momentum). We try to find this association in two ways. First, we argue that the interface may exist at xo and that there are various momenta in the problem, We argue these represent pertinent information which does not appear in ((1)). We suggest that ((1)) seems to be a kind of magnitude equation with p and xo information being part of the direction of vectors i.e. p and xo may exist in a phase exp(i f(p,x)). Together with the conservation of pressure condition, we argue that exp(i f(p,x)) = exp(ipx). This argument, however, implicity involves consider the phase function exp(if(p,x)) as a kind of probability. Alternatively, we consider finding this probability follows from maximizing Shannon’s entropy subject to the periodic constraint i p x P(x), but we investigate why Shannon’s entropy form should be used, i.e. what is its physical significance?
In Part 3, we started with the notion of a free particle probability linked to energy and momentum conservation in two body Newtonian elastic scattering. In particular, we argued that one needed a probability which shows an equal weight for any outcome (ei,ej) and (pi,pj) (momentum vector) set which conserves energy and momentum. This led to a first guess of exp(iC1 E) and exp(iC2 p), but in Part 3, we argued the momentum and energy are defined through a Lorentz transformation of mo at rest and such a transformation must also affect x,t → x’,t’. We concluded one cannot create any probability based on p and E without also considering x and t which transform according to the same transformation. This then allows one to create a Lorentz invariant probability. Here we take an opposite approach, i.e. one which does not involve p and E a priori. It is well known from classical physics that P(x)dx=dx/L, where L is an arbitrary length, describes a uniform distribution for particles at rest. One may, however, view these particles from two frames, one moving with constant v and the other with constant -v. This would seemingly introduce probabilities P(x1’,t1’,v) and P(x2’,t2’,-v) with no notion of energy or momentum present. It would seem that the product of the two for an x1’=x2’ and t1’=t2’ should equal the rest frame result. This suggests an exp(f(x,t,v)) solution, but there is no reason to have an ever increasing or decreasing exponential and so exp(i f(x,t,v)) seems like a solution. There is, however, a problem, because given P(x,t,v=0), and such a solution is then not P(x)= 1/L unless P(x,t,v)=1, which we reject as it assumes a moving particle is identical to one at rest in terms of probability.. In other words, there are two probabilities for the rest frame, P(x)=1/L and P(x,t,v=0) = exp(i f(x,t,v=0). Given that exp(i f(x,t,v) is based on Lorentz frames, one would also expect probability to be conserved and not be created or destroyed simply because a particle with rest mass is viewed from a rest frame or one moving at constant speed. This seems to force one to use relativistic variables linked to a Lorentz invariant, namely E,p as well as x,t. As a result, in Part 3 we argued that one cannot consider a probability of p,E without considering x,t as well and here we suggest that one cannot even consider a probability of x which is conserved in different constant moving frames without introducing t,p and E. Thus, P(x,t,v=0) which seemed like a “P(x)” type of probability in the rest frame is really a different probability, namely one in time exp(i f(x, v=0, t)), i.e. . for this to be Lorentz invariant, one has exp(-iEt+ipx) = exp(-iEt) for any x. In a sense, this is like P(x)=1/L in that each x carries the same weight, but there is still the time variable and the probability actually changes in time unlike P(x)=1/L. This seems to lead to a paradox because the given exp(-imocct) one may formally argue that exp(-px)=1 for p=0 which is the x part of the probability, yet above we argued that in order to obtain this value one should use P(x1’,t1’v)P(x2’,t2’,-v) at x1’=x2’ and t1’=t2’. The difference between P(x)=1/L is that the time measurement has infinite resolution, while exp(-imocc t) implies that it does not because otherwise information about x is the same. Thus, special relativistic considerations lead to the notion of uncertainty in time (hbar/E) and space hbar/p, but in the Newtonian scenario, these do not exist. We argue that the question then becomes: Why would one wish to change from a special relativistically related probability exp(-iEt+ipx) which shows physical uncertainty units of time and space, to one which shows infinite resolution for x and t, through exp(-iEt+ipx)exp(iEt+ipx)? We suggest that exp(-iEt+ipx) or exp(ipx) in time-independent problems is a dynamic probability needed to describe interactions. One may see that cos(px) and sin(px) may have positive and negative values, implying the removal of probability from one place and its addition somewhere else. One the interaction result has been obtained, one wishes to measure spatial results, i.e.count particles in bins in space. There is no need for the interaction information and it may be formally removed using exp(iEt-ipx) exp(-iEt+ipx).
One may consider constant fluxes which exist for all x points in space. These are associated with a constant velocity and a momentum. Consider next a discontinuity which does not change energy. For example, one may have energy E, momentum p and a constant potential V1 which suddenly changes to V2 at x=xo with E remaining constant. This leads to a momentum discontinuity as well as a flux discontinuity, but no work is done. As a result, there exists a pressure balance (related to an incident flux, a reflected one and the new p2 flux for x>xo), i.e. there is no discontinuity in pressure. A single constant flux also has no discontinuity in pressure either. We suggest there should exist a continuous function of momentum and x which allows for/ensures this pressure continuity. The momentum discontinuity may be at any xo, but the pressure balance is the same.As a result, one would expect this function and its first derivative to be continuous at xo, but a combination of the two should eliminate the function (of p and xo) leaving only a pressure balance equation. In other words, two equations need to combine to create a single classical pressure balance equation, suggesting that there may exist a square root flux type probability function. We suggest that pressure balance at momentum discontinuity points (with E remaining constant) may be the only consideration necessary to ultimately obtain the quantum free particle exp(ipx). An advantage of this pressure-only viewpoint is that it accounts for weighted averages of exp(ipx) having the same form, namely exp(ipave(rms) x) and exp(-ipave(rms) x) in a potential with infinite walls. Pressure considerations may be applied to a single particle exp(ipx) or an ensemble, yielding the same form if the same conditions apply. We argue that both a square well potential and infinite well potential are examples of momentum discontinuities, except in this case one has discontinuities in ensemble values (i.e. average values).
It is known that a single photon moving in the x direction and encountering an n1-n2 index of refraction discontinuity at xo will either reflect or refract, with the probabilities for each summing to 1. This problem, however, is solved using a balance of Aexp(ipx)+Bexp(-ipx) = Cexp(ip2 x) at xo and the derivative d/dx, of this equation. The complex conjugate of one equation multiplied by the second yields: AA/c = BB/c + CC/c2 where E=pc and E is constant for the incident, reflected and refracted photon. The two continuity equations apply even in the case of a single photon, but in such a case one has either an incident photon changing to a reflected one or an incident to a refracted one. One never has all three together as in thte two continuity equations. We ask: How is it possible to include both a refracted and reflected photon together with the incident in the same equation? We suggest here that the form exp(ipx) applies to momentum, whether it is part of a particle or part of a potential field. exp(ipx) may be obtained from maximization of entropy -P(x)ln(P(x)) subject to the periodic constraint ipxP(x). We suggest that in a single photon interaction, one has momentum in media n1 and n2 and these allow for the continuity equation with the system interactions taking the place of a photon to create the balance. There is stochasticity in that a single photon either becomes a reflected or refracted one, but exp(ipx) does not simply apply to the photon. it may be applied to the n1-n2 interface momentum which is created during the interaction. This interaction is consistent with the a steady stream picture only involving photons . Thus for a single photon, it appears as if one has an entangled probability exp(ipx)’s for the incident, reflected and refracted photons, but actually the physical photon only plays a role as first an incident one and then either a reflected or refracted one with the n1-n2 interface momentum creating the missing exp(ipx) piece needed for the continuity balance. Ultimately, A,B and C become probabilities AA/c,= 1, BB/c = P(reflected) and CC/c2= P(refracted), but these are not obtained by maximizing entropy subject to a constraint.as one does not have independence of probabilities.We note that the two continuity equations combine to create a pressure balance which one wishes to have physically in the problem. The photon reflection-refraction problem may be reformulated for a particle with rest mass using a particle with E, p in a constant potential V1 reaching a V1-V2 (both constant) interface at xo with E remaining constant.
Classically, it is known that the probability density P(x)=1/L for a static particle confined randomly in a length L. If one has a moving particle, one sometimes sees that argument that the particle spends delta t =constant in each constant dx for a constant speed and so P(x)=1/L as well. In such a picture, there is no quantum W(x)= exp(ipx)/sqrt(L) or W(x)= exp(-ipx)/sqrt(L), accounting for motion in both directions. On the other hand, traditional quantum mechanics uses P(x) = W*(x)W(x) so one would have 1/sqrt(L) in this case for both p and -p. The interesting point is that exp(ipx) and exp(-ipx) are both dynamic (differ for p and -p), while W*(x)W(x) does not. We first argue that by considering dt in dx, one considers an event rather than describing a particle at an instant. dt implies there is an t(initial) and t(final) and one subtracts the first from the second. This is a description of an event and not a description of the moving particle we argue. Even in classical physics, motion is state with the particle being at x at t with v(x,t). Motion to the left or right represent different physical states (described by p and -p) and there is no reason that an instantaneous probability capturing the state of motion should not depend on this direction of motion as well. Furthermore, it should depend on p as this is a variable used in special relativity. The function W*(x)W(x) = 1/L is an AND statement of probability and removes the state of motion from the probability by combining p and -p (as an AND condition). This leaves a stationary state which obeys classical probability, i.e. the probability of a static object with a uniform probability to be anywhere within L. Thus, one has the classical equation: 1/L = F(-v) * F(v), with two solutions: F(v)=constant and F(v) = exp(-iEt+ipx). We describe in detail how we obtain the second solution using invariance of probability under a Lorentz transformation. One might argue that F(p,x)=constant means that the dynamic probability (the probability of the moving particle) is the same for p and -p when in fact the particles are in different states. In such a case, one has F((p,x) = 1/sqrt(L) and all dynamic features are lost. One would expect that such a probability would be useless in trying to describe a dynamic problem. (We discuss this in more detail). The alternative is exp(-iEt+ipx) which suggests that there is periodicity in x and t based on E and p. At first, this is equally strange from a classical point of view, one might argue. We suggest, however, that for x+dx and x-dx, exp(ipx) clearly defines the direction of motion which is essential in a dynamic problem. In other words, we argue that Newtonian mechanics uses dynamic variables x(t), v(x,t) etc to solve a problem. We suggest introducing a dynamic probability which considers instantaneous probaiblies (just as Newton considers instantaneous variables). The big question, from a classical point of view, is then: Why would one use probabilities at all when Newtonian mechanics describes deterministic physics? The answer, we argue, is that P(x)=1/L is an acceptable classical probability and one is allowed products in probability theory. We suggest that there is no reason why one cannot consider a dynamic probability F(p,x) such that F(p,x)*F(-p,x) = 1/L. Unless one can refute the existence of F(p,x) in classical theory, then mathematiaclly one has F(p,x) = exp(ipx) (because F(p,x)=constant makes no sense). The question then becomes whether such an F(p,x) may be used in solving any physical problem with the answer being yes. This formalism solves the 1-dimensional reflection-refraction problem of photons, assuming the photon to be a kind of particle moving at speed c. Newtonian mechanics does not solve this problem, it seems.
In a previous note (1) we argued that quantum square root probabilities follow from the consideration of light reflecting and refracting at a surface along the y direction at x=0. No knowledge of Maxwell’s periodic photon electric field is needed. The photon number and energy are conserved at a single photon level, but flux/intensity may be included through (2) AA/c1 = BB/c1 +CC/c2 where AA is the intensity of the incident beam (AA>0), BB the reflected and CC the refracted. There are two unknowns B,C if A is known and one equation which may be broken into two A+B=C and A/c1 -B/c1 = C/c2. For n1<n2 (n=index of refraction B<0). Thus we argue that there is an underlying square root probability. In this note we first heuristically suggest the form for the square root probability. A key point is that “particle” features such as E and p (for a photon) govern the dynamical wavelike probability. This also applies to a particle with rest mass. We also ask why this square root probability has not appeared in dynamical theories and argue that it appears in special relativity (i.e. a Lorentz boost) for a particle with rest mass mo at x=0 at t=to. As seen from a frame moving at constant speed -v, this particle moves with speed v. Thus x’=vt’, yet a 2x2 Lorentz transformation is applied to (x=0,t=to). Only one relation is needed because x’=vt’ and p’=E’/cc v suffice to yield the variables x’ and p’. Thus we argue that a Lorentz transformation describes more than is necessary for a simple particle. The 2x2 nature of the problem we suggest describes more than a simple particle. It leads to the Lorentz invariant -Et+px ,but x=vt so why should x and t appear as if independent? We argue that -Et+px=A is compatible with dA/dx |constant t = p and dA/dt |constant x = -E. P and E are “particle/photon” properties, but exp(ipx) or exp(-iEt) represent eigensolutions of -id/dx and id/dt. Thus if x and t are independent exp(ipx) may be interpreted as existing at all x points (as there is no time). This is the picture of an unusual probability which is two dimensional, with components which may be positive or negative as in the reflection/refraction problem. Thus we argue that the notion of a wave probability or square root probability not only exists in the reflection/refraction problem, but in the case of special relativity with what looks like an unnecessary 2x2 matrix from the point of view a particle (mo, x=0, t=to) describing an associated square root wave like probability completely dependent on the particle properties E and p.