In (1), we argued that a uniform distribution associated with the placement of static particles in a one-dimensional length L is P(x)=1/L. We then suggested that because a static particle has speed=0, one cannot rule out the existence of a probability F(p,E,x,t) which is Lorentz invariant. For every p, there is a -p and so F(p,E,x,t) complex conjugate * F(p,E,x,t) = 1/L yielding F(p,E,x,t) = exp(-iEt+ipx)/sqrt(L) (see (1) for details). F(p,E,x,t) is a dynamic probability like x(t) or v(x,t) are dynamic variables in Newtonian deterministic theory. We then argued that the F(p,E,x,t) is not simply a math result, but actually solves the 1-D reflection-refraction problem, while Newtonian mechanics cannot. Here we ask the question: Why can’t Newtonian mechanics solve the 1-D reflection-refraction problem? Certainly there is probability in the problem, given by 1= probability(reflect) + probability(refract). It seems that Newtonian mechanics requires that one always know x(t), v(x,t) etc. Imagine that there is a specific deterministic mechanism for reflection and a different deterministic one for refraction and that these turn on with probability(reflect) and probability(refract). One may then describe reflection and refraction separately by deterministic Newtonian mechanics, but one cannot calculate probability(reflect) and probability(refract) because these are built into the physical interfaction interface n1-n2 (where ni is the index of refraction). Experimentally, the probabilities to reflect and refract relative to the incident beams are functions of only n1 and n2. N1 and n2 are known, but that still does not provide one with a method of solving for p(reflection) and p(refraction).Newtonian mechanics is not of use. The question then becomes: What if one introduces a dynamic probability exp(ipx), dropping exp(-iEt) because E is the same for the incident, reflected and refracted beam? How could this possibly help matters? The probability to reflect and refract is based on n1 and n2 alone. We consider the notion that exp(ipx) is a dynamic probability and it seems that it may also be thought of in terms of a steady stream. A steady stream of particles means that one does not know the exact x(t) of any particle, one only has an average sense of the probability flow. This suggests that one can use an extra conservation law that is not available in Newtonian mechanics, namely conservation of dynamic instantaneous probability exp(ipx). We stress that 1-D reflection is not a physical fluid like steady stream problem. We use the notion of a steady stream as an analogy to describe the conservation of exp(ipx), the dynamic probability. Given that exp(ipx) is an instantaneous probability, it must be accounted for at every x point and the viewpoint of a steady stream allows one to bypass single reflection or refraction events because one is viewing the process on average, but at the level of the instantaneous probability which must be defined at each x. This is almost like having probability represent fluid flow, but it is the dynamic instantaneous probability which behaves like the fluid, not a classical flux. Such a new conservation law (and conservation of the dynamical average of momentum) yields two equations which allow one to solve the problem. In a nutshell, Newtonian mechanics does not allow one to combine events that occur at different times in the same equation as if they occurred at the same time. The OR (addition) probability condition allows for this, but 1= probability(reflect) + probability(refract) does not contain the dynamics of the event. One requires a probability which acts like a steady stream probability and we argue that this is the dynamic probability exp(ipx). This forces one to deal with the probability being conserved in x, but also momentum being conserved under this scheme (i.e. using this probability) as exp(ipx) is the instantaneous probability describing a particle with momentum p.
In (1), we argued that a uniform distribution associated with the placement of static particles in a one-dimensional length L is P(x)=1/L. We then suggested that because a static particle has speed=0, one cannot rule out the existence of a probability F(p,E,x,t) which is Lorentz invariant. For every p, there is a -p and so F(p,E,x,t) complex conjugate * F(p,E,x,t) = 1/L yielding F(p,E,x,t) = exp(-iEt+ipx)/sqrt(L) (see (1) for details). F(p,E,x,t) is a dynamic probability like x(t) or v(x,t) are dynamic variables in Newtonian deterministic theory. We then argued that the F(p,E,x,t) is not simply a math result, but actually solves the 1-D reflection-refraction problem, while Newtonian mechanics cannot. Here we ask the question: Why can’t Newtonian mechanics solve the 1-D reflection-refraction problem? Certainly there is probability in the problem, given by 1= probability(reflect) + probability(refract). It seems that Newtonian mechanics requires that one always know x(t), v(x,t) etc. Imagine that there is a specific deterministic mechanism for reflection and a different deterministic one for refraction and that these turn on with probability(reflect) and probability(refract). One may then describe reflection and refraction separately by deterministic Newtonian mechanics, but one cannot calculate probability(reflect) and probability(refract) because these are built into the physical interfaction interface n1-n2 (where ni is the index of refraction). Experimentally, the probabilities to reflect and refract relative to the incident beams are functions of only n1 and n2. N1 and n2 are known, but that still does not provide one with a method of solving for p(reflection) and p(refraction).Newtonian mechanics is not of use. The question then becomes: What if one introduces a dynamic probability exp(ipx), dropping exp(-iEt) because E is the same for the incident, reflected and refracted beam? How could this possibly help matters? The probability to reflect and refract is based on n1 and n2 alone. We consider the notion that exp(ipx) is a dynamic probability and it seems that it may also be thought of in terms of a steady stream. A steady stream of particles means that one does not know the exact x(t) of any particle, one only has an average sense of the probability flow. This suggests that one can use an extra conservation law that is not available in Newtonian mechanics, namely conservation of dynamic instantaneous probability exp(ipx). We stress that 1-D reflection is not a physical fluid like steady stream problem. We use the notion of a steady stream as an analogy to describe the conservation of exp(ipx), the dynamic probability. Given that exp(ipx) is an instantaneous probability, it must be accounted for at every x point and the viewpoint of a steady stream allows one to bypass single reflection or refraction events because one is viewing the process on average, but at the level of the instantaneous probability which must be defined at each x. This is almost like having probability represent fluid flow, but it is the dynamic instantaneous probability which behaves like the fluid, not a classical flux. Such a new conservation law (and conservation of the dynamical average of momentum) yields two equations which allow one to solve the problem. In a nutshell, Newtonian mechanics does not allow one to combine events that occur at different times in the same equation as if they occurred at the same time. The OR (addition) probability condition allows for this, but 1= probability(reflect) + probability(refract) does not contain the dynamics of the event. One requires a probability which acts like a steady stream probability and we argue that this is the dynamic probability exp(ipx). This forces one to deal with the probability being conserved in x, but also momentum being conserved under this scheme (i.e. using this probability) as exp(ipx) is the instantaneous probability describing a particle with momentum p.
In Part 1, we argued that 1-D reflection-refraction cannot be solved using Newtonian mechanics, but can be solved by introducing the notion of dynamic or instantaneous probabilities exp(ipx). We suggested that the problem is discrete in time and space (i.e. reflection occurs separately from refraction) and that to link the two one requires probabilities exp(ipx)s which allow for OR (addition) equations regardless of discontinuities in space and time. In this note, we suggest that one may actually bypass the dynamic probabilities exp(ipx)s, but that this would require extra information. In other words, the exp(ipx)s (i.e. quantum mechanics) seems to be justified on the basis of nature using minimal information as well as on this minimal information being tied to x. In particular, an n1-n2 index of refraction junction at x=0 means that there is discontinuity in space at x=0 and that p2=n2*p while p1=n1*p. For simplicity, consider n1=1. This means that the information present at the junction is n1 and n2. This information, however, is tied to x because n1 holds for x= -infinite to 0 and n2 for x=0 to infinite. N1 and n2 are directly tied to space x and this we argue is important. One could impose the notion of a black box at the junction which assigns fixed probabilities for reflection and refraction independent of n2 for n1=1. In such a case, these independent probabilities could take on any values and so one could not possibly use exp(ipx)s and conservation to calculate reflection and refraction probabilities. There would be no free particle quantum mechanics. It is because one insists that n2 is the only information in the problem that one cannot then associate extra information with the reflection and refraction probabilities. In such a case, one only has n2 to work with and so one has discontinuous values of momentum at different x points. In other words, there is only n1 and n2 as information and these are necessarily tied to position x. A Newtonian approach cannot solve such a case, but a dynamic probability scheme can as shown in Part 1. This scheme, however, is only justified if nature uses a least information scheme with n2 and n1 (the indices of refraction) describing all of the necessary information and this information is tied to x. If it were not tied to x, then one could not use a dynamic probability and the usual probability 1=probability(reflect)+probability(refract) is not sufficient to solve the problem. Given that n2 and n1 are associated with dynamics of the problem, i.e. p1=n1*p , the reflected photon has -p1 and p2=n2*p and that there is a discontinuity in p’s as well as in x and time and the only way to obtain addition equations involving all three disconnected pieces is through a dynamic probability which allows one to combine events that occur at different times and in different x regions, but at the same time allows one to include x as well into the problem as it is linked with the dynamic probability. Actually, the dynamic probability contains x and t, exp(-iEt+ipx), but there is an uncertainty in both, i.e. delta t = hbar/E and delta x = hbar/p. This is how this mechanism bypasses the determinism of Newtonian mechanics and discontinuitites and allows one to have a dynamic probability conservation law Aexp(ipx) + Bexp(-ipx) = Cexp(ip2x) where p=p1 for n1=1. We note that the justification for the existence of a dynamic probability (at least mathematically is given in (1).
In Parts 1 and 2 (and references therein), we tried to make an argument for the existence of a dynamic instantaneous probability which depends on x. This probability exp(ipx) is essentially the quantum free particle picture. The full probability is exp(-iEt+ipx), but in the 1-D reflection-refraction problem considered, E is the same for the reflected, refracted and incident photons. This means that if one establishes OR (addition) equations for probability, exp(-iEt) factors out, leaving only exp(ipx) pieces. In this note, we reiterate that probability already exists in the 1-D reflection-refraction problem as does a conservation of probability equation, i.e. 1= probability(reflect) + probability(refract) ((1)). Why should there be any other probability and other OR equations? We argue that the above equation ((1)) is a time equation which does not focus on space. One does not need to know that the incident and refracted photon move to the right, while the reflected one moves to the left. One does not need to know that the incident and reflected photons are restricted to x<0 for the n1-n2 index of refraction junction at x=0 and the refracted photon to x>0. In other words, one does not need to know anything about space. There are simply two events which are measured in time called reflection and refraction. If ((1)) is a probability conservation equation in time, and spacetime places x and t on equal footing, one may ask: Where is the spatial probability equation? We suggest that one must be careful. Time is positive, but x may be positive or negative. This, we argue, suggests that there should not only be a spatial probability dependent on x, but also two equations which demonstrate the two one-dimensional directions of space. These two directions are associated with motion or momentum and so one could have a spatial probability and its conservation equation, but also a conservation of momentum equation linked with this spatial motion. It is the spatial picture which shows the two directions of p, not ((1)). As a result, we suggest that one should expect the existence of a spatial probability and that it should be associated with two conservation equations, unlike the time probability with its single equation ((1)). This spatial probability viewpoint leads to free particle quantum mechanics, but these quantum equations which solve the 1-D reflection-refraction problem, also yield ((1)) when one combines the two spatial conservation equations, i.e. removes the spatial direction feature in the second equation and focuses only on the two events, reflection and refraction.
We have suggested in previous notes that there is a tendency to consider quantum mechanics as a kind of hybrid probabilistic-deterministic theory because it is supposed to ultimately “merge’ with classical mechanics for high energy levels (tiny wavelengths). Newtonian mechanics is deterministic and so somehow it seems quantum mechanics should also be seen as deterministic, hence the sense of surprise at tunneling when KEave(x)=0, even though vrms(x)=0 does not mean that actual values creating the probability are 0. Here, we wish to argue that exp(-iEt+ipx) (or the exp(ipx) part for spatial interactions) is really a momentum probability scheme which may possibly bypass many aspects of the actual physics to describe a problem in a simpler form in terms of a momentum conserving probability. A key idea is that exp(ipx) is actually the form to use when one has randomness as we will argue for reflection-refraction from an n1-n2 index of refraction junction. We argue that this idea is already known for a particle in an infinite well for which one has sin(pave x), i.e. 1/2i (exp(i pave x) - exp(- i pave x)). In other words, one has the form exp(i pave x), even though it is linked with an average p. Given that one is not considering deterministic physics, there is nothing wrong with exp(i pave x) being composed of Sum over p a(p) exp(ipx), where exp(ipx) holds in all space, not just within the infinite well potential. exp(ipx) is a momentum conserving probability, but on average momentum must also be conserved by a free statistical entity within the well. Similarly, W(x) for a single particle bound state consists of many exp(ipx)s, but represents a fixed E ave and on average this Eave must be conserved in time, hence the time dependent exp(-i Eave t). As a result, one may see that the quantum mechanical exp(-iEt+ipx) is an attempt to describe a system in a simple manner in terms of momentum conserving probability. To see this in more detail, we consider (1) which describes reflection and refraction of a photon at an n1-n2 junction in depth. In a previous note, we have already pointed out that in n2, c/n2 simply means that the photon is interacting on average with atoms/molecules in n2, but has c in between such interactions. It is only on average that one has c/n2 and even E and p2 = p n2. Nevertheless, one may use exp(ip2 x) as a probability which describes average properties of the system, just as it does in the examples in the above paragraph. The question then becomes: Do the microscopic details of the photon physics (the actual interactions) dictate the probabilities to reflect/refract or does the simplified exp(ipx) set linked to momentum conservation do so? Based on (1), it seems that reflection is completely independent physically from “refraction”, the latter requiring re-radiation and affecting all glass molecules. As a result, it appears that it is difficult to suggest that it is the microscopic interactions which dictate probability because reflection and refraction are microscopically so different. On the other hand, reflection-refraction is not a .5-.5 probability random process as experimental results show. The question then becomes: If some kind of systematic probability driver is not supplied by the microscopic interaction, what provides it? We argued in previous notes that one may introduce the probability exp(ipx) into Newtonian scattering to account for an initial p1,p2 (momentum vector set) having equal probability to produce any pi, pj (outcome set) as long as momentum is conserved. We suggest here that it is exp(ipx) that one must use for random interaction if one insists on conserving momentum. Thus, exp(ipx) bypasses microscopic details of interaction and provides an “ any outcome pair of momenta” is equally likely if momentum is conserved. This is a major simplification of the problem, but the question is: What constraints does it impose on physical results? For a free particle, it suggests that it may interact probabilistically with both slits of 2-slit apparatus if the slits are about hbar/p apart. For randomness, one would expect equal probability for each slit and this is the case as long as one also considers exp(ipx) for each slit to a the same point on a screen. In the case of reflection-refraction at an n1-n2 index of refraction junction, we try to see how the simplification exp(ipx) can be consistent with the microscopic details of the interactions and at the same time dictate the probabilities to reflect and refract. We try to consider why an exp(ipx) scheme predicts how nature behaves while a random one with no momentum conservation considerations does not. In particular, we argue that exp(ipx) represent randomness in a two body scattering problem in which momentum is conserved and so this same exp(ipx) must be carried over into problems which involve interactions instead of simply suggesting that one has .5 probability to reflect and .5 to refract because the two processes are independent and hence random. They are random, but with conserved momentum and hence an exp(ipx) scheme is required, we argue.
In a number of previous notes, we argued that free particle quantum mechanics, i.e. exp(-iEt+ipx), follows from the Lorentz invariant A = -Et+px. For t=0,x=0, A=0, but we argued that there is another solution, dx = hbar/p and dt=hbar/E (we use intervals because x=0, t=0 is a trajectory point). We then noted that this solution is linked with the probability exp(-iEt+ipx) which is Lorentz invariant and creates the above intervals in t and x. Finally, we remarked on the fact that exp(-iEt) conserves energy and exp(ipx), momentum, when one uses these for 2-body collisions, i.e. AND situations. Here we point out that there are numerous dt= f(p,E) and dx=g(p,E) (f, g are arbitrary functions) which yield A=0. For example, dt=p and dx=E is an example. Only dx=hbar/p and dt =hbar/E, however, lead to a probability form which conserves momentum and energy and this is something which we did not stress clearly in previous notes. Thus, there seem to be a number of issues involved in obtaining a free particle probability exp(-iEt+ipx) and these are independent requirements, which then lead to a particular solution. We argue that these requirements are: ((1a)) One wishes to preserve Lorentz invariance ((1b)) One desires discrete dx, dt intervals instead of Newton’s intervals which tend to 0. ((1c)) These intervals need to be linked with a particle related probability which defines them. This probability is critical physically because if the theory is to define photons and particles equally well ( as special relativity ((1a)) applies to both), one must account for the probabilistic effects of 2-slit interference and 1-D reflection-refraction at an n1-n2 index of refraction junction. ((1d)) One must have conservation of momentum and energy. We argue that it is the combination of these requirements that ultimately leads to the free particle probability exp(-iEt+ipx).
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).