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.
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.
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 (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 Newtonian mechanics a particle with constant momentum moves deterministically according to x=p/m t. Even a quantum object such as a photon or electron should satisfy this equation on average to give meaning to the notion of the speed of light in a vacuum, for example. Thus it seems surprising that a quantum free particle is also associated with exp(ipx) which only contains momentum and is periodic and complex. Furthermore the physical consequences of exp(ipx) are real as shown in a two-slit interference experiment etc. In this note, we attempt to discover why exp(ipx) exists in the first place by examining the example of the reflection/refraction of light which occurs in the macroscopic classical world. Basically our argument is based on analyzing two important pieces of information in the system, namely the scalar energy (or particle count) and the vector momentum. A main point we make is that energy or particle number considerations are linked to determinism i.e. a specific time of interaction. One may note that the interaction x is fixed by the two medium barrier. This means a single incident photon exists in one time range and the reflected OR refracted photon in another. It is possible to compare the incident photon to either the reflected or refracted or to a sum of probabilities. In particular, a single photon has an energy E and a count of 1. If it reflects, it has an energy E and count of 1 so there is a balance, likewise for refraction. Thinking in terms of probabilities “a” and 1-a to reflect/refract: 1= a + 1-a. Can one perform such a matching for momentum, which is a physical piece of information? We argue one cannot. For example, a single incident photon has a momentum p which is linked to the impulse it may deliver. If it refracts, “all of a sudden” it is linked with momentum of p2 = n2 p where n2 is the index of refraction in the second medium and n1=1. This seems problematic because a physical impulse of 2p is all of a sudden an impulse of 2p2. Thus there exists a big problem which seems to require the notion of probability and all three momentum values p, -p and p2 considered together, unlike the situation of energy/particle number for which a comparison of the incident photon and one possible result yields a balance. Combining all three momenta, however, is also problematic in a deterministic picture because the incident photon exists in one time range and the reflected or refracted in another. How can one possibly compare them together? The solution seems to be to argue that the incident photon is only deterministic on average. In particular, if it has an x probability distribution over a certain length (called a wavelength), one does not know the precise time at which it hits “x” the medium interface. In such a situation, there is time uncertainty which allows all three momenta to be compared at the same x. This, however, leads to a major consequence, namely that so-called deterministic motion associated with constant p is deterministic on average, but contains repeated pockets of x probability distributions such that time is not deterministic within. We try to examine these ideas in this note.
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.
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 Parts 1 and 2, we suggested that various physical scenarios such as -EE + p dot p cc = -momocccc and one dimensional reflection refraction probability conservation 1= P(reflect)+P(refract) are not involved with the direction of motion of a particle. If probability is involved (as in a wavefunction), these equations (even though they act on probability) do not show the direction of motion. In any physical test run, however, there is a specific direction of motion. As a result, we argued that there must exist equations linear in probability (exp(ipx) for a free particle) and id/dt partial, -id/dx partial (energy, momentum) which give rise to the physical non-directional cases. Here we retain the general idea of an equilibrium with no direction present, but argue that matters may be more complicated when different time events appear in space in a time-independent approach using exp(ipx)s. In particular, we analyze scattering and place 1-D reflection-refraction in the framework of scattering. We suggest that to create an equilibrium in both directions, one must consider both W=Aexp(ipx) + Bexp(-ipx) (for an incident p and reflected -p) and its flow -i W* d/dx W, together with the opposite direction scenario W*= A*exp(-ipx) +B* exp(ipx) and i(d/dx W*) W. Averaging yields p ( A*A - B*B), i.e. interference terms disappear in this scheme which directly considers motion in both directions i.e. A is linked with motion in both directions which does not match a physical run, but does match classical equilibrium (pressure). This result is not evident if one uses 1-D reflection-refraction at x=0 because one may then use a nd exp(ipx) and d/dx exp(ipx) set of equations to obtain directly 1=P(reflect) +P(refract), but interference terms creep in if one changes the junction to x1 as seen in Part 1. The averaging of both directions removes this issue. We see that exp(ipx) and exp(-ipx) represent equivalent kinetic energies in the time-independent Schrodinger equation. A math solution thus involves any identical energy directional derivative, i.e. exp(ipx) and exp(-ipx). There is no sense of time, so writing Aexp(ipx) + Bexp(-ipx) with A>B introduces a direction which is not in the Schrodinger equation, but is in a physical trial run. Given that the time-independent Schrodinger equation cannot discern time and probabilities add, a full solution should include various same-energy solutions which represent the particle at different times. There should then be a probability conservation linking these different time states, such as 1= P(reflect) + P(refract), but due to interference of W* d/dx W one has interference terms in x, but these may vanish if one considers motion in the opposite direction, i.e. Aexp(-ipx)+ Bexp(ipx), and classically averages expectations of -id/dx with each as one would do classically.