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.
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*= Aexp(-ipx) +B exp(ipx) and i(d/dx W*) W. Averaging yields p ( AA - BB), 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.
In previous notes, we argued that one may introduce a probability into Newtonian elastic scattering if one assumes that for an incident (e1,e2) energies and (p1,p2) momenta vectors, any (ei,ej), (pi,pj) which satisfy conservation of momentum and energy have equal probability. We then found that for a single particle, the relevant probability is exp(-iEt+ip dot r). This is a unit modulus complex number which is Lorentz invariant, and exp(i p dot r) is time reversal invariant, i.e. p→-p and r→-r yield the same value. The question then becomes: How does one use exp(-iEt+ipx) or for time-independent problems, exp(ipx)? This is not a usual probability linked to quantity, e.g. N P(reflect) = the number of particles which reflect. exp(ipx) is a probability to have a certain momentum, but there is no reason it cannot carry a weight, i.e. a math factor. The question is: What does such a factor mean physically? Secondly, exp(ipx) depends on x (space), but exp(ipx) is not the probability to be at x, but it is somewhat related in that for any x and a given p, the modulus has the same value, suggesting equal probability to be at any x. This mapping, however, really only holds for a single exp(ipx). It seems that one can say two things about exp(ipx). First, if it is a probability for a momentum p, then one should be able to write p exp(ipx) or function(p)exp(ipx). Secondly, one should be able to write AND (multiply) and OR (add) statements. The idea of an OR or addition of exp(ipx)s is important, because one may be able to write an equation for constant kinetic energy: -1/2m grad dot grad W(x) = numberW. The general solution should be a W(x) which contains a sum of all exp(ipx) type solutions which have this kinetic energy. (In the relativistic case, one may consider d/dt d/dt W - cc grad dot grad W = -momocccc W.) The question we consider in this note is: Given that one may write W and grad W, what can only really do with these? They do not directly represent physical observables. Furthermore, if one wishes to impose continuity on W and gradW, which is reasonable mathematically, one should have a physical argument to justify it, and we suggest that this is a little complicated. Ultimately, we try to argue that one may consider continuity of W as well grad W in certain cases based on the association with a classical quantity probability equation. We also suggest one may relate W* grad W which may be generalized to W* grad W + (grad W*) W, as discussed in Part 3 and provides a link to classical probabilities.
In a previous note, we asked the question: How does a single photon approaching an n1-n2 index of refraction junction in a one-dimensional problem know what its probability to reflect or refract is? We argued that this probability does not follow from some stochastic feature of the junction, but rather from a directional probability exp(ipx). We argued that exp(ipx) is a probability linked with conservation of momentum in Newtonian scattering and implies that there is a spatial uncertainty region of hbar/p. We argued that within a delta x region of the n1-n2 junction, the photon is able to establish a “balance” or equilibrium of the incident, reflected and refracted photon, because one does not know which it is during the time spent in dx. We argued that one must describe this by a balance of Aexp(ipx)+Bexp(-ipx) = Cexp(ip2x) at x=0 and a similar equation obtained taking d/dx. It is this balance or equilibrium which determines the weights of B and C relative to A, i.e. essentially solves the problem. If this is in fact the process which occurs, then we argue here that it should also apply to the scattering of a particle from a spherical infinite well potential at ro, for example. In other words, we consider all of the possible directional probability solutions outside of ro together as these participate in the balance or equilibrium at ro, the interaction region. In such a case, one has a general solution of the Schrodinger equation of: Al jl(pr) + Bl nl(pr) ((1)), where jl is a Bessel function, and nl, a Neumann. In order to have equilibrium between all possible exp(ipx) type directional probabilities, one must establish a constraint condition just as in the n1-n2 problem. The constraint condition for the infinite spherical potential at ro is that there is no probability for ro inside. Thus, we argue that the relevant weights in the problem (i.e. the solution of the problem) are established by the constraint condition at the interaction surface. This leads to 0 = Al jl(pro) + Bl nl(pro) ((2)). This yields a relative weight between Al and Bl. The question is then: How does this relate to anything physical? The fact is that even though ((1)) is a general math solution, with Bl now given in terms of Al through ((2)), one may now rewrite ((1)) in terms of a math sum of physical directional probabilities, namely exp(ipz) (incident particle) and f(theta) exp(ipr)/r (scattered particle). Thus, it is the use of directional probabilities (linked to momentum conservation) and constraint conditions which govern the weights of the allowable directional probabilities at an interaction point/region, i.e. ro in the spherical infinite potential case and x=0 in the n1-n2 junction reflection-refraction case. In particular, if one has an incident particle linked with exp(ipz) one wishes to know the weight f(theta) of exp(ipr)/r in order to find out what per cent of particles scatter into angle theta. This means finding a relative weight between the two directional probabilities and this occurs by a probabilistic sampling of all solutions at the interaction point (here ro). Then, only one choice of outcome occurs, but in keeping with the probabilistic balance, just as in the n1-n2 index of refraction junction reflection-refraction situation. We then try to generalize to the one-dimensional quantum bound state situation.
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?
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 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 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.