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.
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.
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.
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.
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 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.
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), it is argued that the Schrodinger cat paradox arises due to a superposition state being considered “physical” instead of an “an expectation over possible future outcomes” . This is consistent with ideas linked to a quantum directional or interaction probability versus a state probability which we have discussed in various notes (2). In particular, classically there exist states, like heads up for a coin, a certain number of a die, an incident, reflected or refracted state for a photon, etc. We have suggested in (2) that exp(ipx) is a directional or interaction probability in that this probability is linked to the probability to have a p in a two-body scattering experiment and is involved in conservation of momentum. A state probability should not depend on x as the state is determined by p (and mo). For example, an incident, reflected or refracted photon probability does not depend on x, rather it is specifically given by AA/c, BB/c or CC/c2, where c is the speed of light in n1=1 (index of refraction), c2= c/n2 and AA,BB,CC are the fluxes of the incident, reflected and refracted photons. Thus, exp(ipx) is linked to the state through p, but through x it is linked to position. This means that for a given p there is uncertainty in x quantified by the wavelength hbar/p. This means there is position uncertainty with respect to an impulse hit making this an interaction probability and a directional one because p and -p have different effects for a conservation of momentum equation. As argued in (2), interactions are based on exp(ipx) because they involve impulse hits and so even though classical state probabilities exist, the physical problem is solved through considerations of exp(ipx)s. In particular, sums or OR situations of exp(ipx)s arise due to uncertainty in x. It is the directional/interaction probability exp(ipx) which predicts the x uncertainty and also appears in so-called superposition equations which represent different possible outcomes which may all exist in a tiny dx during the interaction. One should not think in terms of a state vector, but rather possible interaction probabilities exp(ipx). These are OR or sums of possible probabilities which interact in x and not a mixture of states. In other words, one does not have a dead cat mixed with a living one or an incident photon mixed with a reflected one. Rather, one adds directional/interaction probabilities for different states because one is uncertain which state exists in dx. This leads to math conditions on the exp(ipx)s as discussed in (2) which then ultimately govern the classical probabilities for one to have a particular state or another. The point we make is that exp(ipx) is not a state probability, but an interaction one which is consistent with the ideas of (1).