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 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.
Newtonian mechanics deals with smoothness in time and space through x(t) where the “details” occur within dx and dt which tend to zero. Thus there are no discontinuities in x and t. We have argued that the quantum free particle wavefunction pieces exp(-iEt) and exp(ipx) deal with discontinuities and that smoothness is due to a “modulus” of two dimensional probability e.g. exp(ipx)exp(-ipx) which yields 1 or determinism. Both exp(-iEt) and exp(ipx) contain finite measures in time and space, namely hbar/E and hbar/p and so there may be discontinuities. For example, exp(ipx) contains probability cos(px) and sin(px) within a wavelength so x=vt is only an average type of motion. Thus using exp(ipx) in a problem may result in time discontinuities because the particle is not moving as x=vt within a wavelength. In Part I we examined one-dimensional reflection/refraction of light from an interface of two media with n1=1 and n2>n1. We concluded that the quantum solution of this problem involves two equations, one linked to probabilities and the other to impulse. Multiplying the two equations together (with a special grouping, namely a spatial grouping) yields a single equation which may either be interpreted as a pressure balance equation. Thus one starts with two equations which are both grouped in time, but with the second demonstrating vector directions and obtain a single equation which is only grouped in space (which is linked with spatial grouping). By rearranging this equation, one may obtain a time grouping i.e. energy conservation equation, but the two groupings cannot exist together. This is the problem with the classical pressure picture linked to a steady state scenario and not a single photon process which displays a discontinuity in time i.e. a single photon may either reflect or refract. In this note we try to consider why classical energy conservation/ pressure balance only leads to one equation, while the quantum mechanical approach leads to the two required equations at the price of introducing a square root flux type of probability. We argue that there seem to be two factors. First there is the notion of scalars and vectors and a space versus time picture. The single conservation of energy equation AA/c1 = BB/c1 + CC/c2 where AA,BB,CCC represent the incident, reflected and refracted beams is expressed in terms of time and so vector information (i.e. -p for the reflected beam is not conveyed). If one rewrites this as AA/c1 -BB/c1 = CC/c2 then one obtains a pressure balance type of equation, but time is removed and one is grouping by space. The second and more important feature we argue is time discontinuity. If one considers a single photon, it may reflect or refract, but not both. Thus there is a time period in which to observe both reflection and refraction and more than one incident photon is needed. Newtonian mechanics in the form of steady state treats time as smooth and so everything must happen instantaneously. This is the idea behind pressure. One considers an impulse 2p multiplied by flux i.e. velocity. The flux is used to find the number of particles which strike a surface per second as if this were a continuous smooth process in time. For the case of reflection/refraction it is not. It is discontinuous because either a reflected ro a refracted photon is created each time. Thus the notion of pressure averages this over and loses some of the information in the problem. This, we argue, is why a pressure balance/ energy conservation equation is not sufficient to solve the reflection/refraction problem even though pressure is a physical concept and may be measured (on average) just as x=vt may be measured on average.
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.
A quantum particle (or photon) is described by the two dimensional probabilistic exp(-iEt+ipx), with t and x being treated as independent. exp(-iEt) suggests a stochasticity in time associated with energy E, but none in space. Similarly, there is a stochasticity in x associated with p but none in time. For -Et+px to be invariant as seen in different frames moving with constant average velocities, -EE+pp must also be an invariant i.e. equal momo (c=1). This leads to a contradiction. exp(ipx) suggests a stochasticity in x for a given p, but -EE+pp=momom converts a given p into an E, so there exists stochasticity in x for E as well (which contracts the notion that exp(-iEt) is only stochastic in time). The link between E and p, however, which breaks x-t independence is the notion of average velocity v=x/t, which is physical (as is exp(ipx)). How does one deal with this contradiction? We suggest that p and E are associated with different physical interaction processes. P is related to impulse hits, while E may be associated with loss of energy over a distance if a potential V(x) exists. Even though exp(ipx) suggests stochasticity in x, at the same time one considers fixed objects (targets, slits) and even potentials V(x) at well-defined deterministic x points. This seems to mean that if one wishes to make energy deterministic at each x, one must create an average of exp(ipx)s. We argue that there seem to be two important ideas associated with exp(ipx). First, it is a two-dimensional probability linked to a flux because it is due to p which is due to a v. Probability flux does not get created or destroyed (unless there are sinks or sources). This leads to issues when dealing with a bound state which should be localized. A flux probability is not localized, nor is an average in general (except in the case of an infinite potential). A bound state problem is really a flux problem consisting of a particle trapped in a potential together with it tunneling out or in i.e. one has a dynamic equilibrium picture somewhat like evaporation in equilibrium. (To see the clear difference between a state based probability and a flux-based one, consider the problem of a photon moving in the x direction and reflecting/refracting from an n1-n2 (index of refraction) y-plane at x=0. State probabilities (P(reflect), P(refract)) are not sufficient to solve the problem, but a flux-based one exp(ipx) which allows a single particle to appear as if in a steady stream scenario is.) Secondly, unlike evaporation, the guiding statistical condition for exp(ipx) is that its average makes E deterministic in x space. Thus E is like T (temperature) in that it is a statistical parameter, but it is not based on maximization of entropy (equilibration), but rather in creating E as an average which is then treated as deterministic (ignoring in a sense exp(-iEt) value characterizing an equilibrium system (not based on equilibration). This deterministic E system is linked with discrete values (due to the exp(ipx) equilibrium), but given that E is treated as deterministic, a stochastic process based on maximization of entropy is introduced to describe the absorption and emission of photons (which allow for jumps from one average-deterministic En to another). We argue that the flux-like probability exp(ipx) exists in order to deal with sharp interfaces (e.g. n1-n2 index of refraction) at a deterministic x=0 which probabilistically leads to different p values. It uses the reverse approach i.e. a deterministic p and a stochastic x. In such a way, one may use continuity (and first derivative continuity) at the sharp x with flux-probabilities containing the various allowed p values (due to the impulses). The continuity follows from no sinks or sources existing for the flux-like probability.
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.