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.
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.
Note: Dec. 28, 2022 The form exp(-iEt)exp(ipx) for both the photon and a particle with rest mass may be obtained from the Lorentz in variant -Et+px=A by noting that -id/dx A = p and that exp(ipx) is an eigenfunction of exp(ipx). In a series of notes we have considered the problem of light moving in the positive x direction hitting a medium along the y axis at x=0 and reflecting and refracting. We assumed that light is of the form of a photon which has an energy E which is the same in the incident, reflected and refracted beams. We also considered the notion of a steady state stream. Given a single photon in the incident beam, it has a probability a to reflect and 1-a to refract. As discussed in previous notes, one may consider an intensity AA, BB, CC for the incident, reflected and refracted rays (AA>0 etc) and write energy conservation as: AA/c1 = BB/c1 + CC/c2 where c1 is the speed of light in medium 1 and c2 in medium 2. We noted that the energy conservation equation has two unknowns B,C if A is known, but may be broken into two equations if one considers time removed from the problem (i.e steady state) and one groups by space i.e. (AA/c1 -BB/c1) = CC/c2 i.e. A+B=C and A/c1 -B/c1=C/c2. One may note that one does not have a conservation of flux/intensity equation, but that flux/intenisty which physically exists is directly linked to energy conservation. One may also note that one does not have an equation of energy flux conservation. In this note, we consider the dispersion relationship of light E=pc and the notion of a continuity equation which applies to the “square root” probabilities. We consider exp(-iEt+ipx) as the simplest solution, other than a constant and speculate whether exp(ipx) for a steady state picture represents a kind of square root flux. The idea of a square root flux suggests two sets of equations with square root fluxes which may be multiplied to create a flux equation. exp(ipx) has a kind of wavelike structure with two dimensions and may interfere. It also maps to a constant flux through exp(-ipx)exp(ipx). We consider these ideas and conclude that the wavefunction is like a square root flux which helps to associate a picture with the idea of it being a dynamical probability. This is helpful when considering a W(r vector) not equal to W(r) i.e. not spherically symmetric. This suggests unbalanced flux and rotation. In quantum mechanics, one does not have a container and so the fluxes create a static-appearing density in the bound case.
A one dimensional reflection/refraction problem is by nature probabilistic because an incident photon either reflects or refracts with certain unknown probabilities. The problem is traditionally solved by matching the electric field and its derivative with respect to x at x=0, t=0 where the electric field for a photon follows from Maxwell’s electromagnetic equations and is given by Eo cos(-wt+kx) where p=hbar k. As such, this problem is treated “classically” We argue, however, that this is a quantum mechanical problem (the photon being a quantum particle). Furthermore, no information is needed about the form of the electric field. Reflection/refraction conserves energy. It also occurs at some x point which may be varied i.e. there is translational invariance. A conservation of energy equation may be immediately written. The first medium has an index of refraction n1=1 and the second n2<n1. The conservation of energy equation is: AA/c1 = BB/c1 + CC/c2 where AA, BB and CC are the incident, reflected and refracted intensities = energy per photon * speed * number of photons. AA etc. is written to show that the intensity is explicitly positive. If one considers A known, then B and C are two unknowns, but there is only one equation. Thus energy conservation is not the full reality or information of the problem. This problem may be mathematically solved by using a difference of squares which leads to two equations: (A+B) = C and p1 (A-B) = p2 C. These may be solved for B,C in terms of A. This, however, is not the most general solution in that it does not show invariance of x i.e. the point at which the index of refraction changes. In this note we investigate the consequences of including x in the solution and argue this leads to a kind of square-root probability function which together with its first derivative should be continuous at any x chosen to be the interface of the media. This leads to a solution of Aexp(ipx), instead of A (etc), which is the free particle wavefunction. Thus this wavefunction (or unusual probability) is a direct consequence of the spatial invariance in the problem, but ensures that no x dependence appears if one combines the two square-root probability equations to create an energy conservation equation which has nothing to do with x.
The problem of an incident beam of light traveling to the right in the x direction and reflecting/ refracting at an interface at x=0 may be solved using Maxwell’s electromagnetic equation for the electric field of a photon. In particular, mathematical continuity of the electric field and its first derivative, where the field is of the form Real( exp(ipx - iEt) ) with E energy being constant, but p allowed to change, may be used to establish relationships between weights of the incident, reflected and refracted electric fields. The modulus of each of these waves is then linked to intensity. The above continuity equation, however, is somewhat unusual if one thinks of a single quantum traveling in time. For instance, before it strikes it is an incident photon, but after (in time) it is either a reflected or refracted one. Thus probability naturally appears in the problem and if one performs the usual grouping by time, then the incident photon is one side and the reflected or refracted on the other. This is the opposite grouping used in the continuity approach so we ask: How may this be understood? On average a photon moves with a certain speed, but it also has a wavelength. We argued in (2) that for a particle with rest mass, there is no time within the wavelength region, but rather a probability to be at an x value is given by exp(ipx). In fact, the two dimensional nature of exp(ipx) accounts for the removal of time as one needs two “variables” or dimensions to describe the direction of motion. On average, however, the particle still moves as x=velocity time. If there is no time in the probabilistic approach within a wavelength, then even treating the particle as a point means that there is no clear separation in time of the incident, reflected and refracted rays. If one uses space as a marker (as in the time-independent Schrodinger approach to a bound quantum particle) then this spatial approach within a wavelength governs the solution to the problem. In such a case, one may write Aexp(ipx) + B exp(-ipx) = Cexp(i p2 x) and a similar conservation of momentum equation A p exp(ipx) -p B exp(-ipx) = Cp2 exp(ip2 x) rather than argue that the electric field and its derivative should be continuous at x=0. We use addition because there exists an OR situation, but we do not employ the usual grouping by time, but rather in space with this probability.On average, however, this approach must be consistent with AA/c1 = BB/c1 + CC/c2 where A*A etc are intensity values much like W*(x)W(x)=Probability(x) for a bound particle where W(x) is the wavefunction. Mathematically the above approach is the same as the continuity of the electric field and its first derivative, but we suggest that this view considers the role of time in the problem. The continuity approach allows for the incident, reflected and refracted photon to exist at the same time which only makes sense if time does not really exist within a wavelength, even though there is a mapping to x=vt. Thus both the loss of time and the idea of probability are needed and they complement each other.
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 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 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.