If one considers the reflection/refraction of a steady stream of photons moving in the positive x direction and striking an n1=1 versus n2 interface along the y axis at x=0, one may immediately write a conservation of photon number equation i.e. N(incoming) = N(reflected) + N(refracted) ((1)). This equation, however, does not shed any light on the ratio of N(reflected) to N(refracted) for a given N(incoming). One may also notice that no information about the index of refraction is included. This information manifests itself in: E= p1 c/n1 = p2 c/n2. Taking ((1)) and multiplying by e (photon energy) yields a conservation of energy equation. In both the cases of photon number and energy conservation there is no notion of flux. Taking the energy conservation equation and writing e=p1c1 or e=p2c2 yields an equation relating momenta (proportional to impulses), but this equation involves dynamics i.e. fluxes i.e. c1, c2 values. Thus it is not on the same footing as the photon number or energy conservation equations. We argue that there should exist an impulse balance type equation which does not depend on flux. We suggest this is a key physical idea. (For example, the two-slit interference pattern of p=m1v1=m2v2 are the same.) As a result, the single equation ((1)) is not sufficient for solving the problem. To solve the problem, one needs to recast ((1)) in terms of flux (a notion which exists classically) divided by ci (speed of light) so that one obtains the number of photons. By having 1/c1 appear in the equation, one may multiply by e so that e/ci = pi. One then has the equation: AA/c1 = BB/c1 + CC/c2 ((2)) where AA=incoming flux, BB=reflected flux and CC refracted flux. One must change from a static picture of counting photons to a flux picture, but divide fluxes by ci’s to regain the static picture. Incoming flux is written as AA to indicate it is positive etc. ((2)) still does not solve for B and C in terms of A and n1,n2, but one may bring BB/c1 to the LHS and break ((2)) into two equations using differences of squares. Thus one has an A+B=C equation which is like a nonflux balance of probability or energy (if one multiplies by e) and a static momentum/impulse balance equation pA-pB= p2 C. Solving the problem involves creating not only time independent photon number/energy balance, but also time-independent photon momentum/impulse balance. We argue that the breaking of ((2)) into two equations introduces the notion of quantum mechanics because one uses “square roots” of flux probability. One may ask why one does not simply find a second equation in addition to the photon number/energy balance equation. We argue that the single number balance equation contains all of the necessary information in the problem i.e. flux, photon number, energy or p’s and c’s. Thus a second equation would not be independent. As a result, one must break the single photon balance equation into two independent equations leading to a different kind of physical balance i.e. one involving square roots of fluxes instead of fluxes i.e. quantum mechanics.
If one considers the reflection/refraction of a steady stream of photons moving in the positive x direction and striking an n1=1 versus n2 interface along the y axis at x=0, one may immediately write a conservation of photon number equation i.e. N(incoming) = N(reflected) + N(refracted) ((1)). This equation, however, does not shed any light on the ratio of N(reflected) to N(refracted) for a given N(incoming). One may also notice that no information about the index of refraction is included. This information manifests itself in: E= p1 c/n1 = p2 c/n2. Taking ((1)) and multiplying by e (photon energy) yields a conservation of energy equation. In both the cases of photon number and energy conservation there is no notion of flux. Taking the energy conservation equation and writing e=p1c1 or e=p2c2 yields an equation relating momenta (proportional to impulses), but this equation involves dynamics i.e. fluxes i.e. c1, c2 values. Thus it is not on the same footing as the photon number or energy conservation equations. We argue that there should exist an impulse balance type equation which does not depend on flux. We suggest this is a key physical idea. (For example, the two-slit interference pattern of p=m1v1=m2v2 are the same.) As a result, the single equation ((1)) is not sufficient for solving the problem. To solve the problem, one needs to recast ((1)) in terms of flux (a notion which exists classically) divided by ci (speed of light) so that one obtains the number of photons. By having 1/c1 appear in the equation, one may multiply by e so that e/ci = pi. One then has the equation: AA/c1 = BB/c1 + CC/c2 ((2)) where AA=incoming flux, BB=reflected flux and CC refracted flux. One must change from a static picture of counting photons to a flux picture, but divide fluxes by ci’s to regain the static picture. Incoming flux is written as AA to indicate it is positive etc. ((2)) still does not solve for B and C in terms of A and n1,n2, but one may bring BB/c1 to the LHS and break ((2)) into two equations using differences of squares. Thus one has an A+B=C equation which is like a nonflux balance of probability or energy (if one multiplies by e) and a static momentum/impulse balance equation pA-pB= p2 C. Solving the problem involves creating not only time independent photon number/energy balance, but also time-independent photon momentum/impulse balance. We argue that the breaking of ((2)) into two equations introduces the notion of quantum mechanics because one uses “square roots” of flux probability. One may ask why one does not simply find a second equation in addition to the photon number/energy balance equation. We argue that the single number balance equation contains all of the necessary information in the problem i.e. flux, photon number, energy or p’s and c’s. Thus a second equation would not be independent. As a result, one must break the single photon balance equation into two independent equations leading to a different kind of physical balance i.e. one involving square roots of fluxes instead of fluxes i.e. quantum mechanics.
Note Dec. 8, 2022: B in A+B=C and p1A-p1B=p2C is negative for n1<n2 where n is the index of refraction. Both equations i.e. probability conservation and average momentum conservation are involved in this result i.e. B/A = (c2-c1)/(c1+c2) (1). In Parts I and II we considered the problem of a beam of light A moving in the positive x direction striking a medium at x=0 and reflecting B and refracting C. This is a probabilistic problem because a single photon has a probability to reflect and another to refract. We also noted that conservation of energy holds i.e. I(A)/c1 = I(B)/c1 + I(C)/c2 where I(A),I(B) and I(C) >0 are intensities . The goal is to find I(B) and I(C) given I(A), but there is only one equation and two unknowns. In Part I we argued that an energy conservation equation incorporates two equations representing square root type probabilities. In Part II we argued that a physical motivation for such a situation may be that in a steady state case, one has dynamic directional probabilities not just static ones. The static/classical probability is associated with the conservation of energy and groups by time i.e an incident beam exists during one time range and reflected and refracted beams during another. One, however, may create a steady state system so that all three beams exist at the same time, although the reflected and refracted beams at any instant are the result of incident photons which hit the medium at an earlier time. A steady state system should not rely on time, but still contains probabilities. There is thus no reason to group by time, but rather one groups by space with the incident and reflected beams in one region and the refracted in another. Introducing space grouped dynamic probabilities A,B,C one may write (using conservation of probability and average momentum): A+B = C ((1a)) and p1A -p1B = p2C ((1b)) where p1=n1p and p2=n2p and c1=c/n 1 and c2=c/n2. The second equation has the appearance of a conservation of energy equation if one places the term -p1B on the RHS. The problem is that A+B=C implies B must be negative if n1<n2. Thus dynamic probabilities may be negative (which is unusual) and the second equation cannot be conservation of energy although it is of the correct form. In Part II we noted one may multiply the two equations together and by using a difference of squares preserve the form of an energy conservation equation which now has AA, BB and CC terms which are necessarily positive and may be identified with I(A),I(B) and I(C). In this note we wish to consider the consequences of B being negative. This we suggest implies A and B interfere so that A+B yields a number smaller than A i.e. C for n1<n2. As a result the negative value of B allows for conservation of probability (or photons). At first this may seem a mathematical convenience, but the two equations ((1a)) and ((1b)) lead to physical consequences, namely the probabilities BB and CC in terms of AA. We thus speculate that there might be a physical mechanism associated with the minus value of B, the reflected ray. This physical property (which should exist for individual photons) ensures conservation of probability. If one considers a glass wedge (or thin film) in air one has reflection from the glass (n1<n2) bringing in a minus sign in B (for conservation with the incident beam). The refracted beam becomes the new incident beam within the glass and reflects at an n1>n2 interface so B’ is not negative. B and B’ may now interfere (depending on the thickness of the wedge) creating an interference pattern which must conserve probability overall. In general one would expect B and B’ to be independent, but each carries conservation of probability information (in a physical form) and this information is different causing interference to occur.
In a number of notes (Parts I,II,III) we considered the problem of photons A moving in the positive x direction striking a medium at x=0 and reflecting B and refracting C. In (1) conservation of energy: I(A)/c1 = I(B)/c1 + I(C)/c2 ((1)), where I(A) is intensity which has the form of energy * velocity i.e. a flux, is given. Thus I(A)/c1 = n1 E , I(B)=n2 E and I(C)=n3 E, where E is constant energy of a photon and ni are the number of photons. Conservation of energy/or photon is a conservation law in terms of “particles” and is furthermore a time averaged result. A single photon either reflects or refracts. In both cases conservation of particle number and energy exists and so one may wonder why one needs a time averaged “particle” type conservation equation in the first place. We argue that it demonstrates the role of flux in this problem - a role which differs from that of hydrodynamics in classical physics. ((1)) shows immediately that there is neither conservation of flux i.e. I(A) values or momentum even in a steady state scenario (i.e. time averaged scenario) represented by ((1)). This is different from an incompressible liquid moving from a wide pipe to a narrow one. In such a case flux is conserved yet microscopic particles change velocity, momentum and energy. As noted in Part I, one may write: I(A)/c1 - I(B)/c1 = I(C)/c2 , multiply by E (energy) and use E=pc. This yields a “kind” of momentum conservation equation, but not in terms of particle numbers (i.e. not in a particle picture). Rather I(A)p is momentum multiplied by flux which is unusual. ((1)) is a conservation of particle number and energy, but does not include conservation of flux or momentum flux which one might expect in a steady state flow equation and which appears in classical fluid dynamics. As noted in Part I to obtain these additional conservation/balance equations one is forced to use square root probabilities i.e. I(A)=AA with A representing a square root flux or dynamic probability. Using the idea of differences of squares, one may write: A + B = C ((2a)) and Ap1 -p1B = p2C ((2b)) and solve for B and C in terms of A. In other words a hydrodynamical approach applies to the square root probabilities A,B,C and not to I(A)=AA etc. Ultimately squaring, one obtains I(A),I(B) and I(C) showing that the square root probability or wave/dynamical probability represents balance and conservation. In ((2a)) and ((2b)) i.e. with square root probabilities one has dropped the view of particles and changed to a square root flux/ new dynamical probability picture together with its balances. This is in fact the wave picture because A,B,C are linked with exp(ip1x),exp(-ip1x) and exp(ip2x) (which for photons are associated with the wave picture of the electric and magnetic fields). Thus even though there is particle wave duality, in terms of conservation waves (square root probabilities) yield particle conservation equations. For example, it is the wave picture which allows one to solve for B and C in terms of A. The wave picture may be associated with a steady state (time removed) equilibrium scenario in which dynamical (square root) probabilities add in space (not by time grouping) and it contains both conservation of probability and momentum probability while momentum (of the particles) may not be conserved in the particle picture as is the case for the reflection/refraction problem. Particle number/energy are also guaranteed.
The problem of a photon reflecting or refracting at an n1-n2 index of refraction is solved in (1) using the notion of continuity of the electric field of a photon and its d/dx derivative for the n1-n2 junction at x=0. The photon electric field behaves as cos(-wt+kx) or cos(-Et+px). We argue first of all that the photon is a quantum object and that this problem is really a quantum mechanical one as it may also be solved for a particle with rest mass using two different constant potentials V1, V2 instead of n1,n2. In this note, we consider the photon case for convenience. Given that the problem is really a quantum mechanical one (in our opinion), we ask: How far can one go by trying to solve it classically? We first point out that for a static uniform arrangement of objects in space, P(x)dx =dx/L, where L is an arbitrary length. In the case of particles moving at constant v, one may argue that each particle spends the same amount of time dt in each dx and that this also leads to a constant density in space. We suggest that for this to hold, one must consider a steady state problem and the notion of flux for physical reasons. This leads to writing the equation 1= Probability(reflect)+Probability(refract) as AA/c = BB/c + CC/c2 where AA, BB, CC are the incident, reflected and refracted fluxes and c1=c, c2=c/n2. We have shown in previous notes that one may mathematically write this single equation as: A+B=C and Ap-Bp = Cp2 using E=pc (with E the same for the incident, reflected and refracted photon) and then solve the two equations for the two unknowns B,C, setting A=1. This approach yields a result which matches experiment, but from the standpoint of classical physics, there is no a priori reason to take the square root of flux. It simply arises as a math trick so to speak, we argue, and there seems to be no physical motivation. Here, we suggest approaching the problem differently. We note that if c1-c2<<1, then there should be almost no reflection, while if c2<<1, there should be almost no refraction. We ask: What happens if one suggests that in the c1-c2<<1 case that one insists on BB dropping as (c1-c2)*(c1-c2), i.e. to second power as in a stable equilibrium and for CC dropping as c2*c2 for c2<<1 in the other extreme. This is a possible approach that seems to be linked to some kind of physical principle, and one may consider the simplest solution and compare it with experiment and find that it holds. In other words, it yields the same solution as the above approaches. Moreover, this approach gives a link with the square root approach of the above paragraph, because now one is linking c-c2 and c2 directly with square root fluxes A,B,C so as to obtain the quadratic feature automatically for c-c2<<1 or c<<1 when one ultimately uses AA, BB, CC. In other words, the assumption of quadratic vanishing of BB and CC, automatically leads to c-c2 or c2 vanishing of B and C, suggesting that one may automatically achieve the quadratic vanishing form by working with A,B,C square root flux variables. We suggest that this idea of quadratic vanishing of BB or CC possibly places the approach of using A+B = C, pA-pB=p2C on more physical grounds and points to a new kind of physics which uses A,B,C, namely quantum mechanics.
In Part I we considered light moving in the positive x direction stiking a medium with a different index of refraction with some reflecting and the rest refracting. We argued one may immediately write a conservation of energy equation, but this is a single equation while there are two unknowns, the intensity of the reflected beam and the refracted, assuming the incident intensity is known. (We also argued that one also has the relation E=pc for a photon, but this is not needed to solve for the two intensities.) Our main point was that instead of seeking another independent equation (containing new information), one may write the conservation of energy equation as a product of two equations A+ B=C ((1a)) and A/c1 -B/c1 = C/c2 ((1b)) where AA/c1 = BB/c1 + CC/c2 is the energy conservation equation. Thus energy conservation contains all of the information needed, but one must move to a square root of probability picture in order to obtain the two equations needed to solve for the reflected and refracted values B,C and then the intensities BB and CC. In Part I, we noted that the reflection/refraction problem may be presented as a steady state one in which the scenario is not changing in time. A Maxwell-Boltzmann gas in equilibrium also does not change in time, but here there is a difference, namely a lack of time reversal invariance. In other words there is a distinct direction of motion i.e. the incident and refracted beams move in the positive direction only and the reflected in the negative x. The problem, however, is also a probabilistic one because given a single photon, it either reflects or refracts with certain probabilities. Thus we suggest there exists in this physical steady state problem a dynamic or directional probability. This probability is related to space and may be positive or negative unlike the usual time probabilities linked to events i.e. classical probability theory. The conservation of energy equation is a classical probability one, but dynamic motion probabilities are present in the problem and should also be manifest in equations. In fact the two dynamic probability equations allow for the solving of the reflected/refracted intensities (probabilities), but are also directly linked to the conservation of energy equation which may be written as: AA/c1 -BB/c1 = CC/c2 which points towards flow in two different directions. Thus we argue that a dynamic directional probability is responsible for the decomposition of the energy conservation equation into two dynamic probability ones.
In Part I, we noted that a photon number balance equation , which is equivalent to an energy balance equation, written in terms of fluxes describes all of the information in a photon steady state reflection/refraction problem in one dimension. We also pointed out that one equation cannot solve for two unknowns and suggested that because all information is contained in the classical photon count equation, one may break it into two equations (quantum balance equations for probability and average impulse) in order to solve the problem. Thus the two quantum equations are needed to solve the problem, but collapse into the classical conservation of photon number equation. In this note, we argue that in classical mechanics there are two different, but consistent descriptions of force acting on a particle. The first is impulse Integral dt F = delta p where p is momentum. Two particles with the same momentum, but different velocities, create the same physical impulse. In other words, time is removed (integrated out) in this picture. An equivalent formalism is to write dt = dx/v or F dx = v dp. This form breaks the degeneracy of p’s with different velocities. In other words, time is important. Thus we argue that at the classical level there are two equivalent formalisms. The impulse one does not consider time, while the work-kinetic energy one does. Both are legitimate. For the reflection-refraction of photons in one dimension, both formalisms should apply and lead to consistent results as do the impulse and kinetic energy-work approaches. The latter uses time and is equivalent to three ideas: photon number conservation, energy conservation and pressure conservation, all three being associated with time. The concept of impulse, however, suggests that one should be able to consider space in terms of impulse alone and not time i.e. not velocity. This forces one to introduce a complex probability associated with impulse hits i.e. exp(ipx). This probability and its corresponding average impulse balance yield two independent equations, while the photon count conservation only supplies one. Thus the problem needs to be solved using impulse formalism, even though it is completely consistent with count conservation. The form exp(ipx) leads to new physics (when compared with classical physics) as it suggests interference pattern results.
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.