Abstract Three linear equations are proposed as quantum-mechanical equations for the free propagating photons. The solution of these equations is the vector potential of the electromagnetic field and it is a product of the Gaussian functions for the transverse coordinates and the eigenfunction of the harmonic oscillator for the longitudinal coordinate. This solutions is used to describe the reflection and refraction of photons on the boundary between two dielectrics. The amplitude of the reflected field coincides with the Fresnel formulae for the plane waves, but the amplitude of the refracted field is different. These amplitudes are the probability’s amplitudes of reflection and refraction of photons—like in the quantum mechanics. The photons propagated from the first into the second medium increase (decrease) their transverse size in the plane of incidence, if the refraction index of the second medium is greater (smaller) than in the first medium. The increasing or decreasing grow when increasing the angle of incidence. The difference between the quantum mechanics of the particles with a mass and the photons, as well as the interpretation of the wave function of photons, is discussed.
Abstract Three linear equations are proposed as quantum-mechanical equations for the free propagating photons. The solution of these equations is the vector potential of the electromagnetic field and it is a product of the Gaussian functions for the transverse coordinates and the eigenfunction of the harmonic oscillator for the longitudinal coordinate. This solutions is used to describe the reflection and refraction of photons on the boundary between two dielectrics. The amplitude of the reflected field coincides with the Fresnel formulae for the plane waves, but the amplitude of the refracted field is different. These amplitudes are the probability’s amplitudes of reflection and refraction of photons—like in the quantum mechanics. The photons propagated from the first into the second medium increase (decrease) their transverse size in the plane of incidence, if the refraction index of the second medium is greater (smaller) than in the first medium. The increasing or decreasing grow when increasing the angle of incidence. The difference between the quantum mechanics of the particles with a mass and the photons, as well as the interpretation of the wave function of photons, is discussed.
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 work relates to the fundamentals of quantum physics and photonics, in particular to the formation of processes of reflection, refraction and absorption of waves of ballistic photons. The study of these problems is an important and urgent task for understanding the foundations of the material world, which has not yet been fully resolved. An important property of ballistic photons is their motion in a straight line, which can be curved under the action of physical fields on the photons and the parameters of the medium of their motion. However, if external factors remain constant, their motion continues in a straight line, which allows us to apply the laws of geometric optics to them. The fundamentals of the geometric wave laws of reflection and refraction of light are laid down of the Young and Fresnel. However, they used sinusoidal waves similar to transverse waves in the of musical strings, which contradicts the motion of spherical waves of ballistic photons. Elimination of these shortcomings is the main goal of the work, and its scientific novelty is the substantiation and development of new geometric models of light motion. Research Methods: Since this work has the level of scientific discovery, for finding which strict methods have not yet been created, therefore, general principles of development of the theory of scientific knowledge were used, based on the laws of dialectics and consistency with the basic laws of physics. The author's research methodology was also used, based on the transition to the initial quantum level of the material world. New Results of the Work and their Discussion: It is shown that the first point of contact of photon waves lies on the unified normal to their spherical surface and the contact surface. This shifts the real normal of wave reflection relative to the traditional normal from the point of intersection of the axis of the tube of incident waves with the surface of reflection by the value of -Δ = λmax sin α. The real axis of reflection of spherical waves is associated with the crests most distant from the initial point of their reflection. When photon waves are refractional, their total deviation is formed from the last point of their incidence on the refraction surface. This shifts the real point of the normal of wave refraction relative to the traditional normal from the point of intersection of the axis of the tube of incident waves with the surface of reflection by the value of +Δ = λmax sin α. Geometric models of all these processes at different angles of incidence of photon waves are developed and their simplified schemes are given. Conclusions: The proposed models and schemes of reflection and refraction of spherical photon waves significantly clarify and change previously known schemes and do not contradict the real laws of physics and optics, therefore they have the level of scientific discoveries and should replace previously known processes and schemes.
The laws of reflection and refraction are typically derived using a dispersion relation that assumes that the angular frequency stays the same across a vacuum--dielectric interface and that the angular wave number increases. At the single photon level, this means that the total energy of the photon is conserved but the linear momentum is not. Here I derive the laws of reflection and refraction for a single photon using a dispersion relation that is consistent with the conservation of both total energy and linear momentum. To ensure that both conservation laws are upheld, the optical path length must be considered to be more fundamental than the geometrical path length when reckoning optical distances. As long as the medium is transparent, the optical path length traveled by a photon in a given duration of time is constant. By contrast, the geometrical path length traveled in a given duration of time is inversely proportional to the index of refraction. A photon propagating across a vacuum--dielectric interface is considered to be an indivisible package whose total energy (and angular frequency) and linear momentum (and angular wave number) remain constant, but whose velocity decreases with refractive index as a result of unspecified electromagnetic interactions.
Quantum mechanics is known for its “square root” probability i.e. the existence of a wavefunction W(x) such that P(x) = W*(x)W(x). This begs the question: Why is this the case? In Maxwell’s electromagnetic theory of the photon, a similar situation arises except that one has energy density instead of P(x) which consists of .5C1 Electric field* Electric field + .5c2 Magnetic field*Magnetic field. The “square root” object is a physical observable (Electric field, Magnetic field) which both behave as cos(-Et+px) and so, more accessible than W(x). Alternatively one may use Fermat’s least time principle for light to show that i two dimensional problems (reflection/refraction) d/dx (used in extremization) is really the operator of momentum in that direction i.e. a spatial invariant direction. Thus A=px such that dA/dx=p = x component of momentum which may be generalized to A=-Et+p dot r. (p.r) may be written as -i ln(exp(ip dot r) with exp(i p dot r) representing a “kind of probability”. This probability may be considered a square root probability because for uniform motion (light) all x points along a ray should have equal probability. Thus exp(i p dot r) maps into a constant which happens if one takes the modulus. Both of the above approaches (using continuity at x or a kind of conservation of probability and a momentum) may be used to solve the problem of an incident beam of light moving in the positive x direction hitting a medium and both reflecting and refracting. For example see (1). The goal is to find the two intensity values associated with the reflected and refracted beams assuming the incident intensity is known. In this note, we take a different approach to finding the two unknowns, one that does not use the idea of the electric field proportional to cos(-Et+px) or the probability exp(ipx). We wish to solve the problem using only the idea of information and stress the importance of beginning with energy conservation a priori unlike the two approaches listed above. We postulate that there are two sets of information in this problem. First there is the notion of Energy = p(momentum in a vacuum) n c/n where n is the index of refraction. Thus the incident and reflected light are associated with pc1,c1 and the refracted with pc2, c2 with energy being the same for all. Second there is the idea of conservation of energy i.e. if ten photons with energy E hit the surface and a10 reflect and (1-a)10 refract, then the sum of the refracted and reflected energy equals the initial energy. Energy is then equated with intensity through Intensity/c = energy (1). Using only these two pieces of information we argue one should be able to find the intensity of both the reflected and refracted ray. This leads to an immediate problem because there are two unknowns and only one equation, namely conservation of energy i.e. I(A)/c1 = I(B)/c1 + I(C)/c2 where A is the incident beam, B the reflected and C the refracted. Writing as: I(A)/c1 - I(B)/c1 = I(C)/c2 (i.e. a spatial instead of time grouping) suggests a possible solution might be to employ a difference of squares noting that 1/c = p/E with E being a constant in the problem. Thus one may reduce the problem to A+B=C and p1A-p1B=p2C. This is of the form of a new kind of “square root” type probability, but leads to two equations with two unknowns (A is assumed to be known) which allows the problem to be solved.
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.