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.
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.
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.
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.
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.
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 a previous note (1) we argued that quantum square root probabilities follow from the consideration of light reflecting and refracting at a surface along the y direction at x=0. No knowledge of Maxwell’s periodic photon electric field is needed. The photon number and energy are conserved at a single photon level, but flux/intensity may be included through (2) AA/c1 = BB/c1 +CC/c2 where AA is the intensity of the incident beam (AA>0), BB the reflected and CC the refracted. There are two unknowns B,C if A is known and one equation which may be broken into two A+B=C and A/c1 -B/c1 = C/c2. For n1<n2 (n=index of refraction B<0). Thus we argue that there is an underlying square root probability. In this note we first heuristically suggest the form for the square root probability. A key point is that “particle” features such as E and p (for a photon) govern the dynamical wavelike probability. This also applies to a particle with rest mass. We also ask why this square root probability has not appeared in dynamical theories and argue that it appears in special relativity (i.e. a Lorentz boost) for a particle with rest mass mo at x=0 at t=to. As seen from a frame moving at constant speed -v, this particle moves with speed v. Thus x’=vt’, yet a 2x2 Lorentz transformation is applied to (x=0,t=to). Only one relation is needed because x’=vt’ and p’=E’/cc v suffice to yield the variables x’ and p’. Thus we argue that a Lorentz transformation describes more than is necessary for a simple particle. The 2x2 nature of the problem we suggest describes more than a simple particle. It leads to the Lorentz invariant -Et+px ,but x=vt so why should x and t appear as if independent? We argue that -Et+px=A is compatible with dA/dx |constant t = p and dA/dt |constant x = -E. P and E are “particle/photon” properties, but exp(ipx) or exp(-iEt) represent eigensolutions of -id/dx and id/dt. Thus if x and t are independent exp(ipx) may be interpreted as existing at all x points (as there is no time). This is the picture of an unusual probability which is two dimensional, with components which may be positive or negative as in the reflection/refraction problem. Thus we argue that the notion of a wave probability or square root probability not only exists in the reflection/refraction problem, but in the case of special relativity with what looks like an unnecessary 2x2 matrix from the point of view a particle (mo, x=0, t=to) describing an associated square root wave like probability completely dependent on the particle properties E and p.
It is known that if a photon hits a surface with a different index of refraction (at an angle within a certain range), it has a specific probability “a” to reflect and 1-a to refract. In this note we ask: Where does this statistical nature come from? In previous notes, we have considered Fermat’s principle applied separately to explain two dimensional reflection and refraction. We have also used the idea of a difference of squares and energy conservation in one dimensional reflection/refraction to find values of the probabilities. The question we ask is: Why are there probabilities in the first place? Is there some general idea at work despite the complicated details of the photon interacting with the atoms of the new medium.We argue the general notion is due to the idea of spatial invariance which is linked to the translation operator d/dx which in turn in linked to a p impulse occurring at a given x .Thus the variables p and x are linked through the notion of an impulse existing at some x. The idea that d/dx may operate on a function of constant p and x to yield p, implies the function of p,x should have physical meaning. This is a problem as classically a particle moving with constant p should not be associated with any function of x except a constant. (All x points are the “same”.) Thus we argue that a “probability” in x is established with a constantly moving particle (or even photon). The eigenfunction of d/dx (or rather -id/dx) which exhibits some kind of spatial invariance is exp(ipx). This contains two different x probability distributions cos(px) and sin(px). We noted that d/dx is associated with p due to an impulse hit at say an x point. Thus various exp(ipx)s with different weights should appear together in a physical problem with impulses such as reflection/refraction. The weights of the various exp(ip1 x), exp(ip2 x) etc become linked to probabilities associated with the state 1, 2 etc. Thus the origin of these probabilities (i.e. their specific values) follow from the probabilistic form exp(ipx) which follows from d/dx. (In (1) we have argued that the x distributions in exp(ipx) may be further linked to the force which “created” the p in the first place, with different p values having different distributions cos(px), sin(px)).