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.
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.
The refractive index of a system is often viewed as the collective response of a medium to an electromagnetic field. Nevertheless, even if targeted at a single atom, the incoming light undergoes the dispersion. By studying the propagation of a single photon interacting with a two-level atom, we examine the dispersion behavior, calculate the phase and group velocity of the photon wave packet, and further analyze the dispersion experienced by the photon. These findings are critical for advancing long-distance quantum communications.
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.
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.
Under usual circumstances, i.e. a photon traveling in a vacuum, the notion of a spatial resolution of a photon does not arise (assuming that one does not know about photon interference). One simply focuses on x=ct. If one has an n1-n2 index of refraction junction (as in Part 1) with n1=1, and n2>1, the photon slows down in n2, has the same energy E and a greater momentum p2= n2*p, where p is the momentum in n1. If one considers that a photon in n2 interacts with atoms/molecules, i.e. wastes time that could be used for travel in x, then for a given t, the photon travels a shorter distance, hence c2<c. We suggested in Part 1 that this suggests a sharper resolution in x for p2, due to the shorter distance which we showed was proprotional to 1/p2. We pointed out a mathematical correspondence between the shorter distance x in n2 and 1/p2. Now p2 is a physical property of a particle, but in Part 1, we did not establish spatial resolution as a physical property. We simply noted that in n2, a time t is associated with a shorter x than in n1=1 and that this length in n2 is proportional to 1/p2. Furthermore, we did not suggest any notion of probability, only of spatial resolution. Here we suggest that the notion of probability exists if one considers that in a given time interval t, the time wasted may occur at any point in t or in x for that matter or even at multiple points if one assumes that while the particle is interacting with an atom/molecule, it is not interacting with a force because it is not traveling in x (or is trapped in the dx of the atom and so is not translating overall in one direction. One might argue that a hit from p2 in n2 occurs probabilistically because of these interactions with atoms/molecules, but physically, the idea of a finite fixed number spatial resolution has not been established. We suggest that ideas which apply for p2 in n2, should also apply for a photon moving in a vacuum. Afterall, the photon in n2 is described by E, c2 and p2. One does not really need to know about the interactions with atoms and molecules. This would suggest that the notion of a resolution proportional to 1/p2 should exist in the vacuum even though one cannot assign it to interactions with molecules/atoms. Basically, the interaction with atoms/molecules in n2 served to establish a different dt time for F=dp/dt, i.e. a larger one and hence a larger dp = 2p (for reflection). In other words, the physical implications of a sharper spatial resolution ( x smaller for a given t in n2) was directly linked to 1/p2. We consider E2=n2*E, p2, c for a photon in n1 and E,p2,c2=c/n2 for a photon in n2 and argue that it could make sense to have physical properties of delta t = hbar/E and delta x = hbar/p. We suggest that one may postulate a fixed value dx = constant/p for resolution where constant is a fixed number. There is no longer the idea of simply proportionality. We also suggest that spatial resolution, like p, must be a physical attribute. In n2, we saw that a changed spatial resolution, changed p2 (on average) from F = dp/dt arose. One might argue that one really has a photon with p and c in n2 when it is not interacting, but we suggest that this cannot be resolved. It is the p2 photon with its changed spatial resolution which would emerge in an experiment. If a photon in a vacuum already has a fixed spatial resolution equal to hbar/p, then shifting to p2 in n2 would mean a fixed physical resolution of hbar/p2, but also a resolution of hbar/p2 in a vacuum. In other words, the idea of spatial resolution linked to hbar/p seems to be a statistical approach to handling a problem with the idea that one does not have infinite spatial resolution in x.
In classical statistical mechanics, e.g. a Maxwell-Boltzmann (MB) gas, one tries to find the probability distribution of a particular quantity such as particle energy pp/2m. Thus there is a distribution in a quantity, making it the stochastic variable. This quantity, however, is subject to a conservation law i.e. elastic two particle scattering. In a previous note (1), we considered the case of reflection/refraction and called momentum the stochastic variable and noted that it too was linked with conservation. In fact, we argued that both the MB gas and reflection/refraction followed the same stochastic law: P(variable1) P(variable 2) = P(variable 1 + variable 2) whether variable 1 is a scalar or vector. In (2), however, we noted that exp(ipx), a free particle wavefunction, treats p as a parameter with x being the “probability” variable. This seems a little unusual because p, which is normally a variable in an MB gas and even in a quantum bound state i.e. a*(p)a(p) where W(x)=wavefunction = Sum over p a(p) exp(ipx) takes on the role of a parameter i.e. p=hbar/wavelength. Why this duality? Why is p a parameter for “x” of the probability distribution and yet at the same time a variable associated with conservation of momentum? In this note, we argue that unlike the distribution of a defining property of a particle such as momentum or energy, one may have a distribution associated with probability itself. This we argue is the case for a reflection/refraction problem which applies to both light and a particle. In the case of light, there is an interface of two media with different indices of refraction and in the case of a particle an interface with two constant potentials. In both cases, total energy of the photon/particle is constant, but there are two choices or probabilities, reflection or refraction, but they are linked with different times. A single photon or particle can only undergo one at a time. One knows exactly the two values of momentum, but the probability of each is unknown. Thus unlike an MB gas where energy/momentum is being redistributed because the two particles of the outcome exist at the same time, here a probability of 1 must somehow be distributed to the two events.. Given that the incident momentum is p, the reflected -p and the refracted p2, only considering a specific time event i.e. reflection or refraction, does not conserve momentum. In the MB case, a single time event i.e. scattering conserves both momentum and energy. Thus it seems that one does not deal with events in time. There is a distribution of probability in such a way as to ensure conservation of momentum, but does this probability already exist or get created at the interface point? We argued in (3) that if the scenario were deterministic one would have to follow it in time and treat each event separately. Rather we suggest that a distribution of probabilities allows for conservation of momentum, but this means that all three momenta exist at the interface x point as one is not considering an event at a precise point in time. This then allows for each momentum to have its own x distribution based on momentum p so that one may impose conservation of momentum and not have a deterministic time associated with the interface point.. This last idea of conservation of momentum appearing is not discussed in (3) and allows for p to behave like a parameter for x while at the same time being a variable which is conserved. In short, in an MB gas one distributes energy between two colliding particles in order to conserve total energy while for a quantum particle or photon which reflects/refracts one distributes probability to the two scenarios (reflection or refraction) such that momentum is conserved. Thus the nature of what is “distributed” is very different.