Newtonian mechanics deals with smoothness in time and space through x(t) where the “details” occur within dx and dt which tend to zero. Thus there are no discontinuities in x and t. We have argued that the quantum free particle wavefunction pieces exp(-iEt) and exp(ipx) deal with discontinuities and that smoothness is due to a “modulus” of two dimensional probability e.g. exp(ipx)exp(-ipx) which yields 1 or determinism. Both exp(-iEt) and exp(ipx) contain finite measures in time and space, namely hbar/E and hbar/p and so there may be discontinuities. For example, exp(ipx) contains probability cos(px) and sin(px) within a wavelength so x=vt is only an average type of motion. Thus using exp(ipx) in a problem may result in time discontinuities because the particle is not moving as x=vt within a wavelength. In Part I we examined one-dimensional reflection/refraction of light from an interface of two media with n1=1 and n2>n1. We concluded that the quantum solution of this problem involves two equations, one linked to probabilities and the other to impulse. Multiplying the two equations together (with a special grouping, namely a spatial grouping) yields a single equation which may either be interpreted as a pressure balance equation. Thus one starts with two equations which are both grouped in time, but with the second demonstrating vector directions and obtain a single equation which is only grouped in space (which is linked with spatial grouping). By rearranging this equation, one may obtain a time grouping i.e. energy conservation equation, but the two groupings cannot exist together. This is the problem with the classical pressure picture linked to a steady state scenario and not a single photon process which displays a discontinuity in time i.e. a single photon may either reflect or refract. In this note we try to consider why classical energy conservation/ pressure balance only leads to one equation, while the quantum mechanical approach leads to the two required equations at the price of introducing a square root flux type of probability. We argue that there seem to be two factors. First there is the notion of scalars and vectors and a space versus time picture. The single conservation of energy equation AA/c1 = BB/c1 + CC/c2 where AA,BB,CCC represent the incident, reflected and refracted beams is expressed in terms of time and so vector information (i.e. -p for the reflected beam is not conveyed). If one rewrites this as AA/c1 -BB/c1 = CC/c2 then one obtains a pressure balance type of equation, but time is removed and one is grouping by space. The second and more important feature we argue is time discontinuity. If one considers a single photon, it may reflect or refract, but not both. Thus there is a time period in which to observe both reflection and refraction and more than one incident photon is needed. Newtonian mechanics in the form of steady state treats time as smooth and so everything must happen instantaneously. This is the idea behind pressure. One considers an impulse 2p multiplied by flux i.e. velocity. The flux is used to find the number of particles which strike a surface per second as if this were a continuous smooth process in time. For the case of reflection/refraction it is not. It is discontinuous because either a reflected ro a refracted photon is created each time. Thus the notion of pressure averages this over and loses some of the information in the problem. This, we argue, is why a pressure balance/ energy conservation equation is not sufficient to solve the reflection/refraction problem even though pressure is a physical concept and may be measured (on average) just as x=vt may be measured on average.
Newtonian mechanics deals with smoothness in time and space through x(t) where the “details” occur within dx and dt which tend to zero. Thus there are no discontinuities in x and t. We have argued that the quantum free particle wavefunction pieces exp(-iEt) and exp(ipx) deal with discontinuities and that smoothness is due to a “modulus” of two dimensional probability e.g. exp(ipx)exp(-ipx) which yields 1 or determinism. Both exp(-iEt) and exp(ipx) contain finite measures in time and space, namely hbar/E and hbar/p and so there may be discontinuities. For example, exp(ipx) contains probability cos(px) and sin(px) within a wavelength so x=vt is only an average type of motion. Thus using exp(ipx) in a problem may result in time discontinuities because the particle is not moving as x=vt within a wavelength. In Part I we examined one-dimensional reflection/refraction of light from an interface of two media with n1=1 and n2>n1. We concluded that the quantum solution of this problem involves two equations, one linked to probabilities and the other to impulse. Multiplying the two equations together (with a special grouping, namely a spatial grouping) yields a single equation which may either be interpreted as a pressure balance equation. Thus one starts with two equations which are both grouped in time, but with the second demonstrating vector directions and obtain a single equation which is only grouped in space (which is linked with spatial grouping). By rearranging this equation, one may obtain a time grouping i.e. energy conservation equation, but the two groupings cannot exist together. This is the problem with the classical pressure picture linked to a steady state scenario and not a single photon process which displays a discontinuity in time i.e. a single photon may either reflect or refract. In this note we try to consider why classical energy conservation/ pressure balance only leads to one equation, while the quantum mechanical approach leads to the two required equations at the price of introducing a square root flux type of probability. We argue that there seem to be two factors. First there is the notion of scalars and vectors and a space versus time picture. The single conservation of energy equation AA/c1 = BB/c1 + CC/c2 where AA,BB,CCC represent the incident, reflected and refracted beams is expressed in terms of time and so vector information (i.e. -p for the reflected beam is not conveyed). If one rewrites this as AA/c1 -BB/c1 = CC/c2 then one obtains a pressure balance type of equation, but time is removed and one is grouping by space. The second and more important feature we argue is time discontinuity. If one considers a single photon, it may reflect or refract, but not both. Thus there is a time period in which to observe both reflection and refraction and more than one incident photon is needed. Newtonian mechanics in the form of steady state treats time as smooth and so everything must happen instantaneously. This is the idea behind pressure. One considers an impulse 2p multiplied by flux i.e. velocity. The flux is used to find the number of particles which strike a surface per second as if this were a continuous smooth process in time. For the case of reflection/refraction it is not. It is discontinuous because either a reflected ro a refracted photon is created each time. Thus the notion of pressure averages this over and loses some of the information in the problem. This, we argue, is why a pressure balance/ energy conservation equation is not sufficient to solve the reflection/refraction problem even though pressure is a physical concept and may be measured (on average) just as x=vt may be measured on average.
In classical statistical mechanics, e.g. a Maxwell-Boltzman (MB) gas, one calculates pressure by multiplying 2p (impulse) by v (velocity or flux). This is done because one has multiple particles and so flux is important. One needs to know how many particles strike a surface per sec as well as their impulse. What happens if one has a single photon (or particle)? This is not a steady stream situation, although one may create a corresponding steady stream situation. In particular, a single photon moving in the positive x direction may reach an interface of two media with different indices of refraction and either reflect or refract. In such a case one has a probability to reflect and one to refract which add to 1. Thus there are two unknowns (i.e. the two probabilities) and energy stays constant. One may immediately write a conservation of energy equation and apply it to a steady state scenario. This necessarily utilizes the idea of velocity because there is a flux. This energy conservation equation may be rearranged to give the appearance of a pressure balance equation which also depends on velocity i.e. flux. The problem is that conservation of energy or pressure balance is a single equation and there are two unknowns. One may not simply say that the probability to reflect is .5 and that to refract .5. In fact, in a previous note (1) we solved this problem using the idea of writing the energy equation in terms of two equations linked to differences of squares. We suggested that these two equations which only involve weights are linked to a function which together with its first derivative is continuous at x=0 .In (2) we suggested that the x-distributions cos(px) and sin(px) represent action-reaction impulse type distributions. We argued that an impulse creates momentum p in the first place and that this “impulse”, in the form of x-distributions, is still present because the object may strike a foil and impart impulse at any time. Furthermore, because cos(px) and sin(px) are probability distributions, there is no sense of time within a wavelength. On average, however, the object moves with a constant velocity. In the case of light it is c/n where n is the index of refraction and in the case of a nonrelativistic particle p/mo. The example of a single photon either reflecting or refracting allows one to formulate the problem in terms of impulse i.e in terms of the probability distribution scheme such that one does not have time and hence velocity. This scheme requires two equations. This impulse based x-probability distribution scheme or quantum mechanical approach, however, should map into the classical pressure balance because on average a photon does move as x=c/n t which it does. It is, however, the quantum approach (impulse picture with no time or velocity within a wavelength) which solves the problem i.e. yields the probability to reflect or refract which may then be linked to classical type (steady stream intensities and flux.
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.
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.
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.
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 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.