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.
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.
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 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.
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.
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.
In this note we argue there exist two force delivery schemes which are already well-known in classical mechanics namely: F(x)dx and F(t)dt. The former is associated with work and kinetic energy, is deterministic (yields x(t)) and requires information about velocity (or rest mass). It deals with F(x) where F=Force. The second is associated with an impulse which changes momentum. In this case, one does not need to know the rest mass or velocity. For example, a particle with momentum p (this allows for many mo,v pairs with the same p) strikes a surface and is reflected back with -p. It is also associated with probability as such a scenario exists for light reflecting and refracting from a surface. One does not know a priori the probabilities for these two phenomena so the problem is probabilistic. Classical mechanics primarily deals with the F(x) force delivery scheme, e.g. solving a differential equation(s) which arise directly from dp/dt = F(x) or using Lagrange’s or Hamilton’s formalism. Ultimately one obtains the deterministic result x(t). Although the notion of an impulse is well-known in Newtonian mechanics, it does not seem to be used much in calculations because an explicit form of F(t) is usually not known i.e. the force acts in a very tiny dx region in a very short period of time. As noted above, both schemes i.e. both deterministic and probabilistic situations appear in the macroscopic classical world, yet for some reason it is suggested that Newtonian mechanics is only deterministic. Newton considered a photon to be a particle with momentum p. A single photon reaching a surface with a different index of refraction has a certain probability to reflect and another to refract. This is a probabilistic, not deterministic, scenario governed by an impulse delivery. The probabilistic problem of light reflecting and refracting in a macroscopic situation may be solved by using continuity of the periodic electric field and its derivative from Maxwell’s solution for a photon from his electromagnetic equations. We have suggested in previous notes, however, that no knowledge of Maxwell’s result is needed. A steady state reflection/refraction problem removes time (unlike the x(t) deterministic picture), but seems to require some kind of balance on the right and left hand sides of the medium. Furthermore two equations are needed to solve for the two unknowns (i.e. reflection and refraction probabilities). Intensity/flux does not balance because the incident and reflected rays have positive flux. For n1<n2, it would be good to have a balance function which allows partial cancellation of the incident and reflected fluxes. The solution exp(ipx) allows exactly this and this solution only requires knowledge of p, momentum, not of rest mass (i.e. velocity). Thus there seem to be two force delivery mechanisms in the macroscopic classical world with very different solutions. The first is the F(x) delivery yielding the deterministic x(t) and requiring knowledge of mo (rest mass). The second is the impulse delivery which is probabilistic, makes use of exp(ipx), and does not require rest mass. If rest mass is known, however, exp(ipx) is linked with the average equation x = p/m t. At first the two delivery schemes seem mutually exclusive, but in the impulse situation, we try to show that a series of impulses may create the semblance of an F(x). Thus one has the exp(ipx) from the probabilistic force delivery scheme, but at the same time an F(x) emerges and so an x(t) type solution should also follow. How can this be? exp(ipx) is probabilistic and one may create averages with probabilities. Such an average or root mean square average may exhibit deterministic behavioiur. Thus exp(ipx) may be used together with a potential V(x) to create a solution W(x)= Sum over p a(p)exp(ipx) such that the prms(x) behaves like a Force(x) delivery result, yet at the same time one has the features of the impulse delivery as exp(ipx)s are used. Thus the exp(ipx) impulse approach may be used in strict impulse delivery problems and in F(x) delivery problems. This is exactly what occurs in quantum mechanics.
Newtonian force is based on a change in momentum in time, but momentum is a vector and so may change in magnitude and direction. In many cases, the change is of one form or the other. For example, for a bound particle the change is in kinetic energy i.e. in magnitude, while for a particle moving at constant speed in a circle or pressure in a Maxwell-Boltzmann gas, the change is in direction. For an MB gas, the probability for p and -p are the same so the average momentum at any point x is 0 and thus chunks of the gas do not move, even though a single particle in the gas moves. The pressure, on the other hand, is the average of 2pv (2p for impulse, v for flux) and so pressure for p and -p do not cancel. A quantum free particle is based on exp(ipx) which does not contain time and is a dynamical probability, This is both an eigenfunction of the momentum operator -id/dx and the kinetic energy operator -1/2m d/dx d/dx, yet as pointed out in Part II, there are two pieces (cos(px),sin(px)) with a spatial density and hence a kind of pressure/force. These, however, combine to give the sense of constant motion in one direction. We argue this is due to a dynamical probability, unlike a static balance of pressures/forces found in an MB gas or Newtonian statics. There is a density with pressure distribution cos(px), but this is continuously changing because the particle moves in one direction with constant p. Thus the two pressure distributions “balance” so to speak to create a constant p. Interference of exp(ipx) and say exp(-ipx) may “bring out” a “static” spatial density i.e. 2cos(px), but behind the scene the dynamical probabilities still exist. It is these which “hold” the cos(px) together which appears strange from a static point of view. For example, a quantum bound state has a static W*(x)W(x) spatial density at the same time that the single particle has an average classical kinetic energy KE(x)=En-V(x). Interference also gives rise to the notion of an impulse-pressure in quantum as one ignores the isin(px) and i sin(-px) terms. Creating a pressure scenario with p and -p leads to two considerations. First, there is an impulse-pressure (defined without flux) at each x point and secondly, this impulse-pressure changes with x. The time-independent Schrodinger equation is formulated in terms of energy (i.e. a scalar): KE(x)+V(x)=En where KE(x)= {-1/2m d/dx dW/dx} \ W and W=Sum over p a(p)exp(ipx). Thus, as in Newtonian mechanics, one thinks of -dV/dx solely accelerating a particle. The quantum particle however, is described by an ensemble W(x) which has pressure. This pressure is due to a combination of the form exp(ipx) and V(x), so V(x) does not simply change root mean square speed. exp(ipx), by itself, creates pressure when interference occurs as in a two slit scenario. We try to examine these ideas in this note.