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.
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 mechanics deals with two types of force delivery schemes, one linked to the variable x and the other to t. They may be written as: Integral Fdt (impulse) and Integral Fdx (work) and appear to be symmetric in the two variables. The problem is that expressed in terms of dp one sees an immediate difference linked to discrete versus continuous change. In particular: dp= Fdt ((1a)) and vdp = Fdx ((1a)). ((1a)) allows for a discrete change in p in a tiny interval of time. “X” does not come into play at all in the equation and one might think of the impulse occurring at a point x. Due to the discrete nature of integral dp changes, one may think in terms of probabilities because a given pA may have been obtained from discrete impulses: p1→p2→p3 or p4+p5. etc. The final p state is the same and so not only does there exist probability but also the notion of product probabilities such that exp( pi b) exp( pj b) = exp( (pi+pj) b) where b is a constant. Given that exp(pb) increases exponentially, one might rather use the periodic probability exp(i p b). One may note how impulse type hits and their discrete momentum results lead to stochasticity in a Maxwell-Boltzmann gas. The form vdp = Fdx is very different in that it does not allow discrete jumps of p because v=dx/dt. If one assumes a very large F at x1 and dx–0 such that the product is not 0, one might consider dp changing discretely, but what is then used for v? Instead one considers a range of x with a corresponding range in t such that x(t) in order to compute v. Thus this is a deterministic situation because x and t cannot be decoupled. It leads to the notion of a kinetic energy pp/2m associated with p. This kinetic energy is calculated by considering p change from 0 to p. An interesting feature arises because impulse = Integral Fdt may be thought of as discrete allowing p1 to jump to p2. Nevertheless, the deterministic approach states there is a kinetic energy associated with p1 and p2, namely p1p1/2m, p2p2/2m, even though no deterministic path in time and space was taken. The impulse was said to occur at x in a very tiny time range. Thus there seems to be a contradiction. pp/2m requires a range of x and t with x(t). The impulse only occurs in time, with x decoupled but there must be an associated x range which is then not deterministic i.e. a decoupled x range. This suggests a characteristic length associated with a momentum p in which one does not follow time deterministically. If one does not follow x in t in this special length, then one has a probability distribution. A similar argument occurs for time. Consider Fdx with time absent. If dx becomes very tiny and F(x) very large , one may think that work is done at an instant in time, but there must be a time range for pp/2m. Thus there is a special time period not associated with x in which one has probabilistic behaviour i.e. not x(t). One may note from the notion of impulse that a given p may be obtained through different series of smaller impulses, e.g. {p1,p2,p3} and {p4,p5}. These are associated with kinetic energy series {p1p1/2m, p2p2/2m,p3p3/2m} and {p4p4/2m, p5p5/2m}. The final result is the same. Thus the discrete nature of changes leads to a probability scheme such that P(p1)P(p2) = P(p1+p2) and P(e1)P(e2)=P(e1+e2). This we argue leads to: P(p) = exp(i p b1) and P(e) = exp(i e b2) where b1 and b2 at first seem to be constant. We noted, however, that an impulse implies a probabilistic range in x and Fdx→finite when dx→0 a probabilistic range in t. Thus b1 may be replaced with x and b2 with t. At any given point in time or space, x and t are constants i.e. b1 and b2. Next, we consider the situation of special relativity i.e. a Lorentz boost. We consider an initial state of energy= mo, x=0, t=to and view it from a frame moving at a constant velocity v. Although one may use infinitesimal velocities, a main point we wish to make is that special relativity allows for the use of finite velocities which give the sense of discrete changes like in the impulse case.This should again be associated with probabilities. Special relativity yields the Lorentz invariant A = -Et+px where v=x/t. Nevertheless, x and t are written in the invariant as if they were independent. It may be noted that dA=0 if t changes on its own as hbar/E and x on its own as hbar/p. Thus one may obtain specific x and t probability ranges.This is again consistent with the probabilistic picture or exp(-iEt) and exp(-ipx). It may be noted that even with the probabilistic picture, a particle with constant p moves determinisitcally on average as x= p/m t. If x = hbar/p, then 1/t is not hbar/E and if t=hbar/E then x is not hbar/p. Thus the values hbar/E and hbar/p seem not to be linked with the current deterministic motion, but rather with the creation of p in the first place. E=pp/2m and this is obtained by considering p changing from 0 to p. Thus the “frequency” E seems to capture the information of the creation of p and we argue that the wavelength captures the creation of x as a probability distribution in x which one would usually associate with force/acceleration.
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 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.
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.
In Newtonian mechanics, force and energy are clearly separated. For example, pp/2m = Integral F dx = Integral dp/dt dx = m/2 vv = pp/2m. The equation pp/2m + V(x) = E is considered to be strictly an energy equation. In special relativity, mocc is considered to be strictly an energy. We have suggested in previous notes that mo is proportional to inertia which is a force. Thus, mo and mocc are thought of in two very different manners simultaneouly, i.e. as an energy and as a force. We propose that it might be beneficial to think of both mo and V(x) in terms of energy and force distribution simultaneously. In particular, we consider that V(x) is not only potential energy at x, but is also linked to force or interaction at x through -d/dx V(x). The reason this duality of energy-force may be important is that it is directly linked to quantum mechanics, we argue. In general, it is argued that classical mechanics is a high energy level limit of quantum mechanics, We argue, however, that for a single particle bound state, the energy linked with a momentum p (at any energy level) is pp/2m (nonrelativistic) which is deterministic (and also follows from special relativity) and has nothing to do with quantum mechanics. The difference between classical mechanics and quantum, is that there exists an interaction (impulse hit) probability of exp(ipx) which means that instead of a hit occurring at the center-of-mass point, it may occur within an hbar/p range. Thus, one may have a particle interact and move through either of two slits if they are separated by about hbar/p. If p, however, is changed in an interaction, p1->p2, then the change in energy p2p2/2m - p1p1/2m (one dimension) is classical-deterministic. It is, however, the duality of V(x) with force that force one to consider not only energy pp/2m (deterministic), but also its associated force distribution given by exp(ipx) / W(x), where W(x) = Sum over p a(p)exp(ipx). As a result, one creates the average kinetic energy term: {Sum over p a(p) pp/2m exp(ipx) } / W(x). One may call exp(ipx)/W(x) a probability and so formally one has KEave(x), but physically exp(ipx) represents a force distribution. The balance to V(x) is a combination of a deterministic energy pp/2m with a force distribution exp(ipx)/W(x). This, we argue, is a physical statement and helps to describe why V(x) may be expanded as a series of virtual particles (e.g. photons etc). It is not just energy, but force that must be considered together. It is only in classical physics, pp/2m + V(x) = E that one solely considers the energy aspects, i.e. pp/2m and V(x) only as energy, but the more general picture, we argue, is to consider not only energy, but force distribution together. This is the quantum picture and it incorporates not only probability, but deterministic Newtonian mechanics (or special relativity) through the use of pp/2m.
In (1), we argued that a uniform distribution associated with the placement of static particles in a one-dimensional length L is P(x)=1/L. We then suggested that because a static particle has speed=0, one cannot rule out the existence of a probability F(p,E,x,t) which is Lorentz invariant. For every p, there is a -p and so F(p,E,x,t) complex conjugate * F(p,E,x,t) = 1/L yielding F(p,E,x,t) = exp(-iEt+ipx)/sqrt(L) (see (1) for details). F(p,E,x,t) is a dynamic probability like x(t) or v(x,t) are dynamic variables in Newtonian deterministic theory. We then argued that the F(p,E,x,t) is not simply a math result, but actually solves the 1-D reflection-refraction problem, while Newtonian mechanics cannot. Here we ask the question: Why can’t Newtonian mechanics solve the 1-D reflection-refraction problem? Certainly there is probability in the problem, given by 1= probability(reflect) + probability(refract). It seems that Newtonian mechanics requires that one always know x(t), v(x,t) etc. Imagine that there is a specific deterministic mechanism for reflection and a different deterministic one for refraction and that these turn on with probability(reflect) and probability(refract). One may then describe reflection and refraction separately by deterministic Newtonian mechanics, but one cannot calculate probability(reflect) and probability(refract) because these are built into the physical interfaction interface n1-n2 (where ni is the index of refraction). Experimentally, the probabilities to reflect and refract relative to the incident beams are functions of only n1 and n2. N1 and n2 are known, but that still does not provide one with a method of solving for p(reflection) and p(refraction).Newtonian mechanics is not of use. The question then becomes: What if one introduces a dynamic probability exp(ipx), dropping exp(-iEt) because E is the same for the incident, reflected and refracted beam? How could this possibly help matters? The probability to reflect and refract is based on n1 and n2 alone. We consider the notion that exp(ipx) is a dynamic probability and it seems that it may also be thought of in terms of a steady stream. A steady stream of particles means that one does not know the exact x(t) of any particle, one only has an average sense of the probability flow. This suggests that one can use an extra conservation law that is not available in Newtonian mechanics, namely conservation of dynamic instantaneous probability exp(ipx). We stress that 1-D reflection is not a physical fluid like steady stream problem. We use the notion of a steady stream as an analogy to describe the conservation of exp(ipx), the dynamic probability. Given that exp(ipx) is an instantaneous probability, it must be accounted for at every x point and the viewpoint of a steady stream allows one to bypass single reflection or refraction events because one is viewing the process on average, but at the level of the instantaneous probability which must be defined at each x. This is almost like having probability represent fluid flow, but it is the dynamic instantaneous probability which behaves like the fluid, not a classical flux. Such a new conservation law (and conservation of the dynamical average of momentum) yields two equations which allow one to solve the problem. In a nutshell, Newtonian mechanics does not allow one to combine events that occur at different times in the same equation as if they occurred at the same time. The OR (addition) probability condition allows for this, but 1= probability(reflect) + probability(refract) does not contain the dynamics of the event. One requires a probability which acts like a steady stream probability and we argue that this is the dynamic probability exp(ipx). This forces one to deal with the probability being conserved in x, but also momentum being conserved under this scheme (i.e. using this probability) as exp(ipx) is the instantaneous probability describing a particle with momentum p.