In (1), we argued that classically there are two pictures. The impulse picture delta p = Integral dt F does not distinguish between different velocities if p is the same. In other words, p=m1v1=m2v2 (so time is removed) and v1 and v2 create the same impulse (and quantum mechanically the same interference/diffraction pattern) even though v1 and v2 are physical and different. The second classical picture is the kinetic energy-work one which depends explicitly on time i.e. v dp = Fdx. The two pictures are consistent, but for the one dimensional reflection-refraction problem, a photon count and photon energy balance equation based on time are not independent. Thus one must go to the time-removed impulse picture in order to have two equations, based on the probability exp(ipx), in order to solve the problem. We concluded in (1) that an impulse picture with time removed is necessary to describe interactions in a quantum system in a steady state scenario. Thus this picture should be applied to a bound state. A quantum bound state, however, makes use of a potential V(x) which is associated classically with a kinetic energy at x. Kinetic energy depends on time so how is this described in the impulse picture? We argue that one creates an average of pp based on the impulse probability exp(ipx) a(p) (a(p) is a weight for p) which has time removed. It is the impulse probability exp(ipx) which allows for interference to occur. As a result, <pp> does not have time present and may include p and -p values which classically would occur during different time intervals. Once <pp> is computed in the time-removed picture, however, one divides by 1/2m (in the nonrelativistic case) to create kinetic energy. Thus prms(x), a mathematical average, may be associated with the idea of motion in space and time. pmrs(x), however, may not be considered physical as it is the impulses and exp(ipx) which are physical. As argued in (1), however, classically the two pictures should be consistent. As a result, for high energy levels, the description of <pp> in terms of an impulse picture which does not include time, but then adds 1/2m at the end should be equivalent to a picture in which a particle moves either to the left or right. We suggest that at high energy levels, one may actually assign prms(x) as a physical quantity within a tiny dx and follow the particle in time. Even in the time-removed impulse picture, there is velocity i.e. motion in time. It is just not used to calculate interaction averages in a steady state scenario. We suggest that to move from the quantum picture to the classical, one switches formalisms i.e drops the impulse- time removed picture and uses the time dependent motion in one direction. These two pictures are consistent at the classical level i.e. the classical time picture is a good approximation to the detailed impulse time removed picture which is now occurring in tiny dx regions.
In (1), we argued that classically there are two pictures. The impulse picture delta p = Integral dt F does not distinguish between different velocities if p is the same. In other words, p=m1v1=m2v2 (so time is removed) and v1 and v2 create the same impulse (and quantum mechanically the same interference/diffraction pattern) even though v1 and v2 are physical and different. The second classical picture is the kinetic energy-work one which depends explicitly on time i.e. v dp = Fdx. The two pictures are consistent, but for the one dimensional reflection-refraction problem, a photon count and photon energy balance equation based on time are not independent. Thus one must go to the time-removed impulse picture in order to have two equations, based on the probability exp(ipx), in order to solve the problem. We concluded in (1) that an impulse picture with time removed is necessary to describe interactions in a quantum system in a steady state scenario. Thus this picture should be applied to a bound state. A quantum bound state, however, makes use of a potential V(x) which is associated classically with a kinetic energy at x. Kinetic energy depends on time so how is this described in the impulse picture? We argue that one creates an average of pp based on the impulse probability exp(ipx) a(p) (a(p) is a weight for p) which has time removed. It is the impulse probability exp(ipx) which allows for interference to occur. As a result, <pp> does not have time present and may include p and -p values which classically would occur during different time intervals. Once <pp> is computed in the time-removed picture, however, one divides by 1/2m (in the nonrelativistic case) to create kinetic energy. Thus prms(x), a mathematical average, may be associated with the idea of motion in space and time. pmrs(x), however, may not be considered physical as it is the impulses and exp(ipx) which are physical. As argued in (1), however, classically the two pictures should be consistent. As a result, for high energy levels, the description of <pp> in terms of an impulse picture which does not include time, but then adds 1/2m at the end should be equivalent to a picture in which a particle moves either to the left or right. We suggest that at high energy levels, one may actually assign prms(x) as a physical quantity within a tiny dx and follow the particle in time. Even in the time-removed impulse picture, there is velocity i.e. motion in time. It is just not used to calculate interaction averages in a steady state scenario. We suggest that to move from the quantum picture to the classical, one switches formalisms i.e drops the impulse- time removed picture and uses the time dependent motion in one direction. These two pictures are consistent at the classical level i.e. the classical time picture is a good approximation to the detailed impulse time removed picture which is now occurring in tiny dx regions.
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.
A classical bound state does indicate the direction of motion if one only considers kinetic energy pp/2m. It is the vector quantities, p and x, which indicate direction. In quantum mechanics, a free particle interaction probability (for impulse hits) is directional, i.e. exp(ipx) not= exp(-ipx). Unless one has a single value of pp/2m, KE+= sum over p>0 a(p)exp(ipx) pp/2m / Sum over p>0 a(p)exp(ipx) is not equal to KE- even if a(p)=a(-p). In order to use conservation of energy in quantum mechanics, one is forced to consider both a(p)exp(ipx) +a(p)exp(-ipx) with a(p)=a(-p) which is the correct statistical weighting for a bound state. One may note that exp(ipx) is a probability for interaction and is separate from x=p/m t even though the notion of this velocity is tied to the fact that nonrelativistic kinetic energy is pp/2m. As a result, one must use an average scenario for calculating kinetic energy which seems to be tied to an ensemble average. Such a notion should hold for all energy levels. If one considers the classical limit, then one has p(x)p(x)/2m in a tiny dx. If one were to simply use 1/sqrt(p)exp(ip(x) x)/sqrt(dx) in dx, then Integral W*W dx in dx = C/p. The relative weight of two adjacent dx regions is: p1/ p2 = p1 / (p1+dp). This is equivalent to probabilities proportional to dt= dx/v, where p=mv. Thus, the area represented by W*W captures the number of possible interactions in dx if attempted hits have a fixed period. We argue, however, that given the bound state conservation recipe, one must use a(p) (exp(ipx)+exp(-ipx)) to describe a dx (with p=p(x)), even in the high energy limit. exp(ipx) is simply an interaction pattern (and is a probability) and so if the particle is bound, the correct statistical thing to do for an ensemble seems to be to combine exp(ipx) with exp(-ipx). We then show that quantum mechanics lowers the number of interactions in an area with increased speed more than the classical case. In particular, we find instead of p/(p+dp), 1- dp/p - 2dp/(pp).
Newton’s second law dp/dt = F(x)=-dV/dx may be written in terms of x alone i.e. dp/dx v(x) = -dV/dx or 1/(2m) d/dx pp = -dV/dx leading to the idea of kinetic energy pp/2m and potential energy V(x). In (1) we argued that there is velocity, rest mass mo, and momentum in quantum mechanics. A free particle, however, does not undergo smooth acceleration as in classical mechanics, and we argued there exists a momentum space wave exp(ipx) accompanying the moving particle. In such a scenario, force is considered to be probabilistic momentum hits of the form Vk exp(ikx) leading to a free particle momentum distribution W(x)= Sum over p a(p) exp(ipx). Both of these expressions involve interference due to momentum waves which do not track a particle in time. A mass flux mov or energy flux p=Ev relativistically seems to be the main consideration. Nevertheless, summing Vkexp(ikx) over k to obtain V(x) yields a function varying in x alone. This is a kind of average based on interference and Sum k Vk exp(ikx) is an average linked to momentum hits. We speculate that average momentum hits i.e. -d/dx V(x) are linked to momentum flux changes. In other words, even though an impulse is based on p and does not distinguish v i.e. p=m1v1=m2v2, momentum flux is velocity dependent and appears for an average momentum hit i.e. -dV/dx. In such a case (pave+dpave)(vave + dv ave) = 2 pave dv(ave) . This expression, however, contains both forward and backward motion which is part of quantum mechanics. Moving to an average picture seems to introduce time and direction i.e one uses pave d v(ave). This is already a change from a quantum to a classical scenario albeit for averages (which have not been specifically defined). Thus we consider pave dv(ave). If this is the case, then: d/dx pave 1/m space = -dV/dx or 1/(2m) pave pave = -d/dx V(x). Instead of treating pave pave as two separate averages one considers pave=prms or pavepave=<pp>. Thus the quantum bound picture (at any energy level) becomes a classical type of scenario which includes time and velocity. Note that for a two slit interference pattern, one does not need velocity and time because the interaction with the potentials from both slits is based on exp(ipx). One does not even need the form of the potential. Interference is the same for p=m1v1=m2v2 (even though the process takes place during different time intervals, the pattern is not affected). In other words, we argue that Newton’s second law might follow from a picture of momentum hits in which a flux i.e. p=mov is treated as a quantity which in an average scenario leads to a second flux i.e. an average momentum flux which reintroduces direction, time and velocity.. This is somewhat like a hydrodynamics scenario.
Quantum mechanics is concerned with measurement and we suggest that in a bound problem impulses due to momentum are the form of measurement which occurs in x space. The Lorentz invariant A= -Et+px, which is also the free particle relativistic or nonrelativistic action with v=x/t, associates energy = pp/2m with time and p with x. Creating eigenvalue equations yields id/dt exp(-iEt) = E exp(-iEt) and -id/dx exp(ipx) = p exp(ipx). P (momentum) is associated with physical impulses and with its own two dimensional probability distribution exp(ipx). E is associated with t and has its own distribution in time exp(-iEt) which implies that the same E holds for each x point without any notion of x probability associated with it. In classical physics, a momentum p exists in a tiny dx (dx→0). For exp(ipx), a p impulse may occur within hbar/p with different probabilities, so one cannot restrict the measurement of p to a tiny dx unless dx is of the order of hbar/p. Nevertheless energy exists at all x points. Given that p measurements occur through specific impulses at different x points with exp(ipx) probabilities, one must calculate an average energy using {Sum over p pp/2m a(p) exp(ipx)} / {Sum over p a(p)exp(ipx)} using the probability associated with the measurement of p in space i.e. exp(ipx). This adds to V(x) to yield a constant E at each x point. We suggest that weighted exp(ipx)s are used to calculate average energy at each x, because it is through impulse measurements that one determines p at different x values and hence the average energy. One may note that one does not calculate {Sum over p a(p)exp(ipx) p } / {Sum over p a(p)exp(ipx)}. An impulse hit is associated with a certain probability and this measurement in x yields information about the corresponding energy pp/2m. Each impulse hit also yields information about a time probability exp(-i ppt/2m), but only if the particle is knocked out of the bound system i.e. only if pp/2m is the only energy present, which is not the case for a single particle bound system.
Classical mechanics contains a well-known dichotomy, namely that of work: Integral Fdx = change in KE and impulse, Integral Fdt = change in p (momentum). The former view is concerned with time and velocity, while the latter is not as p=m1v1=m2v2, i.e. the same impulse may occur for two different velocities. When solving classical mechanical problems, however, the time view (conservation of energy equation, Newton’s second law) is utilized and the impulse picture, usually ignored. We suggest that in quantum mechanics, one must retain both dichotomies as they are linked to each other as they are in classical physics. In fact, we argue that in quantum mechanics interactions are based on the impulse picture, which in some problems, such as one dimensional photon reflection-refraction, make it the only scenario which solves the problem. (Actually the impulse scenario makes use of flux values and so is intertwined with the time picture.) In the literature (1) etc, the continuity equation: d/dt partial density = d/dx { velocity(x) density(x,t) } seems to be a common starting point for deriving the Schrodinger equation. We argue that this approach only considers the time-work-kinetic energy scenario a priori and ignores, a priori, the impulse approach to interactions. The impulse picture is a physical description of interactions and so both scenarios should be introduced together a priori. What is usually done is to consider only the time picture and then hypothesize on mathematical grounds that: density(x,t) = W*(x,t)W(x,t) instead of considering the wavefunction form as the representation of the impulse picture which governs interactions. In fact, the time-independent Schrodinger equation is a combination of both scenarios. The work-kinetic energy picture must appear because V(x) is used, but actual interactions occur via impulses and their unusual associated exp(ipx) probability.
It is sometimes stated in the literature (1) that quantum mechanics is based on conditional probabilities. If this is the case, one would expect such conditional probabilities to appear in a single particle bound state. A free quantum particle is associated with a probability to interact of exp(ipx) (linked to 2-slit interference, 1-D reflection-refraction at an n1-n2 junction etc), but the kinetic energy and momentum are pp/2m (nonrelativistic) and p at all x even though each x has a different phase according to exp(ipx). The momentum is p at x, but the probability for an impulse hit at x is not the same even though its modulus is. When different probabilities exp(ipx) add in an OR situation, this leads to a spatial pattern which no longer has a modulus of 1, i.e. translational invariance is broken. Such a pattern also occurs for a single particle bound state W(x)=Sum over p a(p)exp(ipx). W(x) can have 0 points, but the average kinetic energy -1/2m d/dx dW/dx / W is not 0 at such a point. One has, on average (a statistical calculation), no average interaction at such an x point. One might ask why this should have anything to do with average kinetic energy? If one writes a conservation law: KEave(x) + V(x) = 0, then one considers kinetic energy interacting with potential energy. If there is no interaction at some x point for a given p (not W(x) on average), then it should not be part of the conservation of energy at this point. One is only interested in kinetic energy which interacts with potential energy at each x point. Thus, a(p) exp(ipx) + a(-p) exp(-ipx) (with a(p)=a(-p)) may have a 0 point at some x and so it does not have an average interaction at such a point, i.e. pp/2m (a(p)exp(ipx) + a(-p)exp(-ipx)) = 0. As a result, in a bound state, one considers kinetic energy per p, -p multiplied by the average probability (p, -p). Different a(p)exp(ipx) + a(-p)exp(-ipx) (a(p)=a(-p)) weights in this average picture indicate the extent to which the particle interacts. This idea holds for a single p,-p pair, but one must normalize over all such pairs, leading to the idea of a conditional probability. Given that hbar/p must be “created physically” in space so to speak, one must have interaction probability which creates this physical situation even in cos(px)+ i sin(px). This leads to the idea of interaction probability being 0, positive, but <1, or even negative and it is the interaction which allows one to consider conservation of energy with a potential V(x) in the first place. As a result, one cannot include a full pp/2m if there is negative, 0 or only particle interaction. Thus, pp/2m is multiplied by 2 a(p)cos(px) /W(x) for a(p)=a(-p).