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.
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.
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.
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.
Classical mechanics introduces both the concepts of work and impulse i.e.: Work= F dot dr and Impulse = Fdt= dp. As we have noted previously, there seems to be a tendency in classical mechanics to focus only on the work definition which is linked to energy and time. Thus one solves Newton’s equation to find x(t) or writes a conservation of energy equation. This approach seems to be carried over into thermodynamics as the first law is written as: dE=TdS - dW. We have argued in (1) that in the case of quantum mechanics, both the impulse-momentum picture and work-energy-time picture are needed. (In particular, the one dimensional description of a photon reflecting and refracting cannot be solved without the exp(ipx) impulse scenario.) Here we suggest that the notion of entropy should also be associated with impulses. For example, consider the classical case of a particle moving in a circle with a central inward force. The kinetic energy does not change and work is zero, but the momentum vector changes and so there is an impulse. Furthermore, information changes because one needs to continuously know the momentum coordinates. We suggest that this is an example of impulse being associated with entropy. A second example we suggest is a classical particle bouncing elastically off a wall. (One may argue, however, that these notions of entropy are hand-wavy.) In the quantum world, however, we suggest that impulse plays a major role in creating entropy which is clearly defined. We suggest the example of a particle or photon with a fixed momentum passing through a tiny slit, or n-slits. In each case, one obtains a different pattern on a screen which leads to a different entropy expression. The photon or particle does not change the magnitude of its momentum i.e. there is no energy change or work, but there are impulses which are different in the various n-slit scenarios and lead to different entropies. Thus we suggest that impulses and information associated with them may lead to an increase in entropy. This suggests that the first law of thermodynamics is not a complete description of changes in entropy as it bypasses the idea of impulses and only considers work/energy changes.
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 the literature, stochastic approaches employed to derive the Schrodinger equation seem to focus on the idea of a classical spatial density which depends on time and space ie. density(x,t). As a result, d/dt (partial) density appears in various equations such as the continuity and Focker Planck equations. At one point in the derivation the quantum wavefunction is introduced “mathematically” as W(x,t)=sqrt(density(x,t)) exp(i S(x,t)) where S is related to a velocity field through u(x,t)=velocity= d/dx S. From quantum mechanics, -Et becomes a term of S the velocity field. The stochastic/hydrodynamical approaches employ the term du/dt partial = d/dt (partial) d/dx S and then “pull out”” d/dx leaving d/dt (partial) S which yields -E average energy from the -Et term. In this note we argue that for a bound quantum state density(x) so d/dt (partial) density = 0, yet the statistical ideas of quantum mechanics should be present in such a case. Furthermore for the free particle wavefunction exp(ipx-iEt), density=1. In fact, the Schrodinger equation should follow for a free particle even the case of density=1. Stochastic processes seem to be based on equations which consider a change in classical spatial density at a point x in time due to flow in or out or stochastic changes. Quantum processes seem to deal with periodic probability changes from one dimension to another exp(ipx) (2 dimensions) which are somehow related to a physical process of momentum flow. Ultimately one links this process to an energy conservation which does not seem to be the same as a classical stochastic/hydrodynamical approach. In this note we argue that a free particle with a constant momentum p has periodic stochastic processes associated with it through its complex probability change in the region of its wavelength, but that these are linked to creating the momentum p. Furthermore momentum is linked to energy so we argue that the Schrodinger equation is a stochastic equation linking energy and momentum and that time and space are separated i.e. there is no situation of following a change in spatial density in time for a free particle with momentum p. For a bound state, one has a distribution of free particles, but again time is removed from the problem in the sense that it only appears in exp(-iEt) in the wavefunction to again establish a relationship between energy and average momentum and potential. There is again no nonzero d/dt (partial) density term which seems to be key in stochastic equations. Only if one takes a linear combination of energy eigenstates Sum over En an(En) exp(-iEnt) Wn(x) does one obtain the form density(x,t), but we argue that quantum features should already appear in exp(ipx) and the bound state scenario with a zero d/dt (partial) density(x,t). We argue, however, that the Schrodinger equation and fundamental stochastic ideas of quantum mechanics should already be present at the level of exp(ipx).