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.
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.
Classical mechanics defines the concepts of work = Integral dx F = change in kinetic energy and impulse= Integral dt F = change in momentum (vector). We argue that these two concepts are associated with different physical processes and contain different information. For example, a change in momentum is associated with an impulse e.g. the striking (and possibly breaking of a target) and there is a degeneracy in that p=m1v1=m2v2. Kinetic energy, on the other hand, describes a different physical process, namely the “climbing of a potential hill” and one may have different (mo, p) sets yielding the same kinetic energy, just as the same p (same impulse) may be associated with different kinetic energies. Thus there are two distinct physical processes (impulse and interchange of kinetic energy or energy) and each of these are associated with different information. In quantum mechanics, exp(-iEt) describes the time dependent part of a free particle wavefunction (special probability) while exp(ipx) describes spatial. We have argued in previous notes that this probability should be associated with entropy, but in that case there should be an entropy in time and one in space. Given that E and p differ numerically, one needs a reason for the existence of two different entropies. We argue that they are associated with different physical processes and information and so two different entropies may be possible. (We use nonrelativistic expressions here, but the same ideas apply to the special relativistic case.)
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 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.
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 classical mechanics, one assumes one may measure precise momentum and x positions. One may argue, however, that a measurement involves changing the momentum of a particle (i.e. an impulse hit) and that such a change does not occur either in an instant of time or at a point x. For example, a classical particle in a potential V(x) changes its velocity in a region dx, but at the same time is said to have a velocity of v(x) to leading order as dx shrinks to zero. One may argue that dx does not shrink to zero and may not be tiny with respect to an overall length scale. Given that an interaction involves a particle and a potential or another particle, one may argue that a finite dx should be associated with both. Thus a particle with constant momentum should have a tiny dx associated with it, possibly a function of p, but at the same time there is a constant probability for a measurement to take place at any x point.Thus we argue that there are two sets of probability schemes, one for dx and another for a larger length scale and constant probability. This should point to two entropy schemes, each associated with the type of measurement done and its analysis. For example, a two slit system may be used to measure the interaction probability. This gives rise to an interference probability. At the same time, many two slit system may be used to measure the center of mass position of the particle, ignoring the interference pattern. This gives rise to P(x)=1/L and a completely different Shannon’s entropy. Given that there is a dx probability scheme for a constant p associated with interactions, an associated entropy should appear when such an interaction occurs, but this now involves breaking the constant p and considering the associated dx probabilities. Thus there is both momentum and spatial entropy combined as in two slit interference. In the case of a bound state, there are numerous exp(ipx) eigenstates (due to interactions with V(x)), suggesting both internal p and x entropies, but overall one has a single energy eigenstate. In terms of energy one has a fixed state (although exp(-iEn t) suggests there are frequency pockets in time pointing towards a kind of time entropy. If one considers only interactions which change a particle from one En to another, a particle in a single state at T=temperature=0 has entropy 0 (even though the state itself has momentum and spatial entropy). Thus it seems entropy is selective depending on the type of interactions present and what eigenvalues they change. In other words, classically one counts states when computing entropy, but in quantum mechanics one may count states by defining states in terms of quantum numbers. One may consider a bound state to be a single certain state with eigenvalue En and consider entropy being created if interactions (photon/phonon absorption/emission) break the single En. In such a case, one may ignore the breaking of exp(ipx)s within a bound state which implies momentum entropy as well as the spatial entropy linked with even a single exp(ipx). In other words, one may select various different kinds of probabilities linked with different eigenvalues and their measurements when calculating Shannon’s entropy.