Classical mechanics samples space as being uniform i.e. each x point having the same weight. Thus the number of x states in a length L is proportional to L or in 3-space to volume V. The ln of this value then appears in classical entropy calculations. In the case of a particle moving in space the same idea holds, but this means the particle spends equal amounts of time in each dx region. Thus we argue this result of a constant probability in space is based on time sampling. Time sampling is the sampling done for an equilibrium gas. The system does not change on average in time, so it is as if one has a snapshot. Thus one has a constant probability in a length L in time sampling. One may argue that this constant is information so ln(probability(x))=constant1 or P(x)=constant 2. The relativistic/nonrelativistic action may be written as A = -Et+px if one uses v=x/t. This Lorentz invariant form allows one to clearly separate t and x with E and p as constants and may also be considered to be information. In such a case, if one ignores time, then px is the spatial contribution to A and this does represent a constant probability in space. Rather there is scaling of space by p. If x=constant/p, then each p has its own unique length, called a wavelength, in which space sampling is the same i.e. creates the same distribution form, but with 1/p as the wavelength. This we argue is space sampling of a spatial distribution and has physical consequences (i.e. two slit interference, measuring small regions etc). One may note that the wavelength=hbar/p depends on the magnitude of p. In actuality, p has a direction as this is space sampling. If px is considered to be information, then ln(probability(x))= px or probability x = exp(ipx). To obtain time sampling, one needs to freeze out motion i.e. exp(ipx)exp(-ipx) (AND situation). Thus the manner in which one samples an underlying distribution may give rise to a different distribution. Arguing that a particle spends the same amount of time in each dx region and so the spatial probability is constant completely ignores the form px from the classical action. exp(ipx) is associated with a wavelength, thus x samples are within different lengths for different p’s. This suggests an associated entropy which is not additive because one loses information when adding p1+p2. Probability exp(ip1x)exp(ip2x) = exp(i (p1+p2)x), however, is conserved as is the overall impulse p1+p2 which depends only on the net momentum, whereas p1 entropy depends on a specific value.
Classical mechanics samples space as being uniform i.e. each x point having the same weight. Thus the number of x states in a length L is proportional to L or in 3-space to volume V. The ln of this value then appears in classical entropy calculations. In the case of a particle moving in space the same idea holds, but this means the particle spends equal amounts of time in each dx region. Thus we argue this result of a constant probability in space is based on time sampling. Time sampling is the sampling done for an equilibrium gas. The system does not change on average in time, so it is as if one has a snapshot. Thus one has a constant probability in a length L in time sampling. One may argue that this constant is information so ln(probability(x))=constant1 or P(x)=constant 2. The relativistic/nonrelativistic action may be written as A = -Et+px if one uses v=x/t. This Lorentz invariant form allows one to clearly separate t and x with E and p as constants and may also be considered to be information. In such a case, if one ignores time, then px is the spatial contribution to A and this does represent a constant probability in space. Rather there is scaling of space by p. If x=constant/p, then each p has its own unique length, called a wavelength, in which space sampling is the same i.e. creates the same distribution form, but with 1/p as the wavelength. This we argue is space sampling of a spatial distribution and has physical consequences (i.e. two slit interference, measuring small regions etc). One may note that the wavelength=hbar/p depends on the magnitude of p. In actuality, p has a direction as this is space sampling. If px is considered to be information, then ln(probability(x))= px or probability x = exp(ipx). To obtain time sampling, one needs to freeze out motion i.e. exp(ipx)exp(-ipx) (AND situation). Thus the manner in which one samples an underlying distribution may give rise to a different distribution. Arguing that a particle spends the same amount of time in each dx region and so the spatial probability is constant completely ignores the form px from the classical action. exp(ipx) is associated with a wavelength, thus x samples are within different lengths for different p’s. This suggests an associated entropy which is not additive because one loses information when adding p1+p2. Probability exp(ip1x)exp(ip2x) = exp(i (p1+p2)x), however, is conserved as is the overall impulse p1+p2 which depends only on the net momentum, whereas p1 entropy depends on a specific value.
In a previous note (1) we considered a free particle entropy based on the real part of the wavefunction exp(ipx) i.e. |cos(px)|. |cos(px|) may be normalized over a certain length, just as exp(ipx) is. The idea is that there exists an uncertainty in space within a wavelength which should be reflected in an entropy expression. As p becomes very large, cos(px) becomes very narrow and has many peaks in L. Thus probability →0 so p ln(p)--> 0 using l’Hopital’s rule. This coincides with a free classical particle having zero entropy. In this note, we revisit arguments presented previously, suggesting that A= -Et+px, a Lorentz invariant and also the free particle classical relativistic and nonrelativistic action for v=x/t We suggested in the past that A is responsible for both the wavelength hbar/p and the time period hbar/E. Here we suggest uncertainty within a time period should be considered together with uncertainty within a wavelength. For instance, recent experiments (2) have shown that one may have quantum interference in time as well as space. Thus there is uncertainty for a free particle in both space and time. We add the entropies for these two. Next, we consider the case of a bound system. Such a system creates a crest-trough pattern (with different heights and widths) for its spatial wavefunction, but E, average energy is constant and there is the free particle type exp(-iEnt). We suggest that the system is characterized by uncertainty in space and time with En being a known constant at each x. This suggests that various free particle p (momentum) values combine to create the wavefunction such that -1/2m d/dx d/dx W/W is average kinetic energy which in turn adds to V(x) to create a constant En. We are thus concerned with uncertainty in time and space and not in momentum. In the limit of high energy values, a crest fits within a tiny dx and prms is essentially the classical p value which is considered certain, even though in quantum mechanics the range of a(p) weights hold for all n (W(x)=Sum over p a(p)exp(ipx)). Thus we treat a high n limit as being completely certain and not having entropy. This differs from the approach of using C/v(x), where v(x) is velocity given by .5mv(x)v(x)+V(x)=En, as a probability and calculating Shannon’s entropy using this.
The uncertainty principle (linked to p,x or E,t) plays an important role in quantum mechanics. For example, an electron with an exact momentum p is described by a wavefunction exp(ipx). This function exists at all x points and so it is suggested in the literature that there is complete uncertainty as to the electron’s position. (More specifically P(x)=exp(-ipx)exp(ipx) normalization constant=constant and so all x points have the same probability.) Thus if p is completely certain, it is said that position x is completely unknown. We note, however, that exp(ipx) consists of cos(px) and sin(px) which are repeated probability distributions (with zero endpoints). Thus even though they exist in all of space, due to repetition which in turn is due to spatial invariance, there is a specific length number, namely the wavelength. Thus we argue that in classical physics there is a set of points (x,p), but in quantum mechanics ( p , x with its dx interval), with the latter being equal to hbar/p. Then the only uncertainty is within the dx interval i.e. the wavelength and spatial density is linked to momentum i.e. there is dynamical probability. In particular, there is a frequency hbar/E and a wavelength hbar/p which define an internal clock and ruler and the two are independent. Thus within a wavelength one has a probabilistic picture (in order to have endpoints defining the dx interval) and no sense of time, yet on average time exists because of hbar/E and the particle moves in space with a constant v. This begs the question: How do time and space x arise in such a scenario? We speculate the following. Assume there exists a property called energy which defines an object. This object, however, may break into two different objects. As a result, one introduces a parameter called time to distinguish between the two systems (E) and (E1,E2). For a certain time range one has a single object, while for a later time interval, one has two objects. Each of the two objects in the second system, however, needs to be distinguished as well and so another parameter, namely space x is used for this. If the two objects existed at the same x and t, how would one know there were two unless another parameter were introduced? We use t and x as the distinguishing parameters and furthermore give these a measure i.e. dt and dx. We then argue that from these ideas one may show the notion of speed and direction (because there are two objects) as well as conservation of momentum arises. Given the notion of an interval, we argue it must be described by a probability distribution and we link this to constant/p and exp(ipx). We finally note that since spatial density is linked to dynamical probability, one must consider motion associated with a density which may lead even to collective motion.
Classically there seems to be the sense of an isotropy in space and time and so statistically this becomes linked with degeneracies, say in a classical gas in space with no potential. Quantum mechanics of a free particle, however, suggests that different momenta, which represent physically different impulses in space, are not associated locally with isotropy in space or time. There is a wavelength in space and period in time (hbar/p, hbar/E where E=pp/2m nonrelativistically) and probabilities apply to these regions. We have argued that in general one might associate an entropy in both space and time due to the probabilities within a wavelength/period although generally in the literature this does not seem to be done. We have also argued previously that the quantum free particle wavelength and period follow from degeneracy associated with the Lorentz invariant: A = -Et+px. Here we point out that there is an added degeneracy, namely the direction of p in space. Thus the probabilistic exp(ipx) associated with a constant momentum can be further associated with various probability distributions in space if the free particle interacts with an n-slit system. In such a case, the particle has a fixed p and direction approaching the slit, but locally there is no spatial isotropy. This local probabilistic feature converts into a probability distribution for different p vector directions creating a unique spatial distribution on a screen when many identical particles, each with the same p, interact with the n-slit system. Each particle, however, begins with a specific p vector and after the interaction has the same magnitude of p, only a different direction. The local anisotropy associated with the wavelength is now converted into a more global spatial distribution due to extra uncertainties associated with impulse hits with the n-slit system. As a result, what one sees as entropy due to an interaction with a n-slit system is associated with degeneracy and the bias to believing that a free particle behaves in a deterministic manner at length scales of the order of a wavelength.
Quantum mechanics is based on two different probability schemes, namely W(x) =wavefunction and W*(x)W*(x)=P(x), with one being linked to the other. Why should there be two probability schemes? In (0) we argued that there are two spatial behaviour schemes because an impulse p (momentum) is automatically associated with two spatial distributions cos(px) and sin(px) in exp(ipx), yet a particle with constant p moves as x=p/m t. Thus if one is only concerned with motion, each x point has the same weight, and this is physically measurable as a photon or electron travels a certain distance in a certain time in free space. On the other hand, an impulse interaction is associated with exp(ipx) is a completely different spatial picture, and this too is measurable as in a 2-slit experiment. Thus it seems the two schemes are measured differently. We argue that this is an important issue because if there are two x-probability schemes there should be different entropies associated with each. Ttraditionally (1), one sees two quantum entropies: - W*(x)W(x) ln {W*(x)W(x)} and - a*(p)a(p) ln{ a*(p)a(p) } where W(x) = Sum over p a(p)exp(ipx). These are associated with the x=p/m t scheme or x-dependence based on motion or average motion. W*(x)W(x) represents spatial density which should be measured presumably by some sort of scattering, but we ask whether some information is lost in this approach? In particular, we note that a wavefunction may be obtained for a bound state. In the high energy limit, the envelope over the density W*(x)W(x) is said (2) to match C/v(x) from classical physics where .5mv(x)v(x) + V(x) = E. C/v(x), however, is proportional to delta t (with delta x being constant) and has nothing to do with the strength of an impulse hit. In the classical case, there is a single v(x) and so a single p(x) in each delta x thus no impulse entropy within a delta x i.e. at a point x. In quantum mechanics, however, we argue that a constant p automatically has an x probability distribution exp(ipx) associated with it and there might be an entropy linked to this.This entropy is associated with a probability to receive a p impulse at x and is so not linked to interference while the W*(x)W(x) is. In (3) we introduced the impulse or wavelength entropy (for a free particle |cos(px)|/C ln{| cos(px)|/C},, but here we wish to investigate its properties more closely.
In Part 1, we argued that P(x)dx=dx/L (where L is an arbitrary length) applies to a particle at rest. If one views such a particle from a frame moving at constant -v, it is moving. We argued that in such a case, the relevant probability is not simply linked to the position x, but also to the momentum p at x because this delivers an impulse hit. In particular, probability is presumably to be used in a physical problem with interactions and probability and a dynamical particle requires a dynamical probability describing both x,t and E,p. This led to exp(-iEt+px) which actually predicts intervals in space hbar/p and time hbar/E. In other words, a probabilistic view point leads to these results. Here, we ask the question: What if one simply wishes to use deterministic physics to describe a moving particle? After all, Newtonian mechanics is deterministic and one does not really need to focus on P(x)=1/L or its modification for a moving particle. In such a case, a particle moving with constant v (starting at x=0, t=0) is described by x/t=v both in Newtonian mechanics and special relativity. There is no uncertainty in x and t here which begs the question: Why does it appear in the probabilistic formalism of Part 1? In particular, exp(-iEt+ipx) derived in Part 1 implies physical regions: dx = hbar/p and dt=hbar/E and these seem to be absent in x/t=v and this is a problem. We suggest that a probabilistic approach is buried in the Lagrangian formalism which leads to the result d/dt v = 0, i.e describes the particle’s motion in space. This motion is actually x/t=v, but this equation only presents a subset of the description of a particle that is moving with constant speed, even in a deterministic Lagrangian/Action approach. We have already pointed this out in (1). We briefly explain it again here, but the real question is why does one need to consider the Lagrangian in order to describe x/t=v? In other words, the probabilistic approach associated with v=x/t only emerges in the Lagrangian formalism using a relativistic action Lt = -Et+px, with x/t=v. Ignoring the action and Lagrangian leads to an absence of hbar/p and hbar/E and one is only concerned with x/t=v. We note that the Lagrangian is primarily linked to the result: p = dL/dv partial and so argue that one is not simply interested in motion (x/t=v), but also in interaction because p is associated with an impulse hit. Thus, the Lagrangian approach is more than simply a mechanism which describes x,t motion, it is associated with p and E (through Hamiltonian = E = pv-L.) Using the Lagrangian approach suggests that it does not suffice to consider x and t by themselves when p and E are present. One must consider all four variables x,t,E,p which is the same point made in Part 1. The physics of the problem dictates that it is not enough to simply think in terms of x,t,v. The Lagrangian approach, which is constructed to ultimately yield x/t=v, when written in terms of E, p, i.e. Lt = -Et+px demonstrates uncertainty in x, hbar/p and t, hbar/E as shown in (1). This uncertainty must then be written in terms of a probability which is Lorentz invariant and leads to exp(-iEt+ipx). The point is that one would not even consider a probability if one did not consider E,p and x,t together as a complete treatment of a moving particle. In other words, x/t=v is an incomplete description of the state of a particle moving with constant speed, even in a deterministic framework.. At first this might seem surprising, but it is well known that energy and p are linked to interactions. In special relativity, one does not consider x,t,v alone and then compute p and E from these, but considers four vectors and Lorentz invariant equations. It is possible to create a Lorentz invariant equation involving all four state variables x,t,E,p and it turns out that this is the relativistic Lagrangian which describes motion, so this approach is more general than simply stating x/t=v. Thus, delta x = hbar/p and delta t= hbar/E appear in both the probabilistic approach of Part 1 and the deterministic one of the Lagrangian used here.
In previous notes, we have argued that one may obtain exp(ipx) (the spatial free particle probability) from the maximization of P1(x) ln(P1(x)) + i px P1(x), and a similar expression for P2(t). This yields exp(-iEt+ipx). If one integrates i Integral px exp(ipx) dx from px=0 to 2*3.14/p, one obtains i 2*3.14/p as a spatial entropy and i2*3.14/E as a temporal one associated with the complex probability exp(-iEt+ipx). If one considers classical motion of a free particle in space, then there is no uncertainty in time or space and so one might create a classical level entropy of: -E dt + p dx, where dt=hbar/E and dx=hbar/p.This yields -Edt+pdx=0 and one might equate this with a classical entropy dS as done in (1). This overall classical entropy is different from the spatial and temporal wave entropies dx and dt. If one does not wish to work at the classical level, one might try to use the hbar/p, hbar/E entropies in a problem requiring exp(-iEt+ipx), such as 1-D reflection-refraction at an n1-n2 index of refraction junction (x=0). In such a case, it seems that one does not use entropies, but rather their reciprocals which indicate the amount of resolution information present, i.e. L/dx = Lp/hbar. As a result, conservation of E and p may be reformulated in terms of a balance of resolution information which is the reciprocal of dx, i..e p (in the spatial case). For example, a particle at rest may break into two pieces with p and -p. The resolution is proportional to p initial =0 and after one has p and -p (i.e. conservation) of momentum which may be argued to be equivalent to conservation of resolution information. Using the notion of a balance of the reciprocal of entropy (i.e. p which is linked to conservation of momentum), one might interpret: Ap - Bp = C p2 where p2=n2 p in terms of resolution and not momentum conservation, i.e. A-B = C p2/p where p2/p = n2 is the greater relative resolution of the p2 photon. The idea of both a wave entropy (both temporal hbar/E and hbar/p) and its reciprocal resolution information seem to arise as well an overall classical entropy -Edt+pdx = 0 if dt=hbar/E and hbar/hp = 0. dt=hbar/E and dx= hbar/p We have argued in previous notes that one may obtain: dt=hbar/E and dx=hbar/p ((1)) as independent dx and dt fluctuation from the Lorentz invariant. One may show that -Et+px is equivalent to the classical relativistic or nonrelativistic action Lt if Lt=.5movv t (nonrelativistic) or Lt = -mo sqrt(1-vv/cc) t (relatvistic) and x/t=v ((2)) The reason that one recasts Lt as -Et+px is to use relativistic variables. The notion of an action remaining unchanged under a fluctuation delta x(t) , i.e. at t one considers x1 instead to x=vt, leads to: 0 = -E delta t + p delta x = 0 which has a solution of ((1)). In (1), however, the notion of an overall entropy is introduced such that: dS = 0 = -E delta t + p delta x ((3)) Given that one really has a probabilistic scenario (free particle quantum mechanics) associated with ((1)), i.e. exp(-iEt+ipx) = probability which ensures ((1)) and also spatial and temporal invariance by the unit modulus (less information than exp(-iEt+ipx)), then one might consider the notion of an overall entropy.