Historically Newtonian mechanics appeared before the statistical mechanical treatment of a Maxwell-Boltzmann (MB) gas. In this note, however, we initially investigate a MB gas in terms of information which we define as -ln(density(x)). In equilibrium there are no macroscopic changes in time so there is only one variable namely x in one dimension. The density may be changed so it is variable while a potential V(x) is not. Thus we argue that there should be a fixed piece of information based on x in the system i.e. V(x). We then argue (as in (1)) that equilibrium involves minimizing the information so we suggest: -ln(density(x)) = C V(x). This ultimately leads to the form d(x) = C1 exp(-V(x)/T). Next we focus on a Newtonian system, but unlike the MB gas two variables x and t are needed to describe the macroscopic changes for the particle even though x suffices to describe V(x). {In particular one may send in a particle with an initial speed at xo and it will move through x,t in a certain manner. Sending another particle with the same initial speed at xo at a later time leads to the same motion. Thus we argue there is a given V(x) function in the system. } We also suggest there should exist a density d(x) and an information given by -ln(d(x)). One may see from (2) that a Newtonian system may be described by a density(x) even though it is usually only described by x(t), velocity, acceleration etc. The density is linked to a variable quantity of “information” in the system. Given the presence of V(x) we divide the x axis into equally spaced dx units. Thus d/dx the translation generator applies to V(x). For the moving object, however, one has a corresponding dt(x) for each equal dx and the dt’s are usually not the same. In other words the ratio dx/dt = function(x), but it is dt which changes. We place dt in the denominator so that it mimics the time translational generator d/dt. One cannot simply equate function(x) and C V(x) because function(x) does not change as d/dx, rather it changes because dt values change. Thus we suggest using d/dt to express its changes. As a result we propose: d/dt (dx/dt) = C dV/dx ((1)) as both sides of the equation must be vectors. Now d/dt= dx/dt d/dx so both sides of ((1)) may be expressed as functions of x. One then obtains function(x) d/dx function(x) = C dV/dx where function(x) = dx/dt. We call this v(x). v(x) is adjustable so we equate it with information i.e. -ln(d(x)) where d(x) is density as there is no reason to have two different functions of x. Thus d(x)=C/v(x). In other words density(x) is proportional to the “information” dt at each x. In (2) a density is ascribed to a Newtonian system using dt as the definition of density. Here we wish to link density to information= -ln(d(x)). As a result one finds that ((1)) involves information squared unlike the MB gas which involves information -ln(d(x)) to the first power. As a result: d/dx ( v(x)v(x)) = C dV/dx which may be integrated to yield Newton’s conservation of energy: .5v(x)v(x)+ C V(x) = constant. In other words we suggest it may be possible to obtain Newton’s second law using ideas of information. There is further loss of information as the constant applies to all x points so the the information from .5v(x)v(x) combines with CV(x) to eliminate x information. (x information, however, is present in each term i.e. .5v(x)v(x) and CV(x)).
Historically Newtonian mechanics appeared before the statistical mechanical treatment of a Maxwell-Boltzmann (MB) gas. In this note, however, we initially investigate a MB gas in terms of information which we define as -ln(density(x)). In equilibrium there are no macroscopic changes in time so there is only one variable namely x in one dimension. The density may be changed so it is variable while a potential V(x) is not. Thus we argue that there should be a fixed piece of information based on x in the system i.e. V(x). We then argue (as in (1)) that equilibrium involves minimizing the information so we suggest: -ln(density(x)) = C V(x). This ultimately leads to the form d(x) = C1 exp(-V(x)/T). Next we focus on a Newtonian system, but unlike the MB gas two variables x and t are needed to describe the macroscopic changes for the particle even though x suffices to describe V(x). {In particular one may send in a particle with an initial speed at xo and it will move through x,t in a certain manner. Sending another particle with the same initial speed at xo at a later time leads to the same motion. Thus we argue there is a given V(x) function in the system. } We also suggest there should exist a density d(x) and an information given by -ln(d(x)). One may see from (2) that a Newtonian system may be described by a density(x) even though it is usually only described by x(t), velocity, acceleration etc. The density is linked to a variable quantity of “information” in the system. Given the presence of V(x) we divide the x axis into equally spaced dx units. Thus d/dx the translation generator applies to V(x). For the moving object, however, one has a corresponding dt(x) for each equal dx and the dt’s are usually not the same. In other words the ratio dx/dt = function(x), but it is dt which changes. We place dt in the denominator so that it mimics the time translational generator d/dt. One cannot simply equate function(x) and C V(x) because function(x) does not change as d/dx, rather it changes because dt values change. Thus we suggest using d/dt to express its changes. As a result we propose: d/dt (dx/dt) = C dV/dx ((1)) as both sides of the equation must be vectors. Now d/dt= dx/dt d/dx so both sides of ((1)) may be expressed as functions of x. One then obtains function(x) d/dx function(x) = C dV/dx where function(x) = dx/dt. We call this v(x). v(x) is adjustable so we equate it with information i.e. -ln(d(x)) where d(x) is density as there is no reason to have two different functions of x. Thus d(x)=C/v(x). In other words density(x) is proportional to the “information” dt at each x. In (2) a density is ascribed to a Newtonian system using dt as the definition of density. Here we wish to link density to information= -ln(d(x)). As a result one finds that ((1)) involves information squared unlike the MB gas which involves information -ln(d(x)) to the first power. As a result: d/dx ( v(x)v(x)) = C dV/dx which may be integrated to yield Newton’s conservation of energy: .5v(x)v(x)+ C V(x) = constant. In other words we suggest it may be possible to obtain Newton’s second law using ideas of information. There is further loss of information as the constant applies to all x points so the the information from .5v(x)v(x) combines with CV(x) to eliminate x information. (x information, however, is present in each term i.e. .5v(x)v(x) and CV(x)).
In (1) we argued that one may derive Newton’s second law (which is equivalent to an energy conservation equation) by using information arguments. We argued that V(x), a potential, is a fixed piece of spatial information linked to the x axis being divided into equal dx units. The variable spatial information is dt(x) i.e. the time spent in each dx or v(x) =dx/dt (to put this in the same form as the time displacement generator). One may not simply compare v(x) directly with V(x) because they change differently. v(x) changes in a weighted way i.e. v(x)d/dx or d/dt while V(x) changes as d/dx V(x). Thus we suggest d/dt v(x) = C V(x) which ultimately leads to Newton’s second law. Using d/dt → v(x)d/dx and integrating yields conservation of energy: p(x)p(x)/2m + V(x) = E. A similar argument was used in (2) to deal with the special relativistic case. There one cannot use d/dt → v(x)d/dx to obtain a conservation of energy equation. The point we wish to make is that for both the nonrelativistic and relativistic classical equations there are two types of “information” equations. The first is a “force” i.e. -dV/dx equation which is based on how V(x) changes with x. The LHS of this equation is linked to d/dt p(x). The second is a corresponding energy conservation equation. The quantum case is quite different, we argue, because a free particle is described by exp(-iEt+ipx). There is still the notion of p=mo dx/dt(x) as in the nonrelativistic case because p must be defined, but exp(ipx) is additional imposed piece of spatial information. -id/dx exp(ipx) = p exp(ipx) and -d/dx d/dx exp(ipx) = pp exp(ipx). This begs the question: How does one reconcile the two? For the nonrelativistic Newtonian and relativistic cases it is possible to have two equations (one with Force the other with energy) using one x dependent variable p(x). In the quantum case one may take an average to define a kind of complex = Sum over p a(p) p exp(ipx)/ Sum over p a(p)exp(ipx, but this is complex, not real. Furthermore both +/- is included in so there seems to be a problem linking it with changes in V(x) in one direction. Thus it seems that one is forced to consider the second kind of information equation from classical mechanics and define a <pp> = Sum over p a(p) pp exp(ipx)/ {Sum over p a(p)exp(ipx)}. In the nonrelativsitic case this may be multiplied by 1/2mo and then added to V(x) to obtain a constant E. If one only looks at <pp>/2mo as a spatial function then this is identical to KE(x) i.e. kinetic energy(x) for a classical system. Thus in the nonrelativstic case, Newtonian energy conservation is always embedded in the quantum system regardless of the energy. In other words the quantum bound equation i.e. the time independent Schrodinger is not obtained independently from the classical one in terms of information.One may ask: How does information related to exp(ipx) affect the quantum system? Two immediate ways are as follows. First <pp>/2mo = -1/2mo d/dx dW/dx / W where W= Sum over p a(p)exp(ipx). This leads to a differential equation which has discrete En values verified by experiment. Secondly W is linked to spatial density W*(x)W(x) which exhibits crests and near zero regions. Using <pp> suggests that p(x) in the classical case is linked to prms(x) quantum. Thus building quantum mechanics using “informational” ideas seems to involve embedding classical conservation of energy into the quantum equation a priori. This holds both nonrelativstically or relativistically (Klein-Gordon equation). This immediately raises the question: If classical mechanics (in the form of energy conservation) is embedded in quantum mechanics for any energy, why do people seek to take some kind of limit (3) to obtain classical mechanics from quantum? We argue that quantum mechanics has more spatial information than classical, but retains classical conservation of energy so there is no limit to take. One may simply ignore ideas related to the wavelength to obtain classical mechanics, we argue.
Newton’s second dp/dt = Force where p is momentum is equivalent to pp/2m + V(x) =E for a conservative force= -dV/dx with p=mo v(x), v being velocity. In general one may divide the x axis into equally spaced dx units with dt(x) (i.e. time spent) differing in each unit. A main point we wish to make is that dt(x) is the same whether the particle moves to the right or to left. Thus using dt(x) as a kind of density (proportional to C/v(x)) as done in (1) means this spatial density does not distinguish between motion to the right or left. This is one kind of spatial density we argue, one that considers only the magnitude of velocity. Heuristically, however, one might argue that motion to the right or left involves a skewed spatial density in the direction of motion. This we argue represents a second kind of spatial density which follows from Newton’s first law and is associated with quantum mechanical motion. Ideally the two should be linked. Newton’s first law indicates that a particle at rest remains at rest and one in constant motion in a direction continues in this state unless acted on by force. This law is usually not associated with a mathematical equation and so a spatial density is not derived. In (2), however, we argue that Newton’s first law is associated with invariance in a spatial direction if there is no force in that direction, for example x. In such a case, d/dx is the translation generator in that direction and momentum is kind of “information”. We argued in (2) that one may write d/dx f(x) = p to describe this with the simplest solution being f(x)= px for constant p. We further argued that this is in fact linked to Fermat’s least time principle for light. Time =distance/speed of light, but momentum and displacement are parallel. Thus for a two dimensional problem: p dot r = (px)x + (py)y = p distance = p sqrt(xx+YY) and p sqrt( (L-x)(L-x) + YY) for two rays, one being the incident and the other the reflected. Given that there is conservation of momentum in the x direction one may write (px) incident = (px) reflected or d/dx (p distance incident) - d/dx (p distance reflected) = 0 which is Fermat’s principle for reflection with one index of refraction. For refraction p1=pn1 and p2=pn2, but (p1x) = (p2x). f(x)=px satisfies d/dx f(x) = p. This expression depends on x, but also on the direction of motion i.e. one may have p or -p. In (2) we further argued that this expression may be written in terms of: i px = ln(exp(ipx)) with exp(ipx) being periodic, but neither growing or tending to a constant zero for a constant p. As a result we identified exp(ipx) with a second type of spatial density or probability, one which accounts for the direction of motion. exp(ipx) is also associated with a quantum mechanical description of a free particle. Thus Newton’s first two laws give rise to two different spatial densities, but they are related. In particular, the modulus of exp(ipx) equals constant which is P(x) or spatial density from Newton’s second law. In this note, we start with the assumption that a velocity (not velocity magnitude) spatial density should exist and examine what properties of such a density/probability arise.
A Maxwell-Boltzmann gas with no external potential V(x) has a constant spatial density. The value of this spatial density alone does not yield the pressure because this also depends on T through the ideal gas law Pressure = density(x) RT, where T is temperature (associated with a conserved average energy), but also the average single particle pressure. Taking d/density(x) of Pressure yields the single particle average pressure value RT, yet at the same time single particle pressure is a stochastic value evaluated as the average of constant v p where p is momentum and v velocity. The quadratic form of p arises, we argue, because one has more than one particle, thus v distinguishes between the particles and relative speed becomes important. In the case of a single particle (as in a single free quantum particle) moving at constant p (speed as well) there is again a constant spatial density as in the MB gas with no V(x). Given the value of the constant density 1/L (L=length), one has no information about momentum p. In the case of a single particle, impulse and not pressure is the important quantity, we argue, because there is no need for a v as there is no relative motion. One only needs to know impulse which is proportional to p. If one speculates that a free quantum particle may follow the model of pressure in an MB gas, one would require a function F(p,x) such that d/d something F = p where p is the single particle impulse. In the MB case, “something” is density, but this is not appropriate for a single particle as density consists of many particles. Without changing T, one may change pressure in an MB gas by changing density i.e. changing the x value of particles. For a quantum particle, we suggest that the conjugate variable to p is x. One may change x of the single particle. Furthermore an association between d/dx and p exists in Fermat’s Least Time Principle. T Thus we suggest d/dx F(p,x) = p where F is unknown both in form and meaning. Pressure, however, is a statistical quantity so F should also be a statistical quantity i.e. one could have d/dx P(p,x) / P(p/x) = constant p where P(p,x) is a kind of unnormalized probability. This suggests, however, that P(p,x) is an eigenfunction of d/dx and represents a kind of unnormalized probability. This, we argue, leads to the notion of exp(ipx) because one does not want a monotonically growing or decreasing probability in x for a constant p. The “constant” is then i. Thus, we argue that if one mimics the pressure behaviour of a MB gas, and applies its properties to a single particle moving at constant speed, one obtains an unusual unnormalized kind of probability exp(ipx) i.e. the quantum free particle wavefunction. At the same time, a constant spatial probability or density 1/L exists. The two are linked in that exp(ipx) exp(-ipx) =1. The overall idea is that the stochastic force i.e. the entropic pressure in an MB gas has an analogy even for a single particle with constant momentum.
In part I we argued that Newton’s first law implies that constant momentum (i.e. momentum conservation) is associated with spatial invariance i.e. a lack of information in a certain direction. This is equivalent to stating there exists no force in that direction. We argued that one may apply the generator for translation in the spatial direction i.e. d/dx for x and apply it to a function f(x) such that df(x)/dx = (px). The simplest form is f(x)= (px)x. We also argued that it is precisely this function which appears in Fermat’s least time principle because time = distance/speed and one may multiply time by |p| = momentum magnitude. Given that momentum and displacement are parallel vectors for a light ray, p dot r = p distance = (px)x + (py)y for a two dimensional problem (e.g. reflection or refraction). “Distance d” is the key function to use because dd= xx + yy implies that for fixed y, d d/dx = x/d = sin(theta1) (angle measured from y axis) so p d d/dx = (px) which is a conserved quantity if x is invariant. Thus conservation of momentum for two rays is equivalent to extremization. Extremization, we argue, suggests there may be a built-in physical mechanism which allows a free particle to correct itself if perturbed slightly just like a Maxwell-Boltzmann gas which extremizes entropy subject to constant average energy does. This built-in mechanism should be linked to motion in space. Writing (px)x as ln(exp(ipx), one not only obtains an eigenvalue equation i.e. -d/dx exp(i (px) x) = (px) exp(i px x), but the notion of a relative change arises if exp(i px x) is interpreted as a kind of probability. A relative change is natural because an absolute one ignores the idea that probability may differ with x even though (px) is constant. Thus from the idea of invariance in space found in Newton’s first law, which is equivalent to conservation of momentum in that direction, the equation d/dx f(x) = (px) leads directly to -id/dx exp(i px x) = (px) exp(ipx x) where exp(i px x) represents a physical probability type of wave which stablilizes the constant (px) motion because this motion is equivalent to extremizing time or |p| time or p dot r / speed of light in Fermat’s principle. In other words, the quantum mechanics of a free particle is associated with stabilizing wave behaviour exp(ipx x) which was not seen experimentally at the time of Newton or Fermat, but seems to be part of their formalisms, with Fermat’s principle derivable from Newton’s first law.
Newtonian force is based on a change in momentum in time, but momentum is a vector and so may change in magnitude and direction. In many cases, the change is of one form or the other. For example, for a bound particle the change is in kinetic energy i.e. in magnitude, while for a particle moving at constant speed in a circle or pressure in a Maxwell-Boltzmann gas, the change is in direction. For an MB gas, the probability for p and -p are the same so the average momentum at any point x is 0 and thus chunks of the gas do not move, even though a single particle in the gas moves. The pressure, on the other hand, is the average of 2pv (2p for impulse, v for flux) and so pressure for p and -p do not cancel. A quantum free particle is based on exp(ipx) which does not contain time and is a dynamical probability, This is both an eigenfunction of the momentum operator -id/dx and the kinetic energy operator -1/2m d/dx d/dx, yet as pointed out in Part II, there are two pieces (cos(px),sin(px)) with a spatial density and hence a kind of pressure/force. These, however, combine to give the sense of constant motion in one direction. We argue this is due to a dynamical probability, unlike a static balance of pressures/forces found in an MB gas or Newtonian statics. There is a density with pressure distribution cos(px), but this is continuously changing because the particle moves in one direction with constant p. Thus the two pressure distributions “balance” so to speak to create a constant p. Interference of exp(ipx) and say exp(-ipx) may “bring out” a “static” spatial density i.e. 2cos(px), but behind the scene the dynamical probabilities still exist. It is these which “hold” the cos(px) together which appears strange from a static point of view. For example, a quantum bound state has a static W*(x)W(x) spatial density at the same time that the single particle has an average classical kinetic energy KE(x)=En-V(x). Interference also gives rise to the notion of an impulse-pressure in quantum as one ignores the isin(px) and i sin(-px) terms. Creating a pressure scenario with p and -p leads to two considerations. First, there is an impulse-pressure (defined without flux) at each x point and secondly, this impulse-pressure changes with x. The time-independent Schrodinger equation is formulated in terms of energy (i.e. a scalar): KE(x)+V(x)=En where KE(x)= {-1/2m d/dx dW/dx} \ W and W=Sum over p a(p)exp(ipx). Thus, as in Newtonian mechanics, one thinks of -dV/dx solely accelerating a particle. The quantum particle however, is described by an ensemble W(x) which has pressure. This pressure is due to a combination of the form exp(ipx) and V(x), so V(x) does not simply change root mean square speed. exp(ipx), by itself, creates pressure when interference occurs as in a two slit scenario. We try to examine these ideas in this note.
It is well known that a Lagrangian (usually kinetic energy - potential) may be varied to produce Newton’s second law of motion. In (1) it is noted that kinetic energy is proportional to a Fisher’s information of the form b Integral dt dx/dt dx/dt (b=constant) and so in a sense Fisher’s information may be varied together with another piece i.e. -V(x). In (2), it is noted that Fisher’s information, this time with a probability P(x), i.e. b Integral dx P(x) {d/dx ln(P) } {d/dx lnP) } may be varied (with constraints) to yield the Schrodinger equation. (One may note that the two definitions of Fisher’s information are the same if P(x)=x(t)x(t) in the first case.) Here we consider the notion of impulse and pressure which depends on p. We argue that a free quantum particle with constant p undergoes x=p/m t on average, but is associated with a probability distribution exp(ipx) which contains uncertainty in x (within a wavelength) and no time. P, however, represents an impulse which the particle may impart. In the case of potential V(x) one may consider a linear (interfering) sum of exp(ipx)’s i.e. W(x)= wavefunction = Sum over p a(p)exp(ipx). This leads to an average impulse of -idW/dx / W. Classically pressure is created by multiplying impulse by velocity (flux). If time does not exist within a wavelength then there is no time within it. There is, however, still an external clock and p ave in a sense should correspond to an m v(ave) nonrelativistically. Thus WW (bound system) is in a sense like a pressure. One may then suggest that for an equilibrium scenario one extremizes this pressure with respect to constraints (EWW and V(x)W). The variation is with respect to W(x), the wavefunction. Given that a bound particle maps to an rms momentum such that KE(x) = prms(x)prms(x)/2m where prms(x) is the classical value, then at the classical level prms(x) vrms(x) should be pressure and should also satisfy an extremization equation. In fact, the extremization equation using WW should be very closely related to it. One is in fact the d/dx of the other. The classical notion of pressure, proportional to vv, may be extremized subject to constraints. This leads to Newton’s second law which contains d/dt explicitly. Changing to “x” concepts yields v(x)d/dx which may be integrated to yield an energy conservation equation matching the quantum one. Thus we try to think of extremization of Fisher’s information i.e. EPI (extreme physical information) in terms of pressure, in both the classical and quantum cases.