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In (1) the high energy limit of the Wigner transform W1(p,x,n) is investigated for the quantum oscillator. The WKB semiclassical approximation is used and the high energy wavefunction approximated by: 1/ p(x) { exp(i p(x) x) + exp(-i p(x) x) } leading to the result of delta ( PP/2m+kXX/2 - E) where p(x)= sqrt(2m(E-.5kxx)), P,X are classical momentum and position and PP/2m +kXX/2 energy. In this note, we first observe that the WKB approximation may be used for any potential V(x) to find a similar result. We then focus on obtaining the result without using the WKB approximation, but instead utilize: H(p,x) ** W1(p,x,n) = En W1(p,x,n) (from (2)) where H(p,x) is the classical Hamiltonian, ** the Moyal star product involving exp(ihbar/2 Operator) and W(p,x,n) the Wigner function (as used in (2)). The main idea is that in the classical limit, hbar—>0 so exp(ihbar/2 Operator) —>1. Thus H(p,x)W1(p,x,n→large) = En W1n(p,x,n -> large). In (2) it is shown that Integral dp dx W1(p,x,n) = 1, so Integral H(p,x) W1(p,x,n->large) = E. As a result, W1(p,x,n->large) acts as a delta function enforcing pp/2m + V(x)=E.

The quantum oscillator ground state energy may be obtained by solving the time-independent Schrodinger equation and selecting the lowest energy eigenvalue. In (1) it is suggested that ground state oscillator energy appears in an semiclassical calculation of oscillator energy using the Moyal product (**) i.e. (p+iwx) ** (p-iwx) .5 = pp/2 + wwxx/2 + hbar w/2 for m=1. In this note, we try to show that the Moyal product: (p+iwx)/sqrt(2){ d/dx (left)d/dp (right) - d/dp (left) d/dx (right) } (p-iw)/sqrt(2) follows from the fact that Wo(x) and ao(p) where Wo is the ground state wavefunction and ao the ground state momentum wavefunction have the same functional form and that this form is in fact a Gaussian so that Wo(x)ao(p) = exp(-constant( pp/2 + wwxx/2). By using both, one may create a combined time-independent Schrodinger equation with both p and x present at the same time giving the appearance of a semiclassical approach. With respect to the ground state energy, d/dx d/dx acting on a Gaussian exp(-axx-bpp) first brings down 2ax followed by 2a from the second derivative (and similar results for d/dp d/dp). The other part of the second derivative is x dependent and cancels the x dependent potential. Thus one may think in terms of a single derivative d/dx acting on ax or a single derivative acting on bp. Combining in a symmetric form leads to the Moyal product acting on p+iwx and p-iwx to yield the ground state oscillator energy. We argue that this is essentially the same information as the time-independent Schrodinger equation with the special Gaussian form of the ground state wavefunction. Thus derivatives essentially act on pp/2 + wwxx/2 in a product of the x-wavefunction and p-wavefunction and this may be converted into a scheme of single derivative acting on the variable p+iwx and p-iwx because the relative piece of the d/dx d/dx of the Schrodinger equation is: d/dx d/dx exp(-axx) →d/dx 2ax exp(-axx) → 2a (linear in a) with the other term of the second derivative canceling with .5kxx exp(-axx).

In this note we consider, as in (1), finding an oscillator quantum state |z> (called the coherent state) such that yields a classical-like phase space probability distribution in p and x. In (1) the approach is to use the known weights of where Hamiltonian |n> = en |n>. Here we use instead the following ideas. First, we argue that for exp(-Hamiltonian/T) to be used at all, P(e1)P(e2)=P(e1+e2), but also P(z1*z1)P(z2*z2) = P(z1*z1+z2*z2) where z*z = pp/2+xx/2 with k=m=1. This yields a solution of exp(- b z*z) where b is unknown. In the limit of hbar w/T < as a linear superposition of the energy eigenfunctions |n> one may find a solution for |z>, the coherent state which has an average of (pp/2+xx/2)/ hbar w phonons/photons. P(z*z) is called the Husimi distribution.

The Maxwell-Boltzmann probability factor exp(-energy/T) / normalization may be applied to a quantum oscillator using energy levels given by hbar w (n+.5). This leads to the derivative in 1/T of a geometric series and yields an average energy: Eave = hbar w/2 + hbar w / (-1+exp(hbar w/T)). If one were to consider a semiclassical approach using p (momentum and x), energy would be described by pp/2 + xx/2 where for simplicity m=1 and k=1. This would appear in the MB factor with a temperature T and this expression would be expected to hold for high T or w/T small i.e. the quantum phonon jumps would be hardly noticeable compared to the average energy T in this limit. If one tries to write the classical energy, for w/T small, in the form of quantum energy, one has hbar w n in this limit, where n is the number of phonons (hbarw/2 is small in this limit) so: zz= {pp/2+kk/2}/w is the number of phonons which should be multiplied by hbar w/T. One may then ask: What function has the limit hbar w/T? A solution is: {1-exp(-hbarw/T)}. Thus the semiclassical normalized phase space probability: P(p,x) = {1-exp(-hbarw/T)}. Exp{ - zz {1-exp(-hbarw/T)} } yields the small hbar w/T limit quantum average energy. As a result, the average energy in this limit should match for both the classical phase space distribution and the quantum average energy expression. This classical distribution happens to be identical to the Husimi distribution. One may note (as is done in (1)), that {1-exp(-hbarw/T)} is the probability for the quantum oscillator particle to be in the ground state. In the limit of hbar w/T small, this probability is hbar w/T which is a measure of the effect of the quantum phonon leaps as well. Fisher’s information, as noted in (1), is equivalent to this ground state probability, but it is also the average energy which is proportional to the temperature type value, in this case {1-exp(-hbarw/T)}. This follows from the same kind of calculation done for a gas with particles with energy pp/2m, although instead of a factor of .5T one T because there is a contribution from pp/2 and xx/2.

Classical Newtonian mechanics separates a particle (effect) from a force (cause). A particle has properties at a given x (and time t), but if it interacts with a force, this interaction occurs over dt and dx. Given that many forces do not change in time, we argue that a notion of a force is inseparable from the idea of dx. If one gives the value of a force at x, this value acts within a dx, which in Newtonian mechanics tends to 0. We have argued before that a particle with momentum represents a force or impulse itself, but at the same time may receive impulse hits from other sources (e.g. a Newtonian force -dV/dx). If a particle represents a force(impulse) it should be inseparable from the notion of dx, even if it has a constant p (momentum) and is traveling through space. Thus the idea of a momentum defining an internal ruler gradation through a probability distribution in x which we have presented in previous notes is compatible with the idea of describing a particle as a force. If one only considers a particle as an effect, there would be no need to worry about dx until a force(x) appears. An internal dx is not defined by an external ruler and does not need to tend to 0 as in Newtonian mechanics. Rather it is statistical in nature as p is a fixed impulse, but the probability to be at different x points within dx (called a wavelength) differ. In (1) we compared a classical Maxwell-Boltzmann gas to a quantum free particle. We argued that d/d density Pressure = RT = average pressure of a particle. We suggested this is analogous to the quantum free particle situation: (-id/dx probability(p,x) ) / probability(p,x) = p where probability(p,x) is unnormalized. In other words a change in a probability yields a deterministic value p. The notion of a deterministic value p associated with a probability distribution in what would be another deterministic variable x in Newtonian mechanics (except now one has a dx distribution because the particle is a force/impulse) has consequences. If one wishes to have a particle sit at x=0, say in an oscillator system, one needs to use a wavelength which is fairly small and centered at x=0. The wavelength, however, is associated with momentum and even being small spreads out from x=0 meaning there is potential energy .5kxx as well. Thus one may use an ensemble of forces/impulses Sum over p a(p)exp(ipx) to create a “dx” about x=0, but at the same time this ensemble represents an average particle (defined at x with prms(x)) which must satisfy KE(x) + V(x) = En i.e. the classical conservation expression. In other words, one sees again that average statistical quantities may behave as deterministic ones.

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.

Classical mechanics traces a particle’s movement through space and time through x(t). For example, one may consider the case of a particle moving at constant speed (momentum p) in one direction or a particle bouncing back and forth at constant speed in a box. These problems are described in time and have direction and position associated with time etc. It is possible to define a probability P(x,t) in the classical picture based on delta time, the amount of time a particle spends in a little region of space dx. P(x,t), by itself, loses much relevant information present in the x(t) scenario because a particle moving to the right, or to the left, or bouncing back and forth all have the same probability, 1/L, where L is length. Furthermore, this probability is the same for all p (momentum) values. Thus a probability description, without extra information, is a subset of the relevant information in the problem. As a result, to describe the classical picture fully, one might say that a particle has a probability of 1/L, but also a mass mo and velocity +v (i.e. momentum mov nonrelativistically). For the particle bouncing back and forth in a box with constant p, one needs to give the initial position at time=0 and then a subsequent time (as well as the initial velocity and mo) in order to fully describe the problem, even though probability = 1/L. In the literature (see (1)), there have been many attempts to derive the quantum Schrodinger equation from a classical Fockker-Planck equation or a continuity equation (which follows from the Fokker-Planck equation as shown in (1)). The continuity equation contains a probability P(x,t) and a velocity(x,t) and is generally argued to represent classical physics. It does, if one retains appropriate classical conditions such as the discontinuity of density for a particle in a box (i.e. density=0 outside the box, but 1/L inside) and the notion of differing directions within the box. In fact, one may write density=W*(x,t)W(x,t) where W(x,t) is an ad hoc complex function and still retain compatibility with classical physics. If one does not, however, retain the classical conditions completely, nonclassical physics is introduced, even if density(x,t)=1/L. For example, in quantum mechanics density(x) is considered to be continuous in all x. Thus for a particle moving in a box, density outside the box is 0 so at the box wall it must also be 0. In the classical case, the density is 1 inside the box and 0 outside. There is a discontinuity classically and this cannot be changed and this contradicts the quantum mechanical picture. Thus quantum mechanics does not follow form classical Fokker-Planck or continuity equations which retain classical conditions.. Furthermore, because time is completely removed in the FP-continuity approach as applied to quantum mechanics, one cannot distinguish between forwards and backwards motion. As seen by the velocity field given in (1), v is zero for a bound particle which completely removes the sense of time and violates the classical physics conditions. It is, however, not zero for a particle moving in one direction. Thus there seems to be no issue in describing a Fokker-Planck-continuity equation as describing classical physics. One may even introduce density(x,t)= W*(x,t)W(x,t) provided one retains classical conditions allowing for discontinuity in density and different W(x,t) values at different times i.e. a discontinuity in W(x,t) at a box wall. If these conditions, however, are not met, then one no longer describes classical physics, but a new kind of physics (quantum physics). Thus one cannot argue that quantum mechanics follows from classical statistical Fokker-Planck-continuity equations. It may follow from these if one adds a set of extra conditions. Given these extra conditions, however, one could derive quantum mechanics without starting with classical density.

A Maxwell-Boltzmann gas is composed of many molecules which give rise to average bulk properties such as temperature, pressure, density etc. which are measurable. A single particle collides with other particles and when it is not colliding it moves as a Newtonian particle subject to -dV/dx. The gas is characterized by a probability distribution (the Maxwell-Boltzmann MB distribution) P( (ei+V(x)) / T) where ei is kinetic energy and V(x) the potential. T is usually described as a parameter which is fixed by: Sum over i ei exp(-ei/T)/Z = E average where Z= Sum over i ei exp(-ei/T). P(ei/T) may be obtained by maximizing Shannon’s entropy subject to the average energy constraint. P(x) may be obtained from a bulk balance of pressure and -dV/dx density(x). In this note, we wish to consider a single particle view. In that case, P(ei/T) is the probability for the particle to have kinetic energy ei and exp(-V/T)/Normalization is the probability for it to be at x. Instead of maximizing Shannon’s entropy, we consider the idea that P(e1+V(xa)) P(e2+V(xa) = P(e1+e2+V(xa). This is equivalent to considering two body elastic scattering with time reversal balance. If one assumes there is no extra information in the problem than conservation of energy and equal distribution of kinetic energy, then P(1)P(2)=P(3) leads to the MB distribution for a single particle. One could fix T again by considering the average energy a single particle has i.e. E average / N where N is the number of particles. We are, however, interested in the dynamics of the average single particle. The average particle is very different from a single physical particle in one member of an ensemble i.e. a particle which collides and when it is not colliding moves under the influence of -dV/dx. For equilibrium the average particle does not accelerate, but balances the force -dV/dx. One may see this by consider the derivative d/dx acting on P i.e. d/dx P = -dV/dx P. What does d/dx P imply? We argue that -dV/dx is a cause and its action is on a particle which represents an effect. A particle with kinetic energy ei, however, may be a cause of force because ei is proportional to its pressure, but also a recipient. Thus interacting gas particles are both causes of force and recipients (effects). Pressure is not treated in the same way as -dV/dx which simply multiplies P because P inside a little box is the effect. To be consistent, one requires the pressure outside the box to also be a cause, but it differs at each surface of the box (1-dimensional box). As a result, the so-called parameter T is really the average pressure of a single particle. On average this single particle does not accelerate in equilibrium so there is a balance with -dV/dx. If one considers N particles in a box, this balance scales to create the notion of a macroscopic density and a macroscopic pressure balance. The P1P2=P3 probability balance, however, is at the single particle level so in this note we wish to examine why exp(-V/T) should be obtainable via macroscopic balance using the ideal gas law Pressure= density RT and -dV/dx. We argue this follows from considering an average single particle (which behaves differently from a single particle in a member of an ensemble. We further argue that some of these ideas may apply to a single particle bound quantum state. &nbsp

In (1) it is shown how the Schrodinger equation can be “forced” in a sense to follow from the Fokker-Planck classical stochastic equation which contains a classical probability density. Other approaches in the literature which are similar use continuity equations and also a classical density. The key to all of these approaches is to assume, in an ad hoc manner it seems, that classical probability is equivalent to W*(x)W(x) where W(x) is a complex function called the wavefunction. In principle, one may write a density as W*(x)W(x). This is mathematically correct even if, for the sake of argument, it is ad hoc. The problem is that for a constant density (i.e. constant momentum and speed), one does not know if W(x)=exp(ipx) or W(x)=exp(-ipx). We consider here the arguments shown in (1). If a free particle has a density(x)=1/L, it may be associated with W(x) = exp(ipx)/sqrt(L). A free particle density, however, is associated with constant p (momentum) and assumes one knows x(t). If one considers the free particle to be trapped in a one-dimensional box with infinite potential walls, it travels to the right with p and then to the left with -p. The density(x) classically is 1/L, where L is the length of the box. The density is constant in time, but the direction of motion is a function of time. Thus, information is lost considering an equation such as the Fokker-Planck equation which uses a density function which does not track the direction of motion in time. In particular, writing density(x) = W*(x)W(x), which is done in (1) and applying this density to a bound particle in a box with infinite walls leads to an equation linear in W(x). Thus exp(ipx) and exp(-ipx) are both possible solutions if V(x)=0. Given that the conservation type equation obtained from the Fokker-Planck (or continuity) equation is linear in W, exp(ipx)+exp(-ipx) is a solution. Thus there is interference which leads to a density which does not match a constant classical density which should apply to a particle moving to the right and then to the left in a bound state because one assumed a classical picture a priori. Simply suggesting that density classical = W*(x)W(x) mathematically does not mean that the a priori classical picture may be dropped. As a result, we argue that there exists a contradiction in approaches which write classical density as W*(x)W(x) and then obtain the Schrodinger equation. As a remedy, we suggest that W(x) does not follow from a theory of classical density=W*(x)W(x), but vice versa. One needs equation which deal with W(x) a priori, and once it is found, one may create a classical density through W*(x)W(x).

In information theory, ln(Probability) is called information. We argue that in various physical problems this information is linked with a conserved quantity and a variable which changes. For example, in the Maxwell-Boltzmann case the conserved quantity is 1/Temperature and the variable e+V(x) = energy. For a quantum free particle, the conserved quantity is momentum and the variable x. “e” may change subject to the conserved quantity 1/T and x may change subject to the conserved quantity p. Given that probability may be unnormalized, changes may be expressed in relative terms i.e. in terms of a derivative of ln(P) i.e. information. We argue that the derivative of ln(P) with respect to the variable which changes yields the value of the conserved quantity in special cases for which the conserved quantity is the only information governing changes in the probability distribution. Thus one has an eigenvalue equation: d/dvariable ln(P) = conserved quantity. This implies that ln(P) = variable * conserved quantity. This may lead to two different kinds of conservation equations. In the MB case, e is the variable and 1/T the conserved quantity, but ln(P(e1)) + ln(P(e2)) = 1/T (e1+e2) = ln(P(e1+e2)). An eigenvalue equation allows for this conservation. In the quantum free particle case: ln(P(k1,x)) + ln(P(k2,x)) where P=wavefunction = exp(ipx) (i.e. the square-root type probability is used), one has k1+k2=k3 with x the variable canceling out. (In the MB case, it was 1/T the conserved quantity canceling). Again, the eigenvalue equation allows for this conservation, this time of the conserved quantity. We note that the eigenvalue equation is a special case and not only probability distributions are associated with it. For example, a power law distribution’s changes in its variable do not depend only on the conserved quantity 1/T in P(e/T)=power law.

For the case of physical quantities with units, such as mass, time, length etc, a change in the value of the quantity makes sense by itself in addition to a relative change. In other words, one does not need to know the mass of an object to understand what a change of 1 kg is. Probability, on the other hand, is a strictly relative quantity and so needs to be normalized. If it is not normalized, one needs to know relative probabilities. To say that an unnormalized probability changes by a value of 10 does not mean anything. Given that equations often solve for an unnormalized probability function, we argue that the notion of the relative probability should appear in equations. Even in the case of ln(P(y)), called information in information theory, for which there is no derivative, taking a derivative, say d/dy, immediately creates a relative probability derivative. Thus the form ln(P) is in keeping with the notion of a relative derivative. Here we examine the idea of a relative derivative and its association with information ln(P), Fisher information, quantum mechanics and the Fokker-Planck equation as well as classical statistical balance in the presence of a potential.

The Maxwell-Boltzmann distribution is associated with the maximization of Shannon’s entropy - Sum over i P(ei/T) ln[ P(ei/T) ] subject to the constraint Sum over i ei P(ei/T). The entropy form in this case is equivalent to the logarithm of the number of arrangements i.e. ln{ N! / Product over i n(ei)! ) where n(ei) = NP(ei/T) if one uses Stirling’s approximation. This means that each arrangement carries the same weight which is equivalent to the statement: P(ei)P(ej) =P(ei+ej) for allowing all distributions of energy to have equal probability. Thus in the MB case, the notion of maximizing arrangements subject to a constant average E and the idea of entropy dS/dP = ae coincide. In general, however, this is not the case. dS/dP = ae seems to be the recipe used in the literature (1) i.e. there exists a function of P such that its derivative d/dP yields a+be (energy) which is a physical constraint in the problem (average energy is actually used). This, however, need not be linked to equally weighted arrangements because P(ei)P(ej)=P(ei+ej) does not always hold. For example, it does not hold for a power law C(1-ke/T) (power 1/k), but entropy may still be created using dS/dP = 1-ke/T. Furthermore one may consider arrangements and maximize them as well to obtain the same power law distribution, but the constraint is no longer Sum over i ei P(ei/T). The constraint is Sum over i P(ei/T) ln(P(ei/T) which in the MB case yields average energy. In this note, we argue that the Fokker-Planck equation leads to the arrangement maximization form (i,. d/dP or a Shannon’s entropy) with the special constraint needed being linked to the diffusion coefficient.

Note: April 13, 2023. In Parts I and II, when taking derivatives, probabilities are unnormalized. In Part I, we examined how a recipe for MaxEnt (maximum entropy) usually includes the constraint Sum over i ei P(ei/T) and possibly a second constraint Sum over i P(ei). This leads to a recipe for the entropy such that dS/dP = a+bei/T ((A)) where a and b are constants, S being a function of P(ei/T). We saw that in both the Maxwell-Boltzmann and Power Law distributions, dS/dP is strictly a function of ln(P), which is called information in information theory. We then differentiated ((A)) with respect to 1/T to see what distributions appear if S in a function of ln(P) alone and saw the emergence of the MB and Power Law solutions. In this note, we consider the Zwanzig differential equation (from (1)) which includes random noise in the form of a variance function D(x,p). The power law distribution is a solution of this equation. We examine the Zwanzig equation to see exactly what condition needs to hold for any general solution (for a constant coefficient of friction for simplicity) and find the relation: 1= D d/dy ln(P) ((B)) where y = a+bei/T. Thus ln(P) and its derivative which appears in Fisher’s information are key pieces. Instead of differentiating the MaxEnt equation ((A)) by 1/T, we argue that differentiation by ei/T leads to ((B)) i.e. the key solution portion of the Zwanzig equation. Thus we link maximum entropy (albeit a derivative of this equation) to the solution of the Zwanzig equation for the MB and Power Law distributions.

Addition: Reference (1) is: Ran, G. and Du, J. Are power law distributions an equilibrium distribution or a stationary nonequilibrium distribution? (2013 or later) https://arxiv.org/pdf/1403.5441.pdf The idea of extremizing entropy (written as a function of P(e/T)=probability, where e is single particle energy and T temperature, with respect to P(e/T) including the constraint Sum over i ei P(ei/T)=constant leads to a recipe for entropy. This recipe involves solving P(e/T) for e/T and then integrating with respect to dP(e/T) so that d/dP from the extremization cancels the integration procedure. This recipe for entropy is thus based on: dS/dP = e/T or more generally on dS/dP= a+be/T where a and b are constant and the extra constraint Sum over i P(ei)=1 is used. In general, S(P), but we argue that in both the Maxwell-Boltzmann and Power Law Cases, dS/dP is strictly a function of ln(P), which in information theory is called information. We consider how this is linked to taking the derivative d/d (1/T) of dS/dP= e/T or its generalization.

The power law probability distribution P(e/T) proportional to {1-a(e/T)} (power 1/k), where k and a are constants, is experimentally observed in various scenarios, including stellar particles. As noted in (1), it is a solution of a Zwanzig differential equation with a dispersion term related to d/dp’s and a function (1-be/T) as well as a friction term. We note that unlike the conservation of energy i.e. e1+e2=e3 in the Maxwell-Boltzmann case, product probabilities seem to lead to the conservation of ln(1-ae/T), where e=pp/2m + V(x). Writing ln(1-ae1/T)+ln(1-ae2/T) = ln(1-ae3/T) and taking d/dx or alternatively d/dp leads to the conservation of relative terms i.e. -dV/dx / (1-ae1/T) -dV/dx / (1-ae2/T) = -dV/dx / (1-ae3/T) and p1/(1-ae1/T) + p2/(1-ae2/T) = p3/(1-ae3/T). If p is thought of as an impulse, then p/(1-ae/T) is an impulse relative to 1-ae/T. For V(x)-->0, one has p/(1-app/2mT) (nonrelativistically). We also note that relative type derivatives are closely linked to the Fisher statistical form: I = Integral dx P(x) {d/dx ln(P(x))} {d/dx ln(P(x))}. As a result, we transform the dispersion term in the Zwanzig equation into a function of Fisher information (integrand). We find that a term proportional to Fisher’s information and a second term linked to the functional variation of Fisher’s information (integrand) with respect to P appear as well as the friction term. As a result, this is a second differential equation which has a power law probability as a solution, and there might be some physical or statical reasons for writing an equation including Fisher’s information related to relative derivatives.

A quantum single particle bound state involves a single particle, like a classical bound state, but it also involves spatial densities like a Maxwell-Boltzmann gas. For instance, a single particle has potential energy V(x) at x, while a density(x) has potential energy V(x) density(x). As a result, there seems to be a dichotomy. How does one solve the problem of single particle bound state? Does one use the concept of a single particle or of a density(x)? We argue that one focuses on a single particle, but with an added piece of physics. In classical physics, a particle with constant momentum (speed) moves in a deterministic way described by x=vt. On average, a quantum single particle creates its own ruler gradation and a sense of a direction in space through exp(ipx) which maps to 1 via exp(-ipx) exp(ipx). A single quantum bound particle which gives the semblance of an entity moving in a straight line back and forth (in a distance of the same order of hbar/p ) needs to create a semblance of acceleration as an average single particle. As a result, one may think of an ensemble i.e. W(x)=Sum over p a(p)exp(ipx). The difference with a free particle is that one has interference (+p’s and -p’s) which readily display a spatial W(x) which may ultimately be linked to a spatial density through W*(x)W(x). Thus, one now has the idea of spatial density which exists in a Maxwell-Boltzmann gas, This, however, is a consequence of the single particle approach. We argue that the single particle viewpoint leads to the spatial density one and not vice versa because of the way KE(x)+V(x) = E is treated. If one writes V(x)= Sum over k Vk exp(ikx), one does not write: KE(x) = Sum over k bk exp(ikx). We argue the reason is that such a KE(x) is not normalized so as to create the sense of a single particle. We argue that such a normalization leads to: KE(x) = {Sum over p a(p)pp/2m exp(ipx)} / W(x) meaning that on average one has a moving entity. V(x), on the other hand, is not a moving entity.

The microcanonical distribution for a classical gas involves a fixed number of particles N and a fixed total energy E. The energy may be distributed in different ways among the N particles such that its total is E and each distribution or microstate is said to have the same weight or probability. We argue that the idea of equal weight follows directly from Newtonian mechanics. For a dense gas, the collisions in the gas are Newtonian and if there were a greater weight to have e1,e1,e1 (for three particles) than 3e1,0,0, there would need to be a reason for this coming from Newtonian physics. In other words, this is not a statistical issue. Equal distribution of energy among particles has important consequences because one may immediately write P(ei)P(ej) = P(ei+ej), if one relaxes the total energy constraint (canonical distribution). This leads directly to P(ei) = C exp(-ei/T). The underlying idea, however, is present in the microcanonical distribution. The idea that energy distributions carry the same weight also leads to the notion of the maximization of the number of distributions or the maximization of entropy for the gas. Thus we argue that maximization of entropy subject to a constraint on total average energy is not an independent statistical idea, at least for the usual classical gas, but follows from Newtonian mechanics. In (1), it is argued that the equal weights for microstates in a microcanonical gas follow from the notion of virtual work being zero. We argue that the equal weights follow from Netwonian mechanics which does not show any preference for e1,e1,e1 versus 3e1,0,0 for three particles.

The idea of maximization of entropy as a physical maximization, say of disorder, seems to be entrenched in physics. Historically, it seems to follow from analyses of a Maxwell-Boltzmann gas. In particular, a traditional argument seems to be that if one has N particles (where N is very large) and a total amount of energy E, then there is no bias involved in the distribution of E among these N particles. Each “arrangement” carries the same weight. As a result, one wants to include all arrangements which is equivalent to saying that one wants to physically maximize the number of arrangements subject to the constraint E = total energy. Thus there is a physical reason for maximization. This physical maximization is equivalent to the property that P(ei)P(ej) = P(ei+ej) i.e. one only considers energy distribution and does not need to resolve the particles so to speak. {This property is equivalent to: p(e1)p(e2)....p(eN) = p(E) so all distributions of E carry the same weight or probability.) The equation p(ei)p(ej)=p(ei+ej), however, automatically yields the Maxwell-Boltzmann distribution without the need of any maximization calculus. The traditional alternative is to calculate the number of arrangements N!/ Product over i n(ei)!, take ln and extremize it with respect to the constraint Sum over i ei n(ei). N is assumed to be large and Stirling’s approximation is used. Both of these approaches are identical because there is a physical maximization in the problem which leads to the same result: P(ei)P(ej)=P(ei+ej). Although no maximization is needed to solve the above problem, maximization seems to have been adopted as a fundamental physical principle even in problems for which P(ei)P(ej) is not equal to P(ei+ej). An example is a power law distribution, i.e. p(ei/T) = (ei/T) (power g) / Z where g is a constant and Z a normalization factor. The arrangement problem (p(ei)p(ej)=p(ei+ej)) finds a specific form for the function to be maximized, namely n(ei) ln{n(ei)}. This in turn leads to a specific distribution i.e. the Maxwell-Boltzmann. If one starts with the Maxwell-Boltzmann distribution, however, one may work backwards and find a function of probability which when maximized subject to the constraint Sum over i ei p(ei) yields p(ei). This function need not have anything to do with physical arrangements. As a result, this procedure lends itself to a recipe which holds for any probability distribution, even for ones in which p(ei)(ej) is not equal to p(ei+ej). In such a case, it is not at all clear what one tries to maximize physically, but the recipe still holds. The recipe involves taking p(ei/T) and solving for ei/T i.e. ei/T = Function( p(ei/T)). One then integrates this function with respect to dp(ei/T) so that the derivative d/dp(ei), which is the inverse operation, returns ei/T. This matches 1/T d/dp(ei) Sum over i ei p(ei) from the constraint immediately leading to an extremization procedure. The point is that there seems to be no physical maximization involved whatsoever. It is true, however, that one finds a function(p(ei)), with no ei explicitly present, such that small changes to p(ei) with Sum over i d(p(ei)) = 0 and Sum over i ei dp(ei) = 0 leads to 0. Physically, these constraints hold in a system and one may imagine fluctuations leading to p(ei)--> p(ei)+dp(ei). The function of p(ei) might be considered a stability function and thus be associated with equilibrium, but we argue that this does not necessarily have anything to do with any physical maximization. Thus this kind of recipe-based entropy may be better thought of as a stability function than a measure of the maximization of arrangements (sometimes called the maximization of disorder).&nbsp

A one dimensional reflection/refraction problem is by nature probabilistic because an incident photon either reflects or refracts with certain unknown probabilities. The problem is traditionally solved by matching the electric field and its derivative with respect to x at x=0, t=0 where the electric field for a photon follows from Maxwell’s electromagnetic equations and is given by Eo cos(-wt+kx) where p=hbar k. As such, this problem is treated “classically” We argue, however, that this is a quantum mechanical problem (the photon being a quantum particle). Furthermore, no information is needed about the form of the electric field. Reflection/refraction conserves energy. It also occurs at some x point which may be varied i.e. there is translational invariance. A conservation of energy equation may be immediately written. The first medium has an index of refraction n1=1 and the second n2 This problem may be mathematically solved by using a difference of squares which leads to two equations: (A+B) = C and p1 (A-B) = p2 C. These may be solved for B,C in terms of A. This, however, is not the most general solution in that it does not show invariance of x i.e. the point at which the index of refraction changes. In this note we investigate the consequences of including x in the solution and argue this leads to a kind of square-root probability function which together with its first derivative should be continuous at any x chosen to be the interface of the media. This leads to a solution of Aexp(ipx), instead of A (etc), which is the free particle wavefunction. Thus this wavefunction (or unusual probability) is a direct consequence of the spatial invariance in the problem, but ensures that no x dependence appears if one combines the two square-root probability equations to create an energy conservation equation which has nothing to do with x.

In (1) a power law probability p(e) = (e/T)(power -g) / Z is assumed and the first law of thermodynamics dE=TdS utilized to obtain a form for the entropy density as a function of p(e). The main idea of (1) seems to be that dE and dS only change due to changes in p(e) and not due to a parameter (which may alter coefficients as well p(e)). The authors alter p(e)--> p(e)+dp(e) such that Sum over i dp(ei) = 0 ((1)). This form of change is applied to dE-TdS to yield Sum over i (ei- T d/dp (G)) dp(ei) ((2)) where G is entropy density. In (1) it is argued that ei-TdG/dp(ei) =K=constant in order that ((2)) be the same as ((1)) up to a multiplicative constant. This equation may be integrated to ultimately find a solution for G(p(ei)) and ultimately entropy. Temperature in this scenario is a constant which (1) writes in terms of eo,em (the bounds of e) and g. In thermodynamics, T is a parameter which changes and measures the thermal properties of a system. We argue that the approach using dE=TdS, employed in (1), is essentially the maximum entropy principle subject to the main constraint Sum over i ei p(ei), because only p(ei) is varied. In such a case: TdS-dE=0 yields the main term of the entropy of (1). (A second term which is a constant time p(e) is added, but Integral p(e)de=1 so this term is a constant and dE=TdS only defines entropy up to a constant. In (1) certain boundary conditions are used to determine this constant. Thus the main idea is to “maximize S subject to 1/T E by changing p(ei). This maximization process, however, is essentially equivalent to writing ei/T in terms of p(ei) i.e Function(p(ei). This function may be integrated with respect to dp(ei) so that the derivative then yields ei/T, which when multiplied by T, yields ei which, in turn, equals ei from the constraint. Thus S is a mathematical function whose derivative with respect to p(ei) yields essentially ei/T. For a constant T temperature system, it is not clear that dE=TdS is really a thermodynamic equation. In general T may change, but this means that dS involves changes in p(ei) due to a parameter in addition to changes in any coefficients which may be functions of this parameter. We show in the power law case that the coefficient in the function of p(ei), whose derivative with respect to p(ei) yields ei/T, is in fact dependent on parameters which are linked to T. Thus dE=TdS seems to be a maximization with respect to an energy constraint approach and not a thermodynamical one because the first law of thermodynamics does not stipulate that dE and dS change only through p(ei) only. In fact, in thermodynamics one does not need to write E and S in terms of p(ei). p(ei) enters when one thinks in terms of microscopic quantities.&nbsp

Correction: March 31, 2023 We remove the form D = 1-a(p)a(p) in the note as it does not equal the time evolution measure of (1). We wish to only point out that sinc(T/2 |(En-Em)|) in the expression for D has an analogue in the spatial behaviour of a particle in an infinite well where: sinc(T/2 |En-Em\) from D --> a(k) sqrt[3.14/(hbar L)] [n3.14 + kL] / (n3.14) i.e. is proportional to the momentum wavefunction a(k) multiplied by a linear function in k. We point out that even though this analogy exists, the summations involved in D do not map over to the single particle case. Thus there is not a complete analogy. In particular, one could sum over k, but would not generally sum over n, the energy level. --------------------------------------------------------------------------------------- In (1) a measure to determine time evolution of a quantum system is given as: D = 1- Integral dt (0,T) dt’ (0,T) . Using W(x,t) = Sum over n cn exp(-iEnt) Wn(x) (1) shows: D= 1 - Sum (n,m) |cncm||cncm| sinc[wnm T/2] sinc[wnm T/2] The authors argue that this measure may be applied to qubit systems. Their result is based on the form exp(-iEnt)Wn(x) and the idea of a fixed time interval T. Here we argue that an identical form appears in the spatial realm for a system which instead of a finite time interval contains a finite length. The relevant system with a finite length is that of an infinite well potential with length L because the wavefunction is 0 outside this length. In such a case, instead of exp(-iEnt)Wn(x) we relate exp(-ipx)Wn(x) to a(p), the momentum wavefunction, and show that measure of time evolution is essentially of the same mathematical form as: 1- a(p)a(p) for an infinite well of length L. 1- a(p)a(p) has the interpretation of the probability not to find a particle in an infinite well in a momentum state of p for a given energy level with average momentum of n 3.14/L.

Correction: March 31, 2023 We remove the form D = 1-a(p)a(p) in the note as it does not equal the time evolution measure of (1). We wish to only point out that sinc(T/2 |(En-Em)|) in the expression for D has an analogue in the spatial behaviour of a particle in an infinite well where: sinc(T/2 |En-Em\) from D --> a(k) sqrt[3.14/(hbar L)] [n3.14 + kL] / (n3.14) i.e. is proportional to the momentum wavefunction a(k) multiplied by a linear function in k. We point out that even though this analogy exists, the summations involved in D do not map over to the single particle case. Thus there is not a complete analogy. In particular, one could sum over k, but would not generally sum over n, the energy level. --------------------------------------------------------------------------------------- In (1) a measure to determine time evolution of a quantum system is given as: D = 1- Integral dt (0,T) dt’ (0,T) . Using W(x,t) = Sum over n cn exp(-iEnt) Wn(x) (1) shows: D= 1 - Sum (n,m) |cncm||cncm| sinc[wnm T/2] sinc[wnm T/2] The authors argue that this measure may be applied to qubit systems. Their result is based on the form exp(-iEnt)Wn(x) and the idea of a fixed time interval T. Here we argue that an identical form appears in the spatial realm for a system which instead of a finite time interval contains a finite length. The relevant system with a finite length is that of an infinite well potential with length L because the wavefunction is 0 outside this length. In such a case, instead of exp(-iEnt)Wn(x) we relate exp(-ipx)Wn(x) to a(p), the momentum wavefunction, and show that measure of time evolution is essentially of the same mathematical form as: 1- a(p)a(p) for an infinite well of length L. 1- a(p)a(p) has the interpretation of the probability not to find a particle in an infinite well in a momentum state of p for a given energy level with average momentum of n 3.14/L.

The idea of impulse, i.e. Change in momentum = Integral dt Force appears in Newtonian mechanics. Impulse is considered to be a physical observable, for example it may be linked to the damage done by a particle hitting a target. Impulse, however, only depends on momentum and so two particles with the same momentum, but different masses and velocities may cause the same damage i.e. represent the same physical observable. Momentum in nonrelativistic Newtonian mechanics is m v where m is rest mass and v velocity. Velocity requires a clock in order to measure it, yet two particles with the same momentum and different velocities are considered equivalent in terms of the damage done upon collision with a target i.e the physical impulse. Thus we suggest that time must formally disappear from mv, but this implies that rest mass must behave as t or dt. Furthermore, given Kinetic energy = .5mvv, it must behave as 1/dt. In other words, the reciprocal of kinetic energy behaves as a gradation of time. This is not a standard clock as the gradation is based on a specific kinetic energy and is different for another. In other words, two particles with the same p and different v, have different time gradations. This, however, is a quantum mechanical result. Thus we argue that quantum mechanics may be responsible for the idea of p yielding the same impulse hit for two different v. We further argue that these ideas may be extended to include rest mass with v=0 and to indicate that 1/p is linked to dx, a ruler gradation which is again a quantum mechanical result. Lastly, we argue that these ideas lead to the special relativistic result EE = ppcc + mmcccc and so argue that free particle quantum mechanics and special relativity are closely linked and that classical impulse follows from free particle quantum mechanics.

In Newtonian mechanics, one often thinks in terms of x(t) i.e. an association between x and t. Although this is part of Newtonian mechanics, the notion of motion always being linked to time (and so an internal clock being required) is not always the case. For example, two particles with different masses and different velocities such that their momenta are the same may strike a target and reflect with the same magnitude of momenta. The impulse is 2p for both even though they have different speeds. One does not need to consider their speed to discuss the impulse change and so an external clock is not needed. This is further emphasized by the Lorentz invariant A = -Et+px which separates x and t as if they were independent variables (even though implicitly a speed exists in the problem i.e. v=dx/dt.). The point is that there exists a subset of physics (i.e. interactions) which is not dependent on velocity and hence the presence of a clock. The impulse hit discussed above is one example, but interference through 2 slits etc is another. Two quantum particles with the same momentum, but different rest masses and velocities produce the same interference patterns. A particle with momentum, however, has energy and so both time and space are running variables, but there are problems in which one considers p without a clock. Interestingly, even if one thinks in terms of impulses without using a clock, if there is an x interval between impulses then the notion of velocity comes into play even without an external clock because there will be more impulses imparted by a fixed p with a higher velocity than a lower one in the same fixed time period. Without using the idea of a clock, this give rise to the observable called pressure in classical physics, but is also present in a quantum bound state, we argue. Pressure in classical physics is proportional to temperature which in turn is proportional to energy (and a running time). In a quantum bound state, the equivalent to pressure (namely impulse hits with a potential) also require an energy viewpoint which is linked to running time i..e exp(-i En t) where En is the bound energy.

In Newtonian mechanics, time and space are measured by external clocks and rulers in an absolute sense. Special relativity changed this view of absolute time and space, but we argue there is a further issue linked to time and space and that is the idea of “running variables”. In particular, one may have a particle with rest mass mo at x=0. The spatial variable, which is completely arbitrary, remains constant while time is a “running variable” or dynamic. This time may be measured by an external Newtonian clock, but if this clock is removed, does the concept of time disappear? We argue it does not and that mass (or energy) must keep internal time, i.e. somehow create periodic time intervals or an internal clock. The question is: Are these constant time intervals for all energy values? Is there a way to quantify these intervals? We wish to examine this here. One may note that for the rest mass at x=0, time and x are treated differently. Time is a running variable and x is not.For x to be a running variable, a constant speed needs to appear together with a direction. If time as a running variable creates its own internal clock, we argue motion in one direction should create an internal ruler. Are the ruler gradations based on the magnitude of v alone? Interestingly, we note that for x to be a running variable, the magnitude of v is irrelevant. One simply needs a v with a direction. Furthermore, if external rulers and clocks are removed, there should be a physical attribute (which may be measured without them) which accounts for the internal ruler. We argue this attribute is momentum (mov nonrelativistically, mov/sqrt(1-vv/cc) in special relativity). Momentum may be measured without using an external ruler and clock by noting the damage done upon collision with a target. Thus different mo,v pairs lead to the same momentum value and the same internal ruler. Is there a quantitative way to analyse the proposed internal clock and ruler? We argue there is and that it follows from the Lorentz invariant A = -Et+px.

The Kullback-Leibler entropy is defined as: - Integral dx P(x) ln { P(x)/R(x)} where P(x) and R(x) are two classical probability distributions i.e. real, positive functions with values between 0 and 1. In (1), it is shown that the Kullback-Leibler entropy is equivalent to Fisher’s information if R(x) = P(x+delta x) where delta x is a tiny constant. We argue that in this case, the Kullback-Leibler entropy represents either average momentum multiplied by spatial density in a quantum bound state, i.e. Integral dx W(x)W(x) {dW/dx / W} which is 0 because overall the particle does not move in one direction (odd function). This suggests forwards and backwards internal motion which yields 0 overall translation, similar to fluctuations (in x not t) . In addition, it represents the “fluctuations” or average momentum for half the classical probability distribution associated with a quantum free particle exp(ipx), namely C cos(px) cos(px). Again the Kullback-Leibler entropy for this distribution (or sin(px)sin(px)) is 0. (This is equivalent to the result of a particle in an infinite potential well if one considers pave in cos(pave x), the wavefunction.) Thus a statistical quantity, namely Kullback-Leibler entropy, seems to be associated with quantum mechanical motion (average momentum). This motion is linked with zero Kullback-Leibler entropy where P(x) is a distribution and R(x)=P(x+delta x). Furthermore, this seems to show, we argue, that d/dx in Fisher’s information is linked with physical momentum.

In classical mechanics, a free particle moves through x and t with dx/dt=v=constant. Given that x and t may be shifted, it is dx and dt which are important. For fixed dx, dt(x) yields information about the “probability” for the particle to be in dx so to speak. Thus one does not really distinguish between deterministic and probabilistic descriptions in this scenario. A problem with the probabilistic picture, however, is that different v (or p values) have the same constant probability P(x)=constant for free motion. In other words, P(x)=constant does not distinguish either the magnitude or direction of v. Thus, information is lost. Is it possible to restore this information and still maintain a probabilistic picture? The answer, given by free particle quantum mechanics, seems to be yes. The probabilistic approach is to assign a probability P(x) to each x point. This probability must treat each x point as the same and yet give information about v or p. (We use p because it represents impulse and is linked to special relativity i.e. a 4-vector.) The key idea we argue is to consider the first order change in x space because one assigns a P(x) to each x and different p values are linked with different motion from one x to another.. This is the way to distinguish the direction and magnitude of p, but nothing else we argue. In such a case: d/dx f(px) = Constant p f(px) ((1)) ,we argue, where f is an unknown function. We show that this function, which is odd in x, cannot be real i.e. one needs at least a pair of real functions comprising f. We note that the derivative only gives information about p, thus df/dx /f = p. We argue this implicitly means each x point has the “same weight” even though the pair of functions f are not constants. We next analyze the behaviour of the two pairs of functions comprising f and conclude that in order for d/dx d/dx f = pp f, each function of the pair must be converted into itself. Thus: d/dx d/dx f1 = -pp Constant f1 and a similar equation for f2 where f1,f2 are the pair. These, however, are periodic equations. Thus the implicit idea of each x point carrying the same weight is equivalent to f1(x) and f2(x) being separately periodic. This is equivalent to constant modulus problem. How is this linked to Shannon’s entropy or Fisher information extremization? We argue that ((1)) means that ln(P) does not carry enough information in this situation, while for a Maxwell-Botlzmann gas it does. It is d/dx ln(P) which carries information about p and its direction. If one wishes to create a function which does not distinguish the sign of momentum, it still has to incorporate the information of motion in one direction i.e. d/dx ln(P) d/dxln(P). One may take the average of this (called Fisher’s information) instead of the average of ln(P) in Shannon’s entropy. To link with the above ideas, P(x)=f1(x)f1(x) if one considers a bound problem with infinite potential walls. Then Fisher’s information becomes df1/dx df1/dx. The extremization of Fisher’s information subject to Integral f1(x)f1(x)= 1 leads to the periodic condition described above. This means, we argue, that for the case of pp, one cannot sense any change in either f1(x) or f2(x). f1(x) and f2(x) are used to indicate differences to an odd power of d/dx and so for an even, there is no informational change at the f1(x) and f2(x) level. This we argue is what Fisher’s information extremization with Integral f1 f2 =1 gives i.e. periodicity.

The extremization of Fisher’s information subject to constraints usually containing V(x), the classical potential, appears in the literature (1), (2). V(x), however, is a concept introduced by Newton and associated with conservation of energy for a conservative potential. We argue the appearance of V(x) in an extremization approach to obtain an “equation” bypasses the already existing conservation of energy equation. If V(x) is used, we argue that one cannot dispense with conservation of energy and that it should be written a priori. If that is the case, what does extremization of Fisher’s information guarantee? In (1) a probability distribution P(x) is set equal to W(x)W(x) its square root. In (2), a complex function is defined: W(x) = sqrt(density(x)) exp(i S/hbar) where S is the classical action. Here density(x)=P(x). In both cases, the square root of the density appears. The problem, we argue, is that for a free particle with density(x)=P(x)=1, there is no Fisher’s information to extremize, yet the free particle is a key part of quantum mechanics. Its probability (square root probability) is the solution of an eigenvalue problem of the translation operator -id/dx and it is consistent with Newtonian mechanics i.e. conservation of energy with E= pp/2m. From this one may create an ensemble W(x) = Sum over p a(p) exp(ipx). The average kinetic energy is -1/2m d/dx/ dW/dx / W which happens to be the extremized value of Fisher’s information for P(x)=W(x)W(x). In other words, we argue that conservation of energy must occur and that one needs a solution for a free particle and this does not follow from Fisher’s information extremization. Extremization of Fisher’s information produces the form d/dx dW/dx which is taken to be linked to which already follows from -d/dx d/dx exp(ipx). One argument made in the literature (3) is that Newtonian conservation of energy also follows from Fisher’s information extremization i.e. the Lagrangian for a free particle .5m dx/dt dx/dt is of the form of a Fisher’s information with xx = probability (which is unusual). The strategy, it seems, is that one may apply extremization of Fisher’s information directly to a quantum problem by assuming a P(x)=W(x)W(x) and use d/dx instead of P=xx with d/dt. An issue, as noted, is that for a free particle W(x) exists, but extremization of Fisher’ information does not exist. Even if E=pp/2m (nonrelativistically) and this follows from the extremization of Fisher’s information, W(x)=exp(ipx) does not, but it is consistent with E=pp/2m. The reason for this consistency is that it contains p (momentum) which by its very definition must ensure E=pp/2m (nonrelativisitcally). Fisher information extremization, however, yields the bound state solution i.e. may be used to derive the Schrodinger equation. In (1) this is done for a bound state, in (2) for the more general case, but (2) defines W(x) = sqrt(density) exp(iS/hbar) where S= the classical action. For a free particle, the definition of W(x) is exp(-iEt+ipx) which is the solution of the free particle problem without using any equation. Thus the definition is the solution itself for the free particle case in (2). It seems that Fisher’s information, which uses density(x), does not include a scheme for describing motion in one direction in space without time present. exp(ipx) i.e. cos(px) and sin(px) distributions in two dimensions allow for this. This motion is associated with the translation operator d/dx which we argue is key in solving for the square-root probability of a free particle in keeping with E=pp/2m, which (3) argues follows from the extremization of Fisher’s information. Thus, even if one accepts the argument that Fisher’s information is responsible for the energy conservation equation, using Fisher’s information = Integral dt dx/dt dx/dt constant, does that allow one to immediately switch to a problem with P(x)=W(x)W(x) and the derivative d/dx and apply Fisher’s extremization there, we ask? We argue that there is an implicit assumption that: C1 dx/dt dx/dt = C2 d/dx dW/dx / W where C1 and C2 are constants This implies that for a free particle which treats each x point the same i.e. P(x)=constant, W(x)W(x) cannot be P(x). It also implies that d/dx dW/dx = pp W ((1)) where p=mov nonrelativistically. ((1)), however, is the solution to the free particle problem without performing any Fisher information extremization i.e. exp(ipx) is a solution. This solution may be used to create an ensemble W(x)=Sum over p a(p)exp(ipx) which may be used in a conservation of average energy equation. Thus we argue that Fisher extremization used to create the Schrodinger equation is based already on the idea that W(x)=exp(ipx) for a free particle and simply produces the conservation of energy equation which is already known to exist from Newtonian mechanics or Fisher’s information extremization which is argued to be the same as Lagrangian variation in (3).

Special relativity involves the Lorentz transformation matrix which contains velocity v, and acts on 4-vectors. (p,E) and (x,t) are two common examples. Interestingly, we argue, v, p and E are invariant under spatial and time translations, but x and t are not. An interval in time (time period) and interval in space (wavelength), however, are spatially invariant. Thus we argue that the Lorentz invariant A= -Et+px should give rise to time and spatial intervals in order to have overall translation (time and space) invariance. This leads to temporal and spatial intervals linked with specific E and p values, because x and t are not intervals by themselves. The time interval is hbar/E and the spatial hbar/x. By defining these intervals at an E, p level, interactions with intervals existing in nature, e.g. a 2-slit interval, are associated with a wavelength which in turn is dependent on p through hbar/p. Thus p does not simply represent an impulse, but is linked with a spatial type of probability exp(ipx). This idea carries over into the Newtonian picture because one may take the nonrelativistic limit of A= -Et+px, namely -Et+px for a free particle with E=pp/2m and p=mov. This quantity, however, is not a naturally occurring one in Newton’s theory and so the issue of a lack of translational invariance in x and t is not apparent.

Given a coin, one has two complementary pieces of information and a corresponding probability of .5 and .5. Similar considerations apply to a die. As a result, one may get the impression that there exists a unique probability distribution in a probabilistic system and that this single probability scheme is linked to the constrained maximization of Shannon’s entropy or Fisher’s information. We argue however, that there may be different probability schemes in the same problem associated with different pieces of information which are relevant to the physics which occurs in the system. Thus, we argue that it is the physics which selects the probability distribution to be examined. For example, one may loosely think in terms of a detailed probability scheme versus a kind-of coarse grained one in the same system. For example, for a quantum free particle, P(x)=1 i.e there is the same probability for a particle to be at any x point. This, however, is based on velocity, i.e. the relationship between dx and dt. There is another piece of information in the system, namely mo which together with v yields momentum (either relativistic or nonrelativistic) .This need not be measured using a clock (which velocity requires), but by the impact of an impulse on a target. Thus, nonrelativistically p=m1v1=m2v2 give the same impulse, but have different velocities. Quantum mechanics shows that a free particle is governed by exp(ipx) which contains two unnormalized probability distributions cos(px) and sin(px). These are key for impulse interactions (i.e. the physics of interactions such as with a 2-slit apparatus etc), yet exp(ipx)/sqrt(L) maps into the “coarse-grained” P(x)=exp(ipx)exp(-ipx) / L = constant. P(x) is one probability scheme in the problem and exp(ipx) another (for a free particle). Each answers some physical questions, but one needs to know which probability to use (i.e. what information is relevant) to a particular situation. P(x) is relevant to motion in space not involving interactions. exp(ipx) is relevant to interactions (even with a magnetic vector potential causing no magnetic field) or with a 2-slit apparatus. Furthermore, in a quantum bound state, one has W*(x)W(x) = P(x) (W(x)=wavefunction =Sum over p a(p)exp(ipx)) and a*(p)a(p).=P(p). Each probability scheme corresponds to the analysis of a different kind of measurement, but various schemes may also be linked. Thus one may have to start with one to create the others. For example, exp(ipx) is the base for creating W(x)=Sum over p a(p)exp(ipx), which ultimately yields P(x)=W*(x)W(x). As a result, we argue that if probability exists in a system, it is not necessarily the case that there is a unique probability distribution. There may be various probability schemes linked with different information and different measurements/interactions. Thus, physics leads one to focus on a specific probability scheme. As a result, we argue that one must be careful when discussing pure statistical quantities and their extremization, because these quantities, like Shannon’s entropy and Fisher’ s information, contain a classical probability distribution, but they do not indicate which distribution to use in the case that a system has more than one. . As a result, the analysis of a system requires knowledge of the physics (interactions) occurring.We argue against thinking in terms of physics following directly from statistical considerations. The constrained extremization of one statistical quantity may yield a probability distribution related to one piece of information, but not to another. Furthermore, Shannon’s entropy and Fisher’s information pertain to classical probability 0

Momentum and velocity are two well-known physical observables in classical physics (and quantum physics as well because quantum particles have impulse and move through space in a certain amount of time based on an external clock and ruler). They are, however, measured in different ways. Velocity describes motion through x in some time t, while p is measured through impulses. Thus one does not need a clock to measure momentum. It may be measured in terms of the damage done to a foil, compression of a spring etc. Thus p=m1v1=m2v2 etc., i.e. a numerical number associated with momentum does not automatically yield the velocity unless one knows rest mass. In many situations rest mass is known (i.e. one deals with an electron etc) and so there seems to be a tendency to consider momentum and velocity to be almost the same thing when they are distinctly different. In fact, we argue that the notion of wavefunction W(x) and density P(x)=W*(x)W(x) are associated with momentum and velocity respectively. In fact, for a free particle these two measurables are linked with completely different spatial probabilities. Constant velocity impies P(x)=constant while exp(ipx) describes the two-dimensional x-probability associated with p which is based on a constant mo and v. In special relativity for a free particle (i.e. p constant, v constant), there are two well known Lorentz invariants: EE = pp + momo (c=1) and A = -Et + px. One may note that v does not appear explicitly even though one may write E and p in terms of v i.e. E = mo/sqrt(1-vv) and p=Ev. The first invariant describes E in terms of rest mass and p (measured through impulse). The second involves p,E and x,t with x and t appearing as independent even though ultimately x(t) and v=dx/dt=constant = x/t if x and t both start at 0. This independence is crucial because it means one may consider fluctuations which keep A constant separately. For example, if t→ t+ hbar/E and x→ x+hbar/p, then A remains constant. This leads to the idea of the quantum wavelength and a spatial probability independent of t, namely exp(ipx). One average, however, x=vt, but how does this appear in Lorentz invariants? There is no explicit v. We argue that the Lorentz invariant A = -Et + px must convert E from a function of p to one in v and that this occurs through a Legendre transformation. Thus the form of the invariant is itself a Legendre transformation with E being transformed into L, the Lagrangian. There is, however, a second important feature because p=dL/dv (partial) and given that p is constant, dp/dt=0 so d/dt dL/dv partial =0 which is in the form of an extremization. We argue that this is not an accident. dL/dv=p follows from E being constant, so, in other words, it follows from the px term. This same term leads to exp(ipx) which means that one does not have smooth x=vt motion within a wavelength. Thus, we argue that when one transforms to the second measurable v, the notion of this fluctuation behaviour must also be manifested. In other words, v=constant or dv/dt is not enough. The extremization behaviour of the Lagrangian, we argue, is thus not a mathematical convenience, but gives an indication that even though the extreme path x=vt is followed on average, other paths are sampled. We argue that this idea is incorporated into special relativity, in fact within a single Lorentz invariant A = -Et + px. &nbsp

For the case of a free particle, it is known that one may extremize a Lagrangian L=.5mvv, where v=dx/dt, to obtain: d/dt dx/dt = 0. In this classical case, one is interested in x and t or x(t). One may think of moments i.e. powers of derivatives of x(t). Given the first moment is equal to a constant, all higher moments are 0. Each moment yields a piece of information. Thus there is one piece of information, namely the constant velocity v=dx/dt. A second derivative d/dt dx/dt =0 demonstrates there is no further information. The form d/dt dx/dt, however, is associated with an extremization of a function, namely L=.5mvv or Integral .5mvv dt. Thus we argue that extremization is really linked to information found in the problem for the free particle case. Next, we consider .5 Integral dt vv = .5 Integral v dx. This is proportional to Integral p dx where p represents a physical impulse. The equation d/dt dx/dt =0 shows there exists only one piece of information, regardless of the value of v. In other words, all v values satisfy this equation (where v=constant). In analogy, if one wishes all p values to satisfy Integral p dx, p = constant, without distinguishing p, then x= constant/p. This, however, is the quantum wavelength. If it is repeated, in space, one desires a periodic function in px which has a modulus of 1, i.e. an eigenfunction of -id/dx namely exp(ipx). This does not follow from a Fisher’s information type variation, but rather from the idea of information and two quantities of interest, namely a kind of probability exp(ipx) and x. The modulus of 1 is equivalent to the single piece of information present i.e. v=constant. To see Fisher’s information enter the quantum picture, one may consider a particle in a box with infinite walls. In such a case, exp(ipx) and exp(-ipx) are present or cos(px) if a(p)=a(-p). There is one piece of information, so d/dx cos(px) = function(px). function(px) is the one piece of extra information like v. Finding a second moment cannot yield any further information i.e. d/dx d/dx cos(px) = constant cos(px). If it did not, there would be extra information in the system, but v=constant is the only information. d/dx d/dx cos(px), however, follows from the extremization of Fisher’ s information subject to the constraint Integral cos(px)cos(xp) CC = 1. Thus we argue that in the free particle case, extremization really represents moments (shown by derivatives) and information. We also consider the case of a potential V(x) present.

It is well known that a Lagrangian (usually kinetic energy - potential) may be extremized (subject to constraints) to obtain Newton’s laws. In (1) it is argued that the Lagrangian is linked to Fisher’s information and that there exists a general extremization process of Fisher’s extremization subject to constraints i.e. Extreme Physical Information (EPI) which accounts for the Lagrangian process. Here we note that special relativity yields EE= pp + momo (c=1) for a free particle and (E-V)(E-V) = pp +momo for one with a potential V(x). From this follows Newton’s conservation of energy equation E= pp/2m + V. We thus ask: Does special relativity guarantee the existence of a function L (called a Lagrangian) such that its extremization leads to a dynamical equation of motion? We argue that it does. In particular, we argue that special relativity ensures p= dL/dv if one defines L using the standard variable switch approach (used in thermodynamics) i.e. E = -L+pv. This, together with v=dx/dt, guarantees that the function L, when extremized, yields the equation of motion. Thus we argue that special relativity is behind the extremization/variational approaches of the Lagrangian and EPI.

In Part I, we focused on the special relativistic Lorentz invariants EE = pp + momo (c=1) and (E-V)(E-V) = pp + momo (c=1). We argued that these together with the variable switching forms (familiar in thermodynamics e.g. E(S,V) but F=E-TS → F(T,V)) i.e. E = -L1(v) + pv and (E-V) = -L1(v) + pv lead to p=dL/dv (partial). From this, one may show using the variable switching forms that an equation of motion follows from the extremization of L. Here we focus instead on the Lorentz invariant -Et+px (for a free particle) which we write as: L1(v)t = t(-E+pv) and L1(v)t = t( -(E-V) + pv) for a particle with a potential. We show again that dL/dv partial = p from which the extremization of L =L1-V follows. Interestingly, this same Lorentz invariant with another partial derivative linked to p, namely Action = Lt = -Et+px → dAction/dx = p. In this case, the independent variables differ. Forming an eigenvalue equation: -id/dx exp(iA) = p exp(iA) leads to quantum mechanics. This time, in the case of a potential one cannot simply use E→ E-V(x) as many exp(ipx)s are stirred up to create the distribution W(x)=Sum over p a(p)exp(ipx). If one, however, uses -id/dx as a momentum operator (which follows from -id/dx exp(ipx) = p exp(ipx)) and applies this to the Lorentz operator: (E-V)(E-V) = pp + momo (c=1) then one obtains the Klein-Gordon equation (id/dt partial-V) (id/dt partial -V) W = -d/dxd/dx W + momo W. Thus a quadratic form holds with E-V, but the linear form -Et+px only holds for a single p in the quantum (but not the classical) case.

The notion of extremizing Shannon’s entropy subject to constraints is well known in classical statistical mechanics (e.g. in the case of a Maxwell-Boltzmann gas). There, however, the maximization of the number of arrangements is the key physical idea. Alternatively, in dynamics there exists the extremization of Extreme Physical Information (EPI) (which makes use of Fisher’s information) (1). This may be used to derive Newton’s equations (from the Lagrangia ) and the time-independent Schrodinger equation. We suggest there exists a unifying idea, namely the extremization of pressure subject to constraints, which accounts for both the extremization of entropy and (EPI). We argue that this is the key idea behind the two seemingly different extremization approaches.

Note March 9, 2023 At the same time that Fisher's information is linked with variance of an estimator, it is also linked with a classical notion of pressure for a particle in a box with infinite potential walls. Thus minimizing the variance subject to Integral P(x)dx=1 may also be thought of as extremizing pressure with respect toIntegral P(x)dx=1.We argue that the cos(px), sin(px) distributions are linked with the impulse p and at the same time the particle moves with an average speed p/m (nonrelativistically). A quantum free particle is represented by the free particle wavefunction exp(ipx). This consists of two periodic distributions (in 2 dimensions) such that their modulus is 1. sin(px) and cos(px) are a possible solution set. sin(px) and cos(px) are, however, linked to a unit radius which rotates in 2 dimensions. Alternatively one may note that d/dx cos(px) = -p sin(px) and d/dx sin(px) = p cos(px). Thus one may determine the form of a free particle “probability” using the notion of periodicity, 2 dimensions and a constant modulus. No concept of extremization is needed. It is interesting to note, however, that d/dx d/dx cos(px) = pp cos(px) and d/dx d/dx sin(px) = pp sin(px). Each distribution sin(px), cos(px) by itself represents a solution of extremization problem, namely the extremization of Fisher’s information defined by I= Integral dx P(x) {d/dx ln(P)} {d/dx ln(P)} where P(x)=W(x)W(x) and W(x)=cos(px) or sin(px). Then I = Integral dx dW/dx dW/dx. Extremizing with respect to the constraint Integral P(x) = 1 yields: d/dx d/dxW = constant W. cos(px), sin(px) and exp(ipx) are solutions of this extremization equation. In order to have a solution which treats all x points the same in a certain sense, one may choose exp(ipx). The point is that even for a free particle, the two distributions involved are linked with extremization of Fisher’s information (with Integral W(x)W(x)dx=.5Length), although one may obtain exp(ipx) with no extremization and no knowledge of Fisher’s information. Why should Fisher’s information be extremized? Fisher’s information is a statistical quantity which appears in the Cramer-Rao bound: var(estimator) >= 1/ I. One may take the inverse of this equation and consider variance(estimator) to be linked to a spread or uncertainty in x. If one wishes to minimize this, one may maximize I subject to constraints. Thus there seems to be a purely statistical approach (i.e the extremization of Fisher’s information) even for a free particle. This extremization approach, however, is linked directly to the fact that one has two probability distributions in a 2-dimensional object with a constant modulus. This does not have the appearance of any extremization, but is in fact equivalent to one for each distribution function. In fact, we argue that if P(x)=W(x)W(x) with Integral P(x)dx=1, then if W(x) is cyclical/periodic i.e. d/dx d/dx W = constant W, then W may be obtained through the extremization of Fisher’s information subject to the constraint Integral P(x)dx =1. Thus periodicity and extremization are linked, but in a somewhat unusual manner.

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. &nbsp

In (1), a variational approach is applied to Fisher’s information (with constraints) to obtain a differential equation in P(x)=probability to be at x. It is suggested the equation may be simplified using P(x)=W(x)W(x) where W(x) is real. The result is the nonrelativistic Schrodinger equation with mass=1. As a first observation, we note that Fisher’s information, which is a purely statistical quantity, should not necessarily yield a nonrelativistic equation which is an approximation of relativistic results. In particular, we argue that Fisher’s information is used to find not necessarily 1/2m , but this is a minor point. A more important issue, we argue, is the meaning of P(x) in a bound system. A classical bound system is sometimes said to have a P(x)=C/v(x) i.e. P(x)dx is proportional to dt, the amount of time a particle spends in each equally sized dx. In the high energy limit, the envelope of a quantum bound probability P(x)=W(x)W(x) (W(x)=wavefunction) is said to roughly match C/v(x), yet this density does not follow from the extremization of Fisher’s information (together with two general constraints which yield E and V(x)=potential). This begs the question: What is P(x)? If one considers a particle moving at constant speed (i.e constant momentum) then P(x)=constant because each dx point carries the same probability. Classically a particle placed in a box with infinite potential walls would have a P(x) =constant and p and -p motion would be separated in time. Thus if P(x) is to have meaning in a box, one must conclude that the probabilities associated with p and -p are not constant, but rather functions of x. This then leads to the notion of a complex new kind of probability (which maps to P(x)=constant) for a free particle i.e. W(x) =cos(px) + i sin(px) so that the modulus W*(x)W(x)=P(x)= 1. These ideas seem to go beyond the notion of Fisher information. What then is the role of Fisher information? We argue it is related to two very specific moments of the distribution cos(px) or rather W(x) = Sum over p a(p)cos(px) with a(p)=a(-p), namely and . In classical physics there exists p and pp at each x point (but one is the square of the other), but with distributions does not equal to in general. Both and are linked to physical concepts which exist in Newtonian mechanics, namely impulse and work. With the presence of cos(px) one has a combination of Newtonian mechanics and a statistical distribution. We argue that the role of Fisher’s information is to allow one to transform from to . In particular, -idW/dx appears in Fisher’s information (and is linked to ), but the specific role of a variation is to transform it into which appears in a conservation of energy equation. We argue that this is why Fisher’s information may be used to derive the Schrodinger equation.

In (1) the Schrodinger equation is derived using the extremization of Fisher information in a certain way formulated in (2) i.e. Extreme Physical Information. Fisher information, however, is defined in terms of P(x) which in quantum mechanics is W*(x)W(x) where W(x) is the wavefunction. As a result, the Schrodinger equation is derived in terms of P(x) and then the mathematical substitution P(x)=W(x)W(x) for a bound state “simplifies” the form of the equation. Otherwise there would be no need to introduce the wavefunction. Furthermore P(x) = constant for a quantum free particle which is also a solution of the Schrodinger equation. For P(x)=constant, however, one would normally not consider using the Fisher information. We further note that in a physical example, namely that of the reflection/refraction of light from an interface of media one is forced to introduce the concept of a wavefunction i.e. the square root of flux, otherwise the problem cannot be solved. Thus in this situation one does not solve the problem in terms of P(x) and then introduce W(x) to simplify the form. In particular, if one considers a single photon moving to the right, it either reflects or refracts (with probability), but the ensuing equations involve the weight of the square root fluxes (wavefunctions) of all three. How can this be unless there is uncertainty in time and space which allows one to have all three together? Thus the reflection/refraction problem depends on square root probability or flux and introduces the idea of uncertainty in x, t. Given the translational invariance in the problem, it is natural to introduce the translation operator d/dx. In fact, one may define the uncertainty in x such that one has an eigenfunction of -id/dx i.e. exp(ipx). This is complex and so its modulus respects P(x)=constant which would yield a Fisher’s information of 0, meaning that there is no probability and hence no need for Fisher’s information. There is, however, probability in x, but a different level than the deterministic x=vt where v=constant. Thus we argue that one may obtain exp(ipx) strictly from the notion of translational invariance. In other words, Fisher information is not required to find this form and the quantum physics associated with it. Nevertheless Fisher information and EPI (Extreme physical Information) may be used to derive the Schrodinger equation (applied to a bound state) and one may deduce exp(ipx) from this. We note, however, that Fisher information makes use of d/dx which is the translational generator and so the idea of translational invariance and Fisher’s information/EPI may be linked.

In Part I we noted that in quantum mechanics, and even in the case of special relativity, there is an arbitrariness associated with x and t. For example, momentum in one dimension is exactly 0. One cannot arbitrarily shift it. If x=0, however, this is arbitrary, it may be shifted. We argued that this arbitrariness and the independence of x and t in a rest frame mo, x=0, t=to should carry over into the results seen in a moving frame, but in that case x’/t’=v because a Lorentz transformation links (x=0,t=0) to (x’=0, t’=0). Thus it appears that the arbitrariness of x,t and their independence have been broken, but we argue that a special Lorentz invariant -Et+px restores it by creating a time period hbar/E and wavelength hbar/p. In other words, a mathematical point is an abstraction representing a wavelength region defined by p through a complex probability exp(ipx) i.e. the wavefunction (and a similar exp(-iEt for time). In this note, we focus on these points in further detail. A point x is associated with p or -p in one dimension, but a distribution, say cos(px), combines a fixed p with probabilities and gives the impression of positive and negative momentum values when the slope is positive or negative. Thus cos(px) by itself shows no overall translational motion, but contains regions of positive or negative momentum i.e. zero point motion. One must superimpose on this zero point motion (which exists for all energies) actual “translation” to the right or left. Given that positive and negative p’s already exist in one dimension which have nothing to do with translation, one is forced to move into a second dimension to create the translational motion which is associated with a probability such that each x point has the same value. This then leads to exp(ipx). Next we consider a bound state situation. In the case of exp(ipx), cos(px) and sin(px) are linked to the positive/negative zero point type energy which has nothing to do with translation, while p has everything to do with translational motion. Thus if one is only interested in the translational motion (as Newton was) one may map exp(ipx) to p at x. An interesting feature, however, that appears in quantum mechanics is that is the zero point type motion which creates a very specific kind of averaging combines with pp/2m values to create the notion of translation(to the right and left at the same time so to speak because time is removed from the bound problem. This is very different from the quantum free particle which has p and pp/2m at each x point and is due to the two dimensional nature of exp(ipx) which leads to pexp(ipx) - pexp(-px) not equal 0. Thus the nature of “acceleration” in a bound quantum system is closely linked to zero point energy (i.e. the back and forth action-reaction momentum) which is already present in a free particle. In fact, for a bound system, this zero type motion is linked to the average -idW/dx / W (W=wavefunction) which is not the same as prms(x) obtained from -1/2m d/dx dW/dx /W.

In previous notes we have argued that the notion of quantum period and wavelength follow from the Lorentz invariant A = - Et+px. In this note, we wish to investigate these ideas more closely. For example, hbar/E and hbar/p yield -Et+px = 0 so one may shift to, xo, but such a shift does not hold for another Lorentz invariant: xx-tt = constant. As a result, the notion of frequency and wavelength are governed by a constraint equation linked to p and E which are considered exact numbers calculated using mo (rest mass) and v velocity. We argue that given a rest state with mo, x=0 and t=to there is actually decoupling of space and time and arbitrariness of their values. In other words, there is a dualism of determinism and indeterminism already in the rest frame with p=0, v=0 being deterministic, but x,t values being independent and arbitrary. This dualism should be preserved if one views the system from a frame moving at constant speed -v. Then the particle is seen to move at speed of v which is deterministic as are E and p. The arbitrariness of x,t in the rest frame and their independence have now been supposedly lost as one has a ‘x/t’=v. In other words, “information” has been lost. We argue that one should seek to regain this information. E’,p’ are precise values and are linked with x’,t’ through the Lorentz invariant A=-Et+px. This invariant allows for independence of x and t, and also for shifted values whose ratio “x”/ “t” do not equal v, yet keep A constant. Thus we argue that seeking independence of x and t and a notion of arbitrariness in x’, t’, both features which exist in the rest frame, leads to the notion of a quantum period and wavelength. This, however, begs the question: Why does a shift of period and wavelength leave -Et+px unchanged, but change the Lorentz invariant xx-tt?

In general, x and t are considered free variables taking on all real values (with t>0). A particle’s velocity may be calculated from x(t). For the Lorentz invariant A= -Et+px one may consider E for a free particle to follow from Integral F dx and p from Integral Fdt. If one wishes to put these on an equal footing, then Fdx requires a dt and Fdt a dx. This means t is replaced with dt and x with dx in -Et+px. What does it mean to have t and x intervals associated with a specific E,p and continuous x(t) motion? There does not seem to be either an x or t interval. There are, however, dx and dt intervals associated with the creation of E and p from p=0.In other words, these dt, dx intervals are physically real properties just like p and E and are manifested experimentally. If one imposes time reversal symmetry, not only should these intervals represent a kind of probability, but they should be symmetric. Thus, x=0 and t=0 are starting points such that A=0. A is a Lorentz invariant and so the end point should also have A=0 for time reversal to apply. This time reversal leads to two symmetric distributions, one in time with hbar/E as a period and one in space with hbar/p as a wavelength.This, however, does not show the direction of motion, say in the spatial distribution. Thus the symmetric x-probability distribution should ultimately be linked to both p and -p i.e should have a vector nature, although this has to be formally proved. A full x-probability distribution linked with p should show the direction of motion. One may write F(px) for the even symmetric function described above and G(px) and G(-px) for skewed full probability distributions such that G(px) + G(-px) = 2F(px). This implies G(px) = F(xp) + B(px) where B is an odd function. We discussed the idea of G(px) being the sum of an odd and even function in previous notes, but did not try to give a reason why F(px) should be an even function with probability endpoints of 0. In this note, we argue that this is due to time reversal invariance. A free Newtonian particle follows x= vt and this should hold on average for a quantum particle which travels through space measured by an external ruler and clock. Above, however, we discussed probability distributions due to p. In fact their sharpness d/dx is proportional to p. How does one eliminate x dependence? As we argued in a previous note (2), one may take the AND condition G(px)G(-px) which should remove all motion.This should be a constant. It is not just motion which is removed, but overall impulse which is the cause of the x-distributions. The square nature of G(px)G(-px) explains why quantum mechanics deals with square root probabilities. These are the probabilities which are linked to impulse, hence to interactions, but the spatial distribution cos(px) from exp(ipx), does not describe the motion x=vt which is also real.The problem, as we show, is that G(px) cannot be real. This forces one to adopt a two dimensional probability and ultimately obtain exp(ipx) i.e. the two dimensional vector one might expect.

Newtonian mechanics deals with smoothness in time and space through x(t) where the “details” occur within dx and dt which tend to zero. Thus there are no discontinuities in x and t. We have argued that the quantum free particle wavefunction pieces exp(-iEt) and exp(ipx) deal with discontinuities and that smoothness is due to a “modulus” of two dimensional probability e.g. exp(ipx)exp(-ipx) which yields 1 or determinism. Both exp(-iEt) and exp(ipx) contain finite measures in time and space, namely hbar/E and hbar/p and so there may be discontinuities. For example, exp(ipx) contains probability cos(px) and sin(px) within a wavelength so x=vt is only an average type of motion. Thus using exp(ipx) in a problem may result in time discontinuities because the particle is not moving as x=vt within a wavelength. In Part I we examined one-dimensional reflection/refraction of light from an interface of two media with n1=1 and n2>n1. We concluded that the quantum solution of this problem involves two equations, one linked to probabilities and the other to impulse. Multiplying the two equations together (with a special grouping, namely a spatial grouping) yields a single equation which may either be interpreted as a pressure balance equation. Thus one starts with two equations which are both grouped in time, but with the second demonstrating vector directions and obtain a single equation which is only grouped in space (which is linked with spatial grouping). By rearranging this equation, one may obtain a time grouping i.e. energy conservation equation, but the two groupings cannot exist together. This is the problem with the classical pressure picture linked to a steady state scenario and not a single photon process which displays a discontinuity in time i.e. a single photon may either reflect or refract. In this note we try to consider why classical energy conservation/ pressure balance only leads to one equation, while the quantum mechanical approach leads to the two required equations at the price of introducing a square root flux type of probability. We argue that there seem to be two factors. First there is the notion of scalars and vectors and a space versus time picture. The single conservation of energy equation AA/c1 = BB/c1 + CC/c2 where AA,BB,CCC represent the incident, reflected and refracted beams is expressed in terms of time and so vector information (i.e. -p for the reflected beam is not conveyed). If one rewrites this as AA/c1 -BB/c1 = CC/c2 then one obtains a pressure balance type of equation, but time is removed and one is grouping by space. The second and more important feature we argue is time discontinuity. If one considers a single photon, it may reflect or refract, but not both. Thus there is a time period in which to observe both reflection and refraction and more than one incident photon is needed. Newtonian mechanics in the form of steady state treats time as smooth and so everything must happen instantaneously. This is the idea behind pressure. One considers an impulse 2p multiplied by flux i.e. velocity. The flux is used to find the number of particles which strike a surface per second as if this were a continuous smooth process in time. For the case of reflection/refraction it is not. It is discontinuous because either a reflected ro a refracted photon is created each time. Thus the notion of pressure averages this over and loses some of the information in the problem. This, we argue, is why a pressure balance/ energy conservation equation is not sufficient to solve the reflection/refraction problem even though pressure is a physical concept and may be measured (on average) just as x=vt may be measured on average.

In classical statistical mechanics, e.g. a Maxwell-Boltzman (MB) gas, one calculates pressure by multiplying 2p (impulse) by v (velocity or flux). This is done because one has multiple particles and so flux is important. One needs to know how many particles strike a surface per sec as well as their impulse. What happens if one has a single photon (or particle)? This is not a steady stream situation, although one may create a corresponding steady stream situation. In particular, a single photon moving in the positive x direction may reach an interface of two media with different indices of refraction and either reflect or refract. In such a case one has a probability to reflect and one to refract which add to 1. Thus there are two unknowns (i.e. the two probabilities) and energy stays constant. One may immediately write a conservation of energy equation and apply it to a steady state scenario. This necessarily utilizes the idea of velocity because there is a flux. This energy conservation equation may be rearranged to give the appearance of a pressure balance equation which also depends on velocity i.e. flux. The problem is that conservation of energy or pressure balance is a single equation and there are two unknowns. One may not simply say that the probability to reflect is .5 and that to refract .5. In fact, in a previous note (1) we solved this problem using the idea of writing the energy equation in terms of two equations linked to differences of squares. We suggested that these two equations which only involve weights are linked to a function which together with its first derivative is continuous at x=0 .In (2) we suggested that the x-distributions cos(px) and sin(px) represent action-reaction impulse type distributions. We argued that an impulse creates momentum p in the first place and that this “impulse”, in the form of x-distributions, is still present because the object may strike a foil and impart impulse at any time. Furthermore, because cos(px) and sin(px) are probability distributions, there is no sense of time within a wavelength. On average, however, the object moves with a constant velocity. In the case of light it is c/n where n is the index of refraction and in the case of a nonrelativistic particle p/mo. The example of a single photon either reflecting or refracting allows one to formulate the problem in terms of impulse i.e in terms of the probability distribution scheme such that one does not have time and hence velocity. This scheme requires two equations. This impulse based x-probability distribution scheme or quantum mechanical approach, however, should map into the classical pressure balance because on average a photon does move as x=c/n t which it does. It is, however, the quantum approach (impulse picture with no time or velocity within a wavelength) which solves the problem i.e. yields the probability to reflect or refract which may then be linked to classical type (steady stream intensities and flux.

A classical bound system may be said to have a spatial density proportional to the amount of time dt a particle spends in a constant dx region i.e. C/v(x) (1) We stress that in this scenario there is a well defined and constant dx so that “dt” may be linked to velocity v(x). We also note that there is no spatial representation of action-reaction here even though this exists in Newton’s theory. In a classical Maxwell-Boltzmann gas there are many particles which each act dynamically according to .5mvv+V(x) = E (where v and E are particular to the particle when it is not colliding). At the same time there is a spatial density which “represents” macroscopic static chunks of matter which are balanced through pressure differences (P=spatial density(x) RT) and -dV/dx density where V(x) is the potential. From the pressure equation, T is proportional to energy which in turn is proportional to RMS speed squared. RMS speed may then be linked to a dt if a fixed dx is understood. In the classical statistical case, however, spatial density is directly linked to the number of particles in a region. Time enters in the pressure expression through a term proportional to the average kinetic energy i.e. pv where p is like an impulse and v a flux. What happens in a single particle quantum bound state? In a sense it is like a classical bound state as the equation: prms(x)prms(x)/2m + V(x) = En holds for any energy level. Can one then take vrms(x), choose a fixed dx and argue that spatial density should be equal to dt or C/vrms(x)? We argue that one cannot due to the lack of defined “dx” quantity. Why is dx problematic to define? We argue that it is because quantum mechanics is based on action-reaction x-probaiblity distributions linked to impulse. For example, exp(ipx) for a free particle captures only p or impulse (not velocity) and cos(px) and sin(px) are its action-reaction regions.These lead ultimately to a density is given by the well-known W*(x)W(x) where W(x)=Sum over p a(p)exp(ipx), although kinetic energy with pp/2m = .5pv is used to find W(x) as well. . W*(x)W(x) has crests and troughs similar to cos(px) in exp(ipx), except that these now have varying heights and widths. We try to argue that even though each x point maps to a different KEave(x), the hump physically represents action-reaction in the system (just as the humps in cos(px) do except for a single p) while KE(x) represents an average of p’s. This action-reaction is linked with impulses. In general, if one takes d/dx of the time-independent Schrodinger equation one finds an average impulse term, -dV/dx and KE(x) dW/dx / W where dW/dx / W is the average impulse. At peaks, dW/dx =0 and so an average force through time*impulse term balances -dV/dx=Newtonian force. These peaks are also associated with W*(x at peak)W(x at peak) which is the spatial density. We interpret this as representing the average time the single particle spends near x at peak. This value, however, follows from ideas of impulse and not from speed as there is no well defined dx even though there is a well defined vrms(x). Thus the ratio of W*(x)W(x) at two peaks is not the inverse of the ratio of vrms(x) except apparently when the energy level n becomes very high. We investigate these ideas. &nbsp

We have argued in previous notes that the quantum free particle wavefunction exp(ipx) represents two spatial distributions in two dimensions (i.e. at different times). We have further argued that a spatial distribution is linked to p, the impulse, needed to create the momentum in the first place. Thus a free particle contains a kind of stored impulse/force manifested by the distributions cos(px) and sin(px). Two dimensions are needed to give the semblance of smooth motion i.e. x=p/m t on average. In (1) we argued that the even in the case of a Newtonian impulse Integral F dt which one might assume occurs at a point x (as in two body collisions in a Maxwell-Boltzmann gas), there must be a dx range as the energy pp/2m follows from Integral Fdx which requires such a dx. In Newtonian mechanics everything is smooth, so there is constant acceleration within dx, but we argue that such a smooth picture violates the idea of action-reaction and inertia. Thus we argue that Newtonian mechanics should lead to the notion of an x-distribution within dx even if the particle is a point. This x-distribution which exists as the particle is under force does not disappear when the external force is removed because a particle with momentum p represents an impulse (which in turn includes a force). With the external force removed, one might expect there is nothing to maintain an x-distribution i.e. action/reaction has stopped. In one sense this is true as the particle moves through space as if each x point carries the same weight. On the other hand, the Newtonian idea of inertia, even with no momentum suggests one does not simply have a translating object. The idea of inertia i.e. rest mass should be linked then to the spatial distribution associated with the wavelength hbar/p which we argue happens in special relativity. &nbsp

In Newtonian mechanics, a space-dependent potential V(x) is associated with both a change in energy in x and a conservative force -dV/dx. There is no notion of an interaction through a resonance, even though classically resonance situations exist. A quantum free particle is represented by the space and time two-dimensional probability distributions exp(ipx) and exp(-iEt) with x and t representing independent variables, even though on average there exists the relationship x= p/m t. In the case of the time-independent Schrodinger equation, one uses a potential V(x) taken from “classical” physics which is combined with the nonclassical exp(ipx) (i.e. with time removed). An immediate issue, we argue, is that V(x) implies different potential energy values at each x, while exp(ipx) → cos(px) describes probabilities at different x points associated with the same p. According to classical mechanics, p should change from one x point to another in a deterministic manner governed by V(x). Thus there exists a contradiction. We argue that the probabilistic form cos(px) remains intact, but that the interaction between an exp(ipx) and V(x) involves the formation of a probability resonance which involves other exp(ipx)s in the problem. In other words, V(x)=Sum over k Vk exp(ikx) suggests that the potential also behaves in a probabilistic manner and one may have Vkexp(ikx) a(p-k) exp(i (p-k) x) = Vka(p-k) exp(ipx). One may argue that Sum over k Vk expIkx) is simply a mathematical formalism (namely a Fourier series) without any physical manifestation. In this note, however, we argue against this and suggest that the “resonance” involved is linked with physical probability distributions which exist in both V(x) and external energy-supplying objects (such as a photon). According to the time-independent Schrodinger equation there are discrete energy levels associated with this differential equation, so an entity which may provide energy is required to lift a quantum bound particle from one energy level to the next. Electrons in an atom absorb photons to move to higher levels. Thus the photon is an energy-supplier and the fact that it has a spatial wavelength does not seem to directly enter into the solution of the time-dependent Schrodinger equation. If, however, a particle may only interact through a probability resonance, we argue that it is not only the potential V(x) which must physically be able to interact in a probabilistic way, but also any energy entities used to supply energy. Thus we argue that a photon must have a spatial distribution i..e wavelength to allow it to be part of the formation of spatial resonances in the particle - V(x) system which it in fact does. Thus we argue that the wavelength of a photon is experimental evidence for x-probability distribution resonances which occur between V(x) and a bound particle.

In a previous note (1) we argued that a quantum free particle, represented by exp(ipx), carries the information of the force that created it and which it may also impart when striking a target in the distributions cos(px), sin(px) of its wavefunction exp(ipx). In this note, we try to quantify this argument by first examining a second collective force in a quantum bound state other than -dV(x)/dx, namely KE(x) (-dW/dx / W) where W is the wavefunction. We argue this force should be present in a particle in a box with infinite walls and ultimately in a quantum free particle.Thus -id//dx, related to the translational operator, is linked to momentum and its conservation, but also ultimately to impulse carried by such a momentum, namely p. The force is found from pp/2m p where pp/2m is the frequency. Thus p is enough information for both momentum and force.. A high resolution (i.e. small wavelength) of a high momentum particle is directly linked to the high force it may impart when striking a target. Alternatively, the notion that an impulse Integral F dt occurs at a point x, as in two-body scattering in a Maxwell-Boltzmann gas, is only achieved in the high p limit, but an MB case is considered to be a high energy, high p limit i.e. a classical situation.

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.