In a previous note (1) we argued that Fermat’s principle of least time is present in the photon wave dispersion relationship E=pc which may be written as 0 = -Et+px with x/t=c. We argued that a nonzero constant = -Et+px applies to a free particle with rest mass, but that Fermat’s principle is still present. Furthemore the form -Et + px implies the existence of a frequency and wavelength for both a photon and free particle with rest mass. (Interestingly -Et+px is both the free particle relativistic action -mo t sqrt(1-vv/cc) and nonrelativistic .5mo t vv with c=1 and v=x/t. ) Fermat’s principle implies that d/dx → momentum (x component) as discussed in(1). An eigenfunction exp(ipx) demonstraties wave behaviour of a free particle with rest mass. In (1) we argued that if one knocks out a quantum bound particle it will be in a state exp(ipx). Thus this wave type information should also be included in the bound state so wepostulated a bound state represented by W(x) = Sum over p a(p)exp(ipx) (i.e. a wavefunction) such that the Fermat result d/dx → p still holds. In this note we wish to consider the idea of a statistical equilibrium and the concept of a constant energy. In the case of refraction it seems a complicated statistical scattering process occurs which slows down the motion of a photon i.e. c changes to c/n where n is the index of refraction. Thus one piece of information, namely n, describes the complicated statistical process as in thermodynamics. If E (energy) remains constant p→ pn. Thus in this complicated statistical scenario a cos(-Et+pn x) electric field pattern remains but p → pn i.e the wavelength has changed. The idea of a constant energy may even be applied to light scattering from a mirror moving with a constant v in the negative y direction. The actual energy and momentum of the incident and reflected rays are not equal, but for E1 special = p1 v(relative 1) and E2 special = p2 v(relative 2) with v(relative 1) = c- vcos(A) and v(relative 2)= c+vcos(B) where A and B are angles of incidence and reflection measured with respect to the y axis. E1 special t=E2 special. In this note we retain the idea that a particle knocked from a quantum single particle bound state is described by exp(ipx) in keeping with Femat’s principle. We consider the bound state to still have waves as Fermat’s principle suggests d/dx →p, but they cannot be of the same height and width because of the changing value of V(x) with x (i.e. the potential). According to Newton’s law one has conservation of energy i.e. pp/2m + V(x) = E, but we consider this to apply to a p (rms) = m v(rms) in the Schrodinger equation. If the forms exp(ipx)’s are to be retained within the bound quantum state (i.e. this information is not lost) and one has a constant E (energy) then the potential must be modified statistically to allow for the idea of conservation of energy for each p, but must also represent a constant potential to retain the form of exp(ipx). Thus in short the Schrodinger equation becomes a weighted sum of exp(i p(j) x) functions with each having a different wavelength (like refraction), but interacting with different constant potentials to ensure that E remains constant for each p(j). In other words we think of a potential V(x) as a statistical set of refraction discontinuities i.e. impulses which produce exp(ipx)’s which are like light moving through a homogeneous medium i.e. a momentum eigenstates. For this to happen, however, the various exp(ipx)’s need to combine with V(x) = Sum over k Vk exp(ikx) terms to create a Vmodified(p) exp(ipx) where Vmodiied(p) is a constant because each free particle exp(ipx) is associated with a different effective potential and a kinetic energy pp/2m, but the same energy En. We argue that this results from applying ideas of Femat’s principle to a particle with rest mass. Newton’s picture of a particle accelerating i.e. x(t) emerges if one considers rms values for which all information of exp(ipx) and crests/troughs of W(x) are ignored. In particular, W(x)=wavefunction may have crests and troughs, but -1/2m d/dx dW/dx/ W = KE(x) Newton which shows no crests or troughs, but simply balances V(x) in KE(x)+V(x) = E.
In a previous note (1) we argued that Fermat’s principle of least time is present in the photon wave dispersion relationship E=pc which may be written as 0 = -Et+px with x/t=c. We argued that a nonzero constant = -Et+px applies to a free particle with rest mass, but that Fermat’s principle is still present. Furthemore the form -Et + px implies the existence of a frequency and wavelength for both a photon and free particle with rest mass. (Interestingly -Et+px is both the free particle relativistic action -mo t sqrt(1-vv/cc) and nonrelativistic .5mo t vv with c=1 and v=x/t. ) Fermat’s principle implies that d/dx → momentum (x component) as discussed in(1). An eigenfunction exp(ipx) demonstraties wave behaviour of a free particle with rest mass. In (1) we argued that if one knocks out a quantum bound particle it will be in a state exp(ipx). Thus this wave type information should also be included in the bound state so wepostulated a bound state represented by W(x) = Sum over p a(p)exp(ipx) (i.e. a wavefunction) such that the Fermat result d/dx → p still holds. In this note we wish to consider the idea of a statistical equilibrium and the concept of a constant energy. In the case of refraction it seems a complicated statistical scattering process occurs which slows down the motion of a photon i.e. c changes to c/n where n is the index of refraction. Thus one piece of information, namely n, describes the complicated statistical process as in thermodynamics. If E (energy) remains constant p→ pn. Thus in this complicated statistical scenario a cos(-Et+pn x) electric field pattern remains but p → pn i.e the wavelength has changed. The idea of a constant energy may even be applied to light scattering from a mirror moving with a constant v in the negative y direction. The actual energy and momentum of the incident and reflected rays are not equal, but for E1 special = p1 v(relative 1) and E2 special = p2 v(relative 2) with v(relative 1) = c- vcos(A) and v(relative 2)= c+vcos(B) where A and B are angles of incidence and reflection measured with respect to the y axis. E1 special t=E2 special. In this note we retain the idea that a particle knocked from a quantum single particle bound state is described by exp(ipx) in keeping with Femat’s principle. We consider the bound state to still have waves as Fermat’s principle suggests d/dx →p, but they cannot be of the same height and width because of the changing value of V(x) with x (i.e. the potential). According to Newton’s law one has conservation of energy i.e. pp/2m + V(x) = E, but we consider this to apply to a p (rms) = m v(rms) in the Schrodinger equation. If the forms exp(ipx)’s are to be retained within the bound quantum state (i.e. this information is not lost) and one has a constant E (energy) then the potential must be modified statistically to allow for the idea of conservation of energy for each p, but must also represent a constant potential to retain the form of exp(ipx). Thus in short the Schrodinger equation becomes a weighted sum of exp(i p(j) x) functions with each having a different wavelength (like refraction), but interacting with different constant potentials to ensure that E remains constant for each p(j). In other words we think of a potential V(x) as a statistical set of refraction discontinuities i.e. impulses which produce exp(ipx)’s which are like light moving through a homogeneous medium i.e. a momentum eigenstates. For this to happen, however, the various exp(ipx)’s need to combine with V(x) = Sum over k Vk exp(ikx) terms to create a Vmodified(p) exp(ipx) where Vmodiied(p) is a constant because each free particle exp(ipx) is associated with a different effective potential and a kinetic energy pp/2m, but the same energy En. We argue that this results from applying ideas of Femat’s principle to a particle with rest mass. Newton’s picture of a particle accelerating i.e. x(t) emerges if one considers rms values for which all information of exp(ipx) and crests/troughs of W(x) are ignored. In particular, W(x)=wavefunction may have crests and troughs, but -1/2m d/dx dW/dx/ W = KE(x) Newton which shows no crests or troughs, but simply balances V(x) in KE(x)+V(x) = E.
In Part I we noted that Fermat’s least time principle was equivalent to conservation of momentum in the direction of an invariant spatial variable and that this principle is contained in the dispersion relation E=pc for a light which is strictly a wave relationship. We then argued that this wave relationship should somehow be extended to a particle with rest mass because such a particle may bounce off a mirror like light. To make the connection we wrote 0 = -Et+(px)x +(py)y for light with r/t=c and constant = -Et+(px)x + (py)y for a free particle with rest mass. In both cases Fermat’s principle is contained and is equivalent to conservation of momentum in the x direction. The form -Et+px, however, suggests a wavelength proportional to 1/p and a frequency proportional to 1/E for both light and a free particle with rest mass. We argue in this note that the wavelength represents spatial information which may show conservation (e.g. of momentum in a particular direction). Thus the focus of Part I was to link a free particle with rest mass to the wave formalism of light. Newtonian mechanics describes a free particle with rest mass as well as one moving in a potential without any need for a wavelength or frequency. In fact, for light E=pc means p→pn and c→c/n where n is the index of refraction means that E remains constant. For a particle with rest mass (nonrelativistic) E = pp/2m + V(x). A change in p means one needs the potential to absorb or yield energy to keep E constant which is different from light. Given that Newtonian mechanics describes particles with rest mass one may ask: Why bother to try to link a free particle with light and Fermat’s principle as done in Part I? The answer is that a free particle (with rest mass) diffracts or shows interference just like light in experiments. Furthemore -Et+px which suggests a wavelength proportional to 1/p and frequency proportional to 1/E is equivalent to the free particle action: -mo t sqrt(1-vv/cc) (relativistic) and .5 mo t vv (nonrelativistic) with v=x/t. Thus if Lagrangian/Hamiltonian formalism is identical to Newtonian mechanics we argue that the idea of wavelength and frequency is already present. It is not manifest because one deals with averages in pp/2m + V(x) = E. To see how wavelength may be incorporated in Newtonian conservation of energy, consider a free particle in a potential i.e. E= pp/2m + V(x). Next consider knocking out the particle so it has constant momentum. We argue that such a particle should exhibit wave-like behavour i.e be described in a manner linking it with light/Fermat’s principle e.g. be described by exp(ipx). Fermat’s principle already suggests that d/dx → p so -id/dx exp(ipx) = p exp(ipx) is an eigenfunction. We then argue that if exp(ipx) describes the particle when it is knocked out of the bound state, this information should still exist while it is within the bound state. Thus we suggest the form: W(x)= Sum over p a(p)exp(ipx) as describing the particle within a bound state. On average one has Newton’s conservation of energy result, but the idea that a particle may be knocked out and should behave as a wave is also present. Furthermore the various exp(ipx) interfere so the wave nature is also present in the bound state.
Addendum Oct. 26, 2022 In ((3)) one should formally subtract i.e. d/dx{ -E1 t + (p1x) x +E2t + (p2x) x) = 0. Then p1x-p2x=0 and -E1 dt1/dx + E2 dt2/dx = 0. Thus conservation of momentum implies Fermat's principle. The minus sign between dt1/dx and dt2/dx follows from t1 being a variable of x and t2 of (L-x) where L is the length of the mirror such that initial and final y values have the same magnitude. (This does not change the conclusions made in the note we argue.) In a previous note (1) we argued that Fermat’s least time principle applied to light involved finding an expression for time in terms of spatial variables and the taking of d/dx where x is an invariant spatial variable in the problem. For example, for reflection from a stationary or constantly moving mirror in the y direction or refraction with different n(i) (indices of refraction in the y direction) x is invariant because there is no medium discontinuity in the x direction. We argued that Fermat’s principle is equivalent to a conservation law in the direction of the invariant variable and that this conservation is in fact that of momentum in this direction. In (2) we argued that E=pc with c=x/t i.e. the wave dispersion equation for light contains Fermat’s principle because one may write it as 0= -Et+px and apply d/dx using t(x) with x being the invariant variable. With conservation of momentum in the x direction and E1 = E2 for the two rays: dt1/dx + dt2/dx = 0 for a two ray problem. Fermat’s principle and E=pc pertain to light. The latter may be written as 0= -Et+px ((1)) for light with x/t=c which as we argued above leads to Fermat’s principle if one takes d/dx for two rays. One may note, however, that one also obtains Fermat’s principle for constant= -Et+px with momentum conservation in the x direction. Furthermore a particle with rest mass bouncing off a floor may behave like light. This we argue leads to the idea of considering two different frames. For light -E1 t1 + p1 x1 = 0 = -Et + px with x/t = x1/t1 = c. For a particle with rest mass -mo t1 + 0 = -Et+px with x/t=v where c=1 and mo is energy in a rest frame. (This is in fact the idea of a Lorentz invariant.) Thus Fermat’s principle for a free particle seems to appear in the form -Et+px = constant. The constant is 0 for light. The form -Et+px is consistent with a wavelength proportional to 1/p and a frequency to 1/E not only for light, but for the particle with rest mass. We argue that light moving in a medium is a complicated statistical process of scattering, but that the problem is treated in a “thermodynamical” manner by characterizing the medium by a single number i.e. single piece of information, namely the index of refraction i.e. c→c/n and p→pn for light. We argue that movement within the medium is like moving in a kind of constant potential for light. -Et+ (pn) x still holds and there is still conservation of momentum in the x direction for two media. Thus the idea of d/dx → p is already present in the photon form -Et+px which follows from Maxwell’s equations, but could also be derived from Fermat’s principle as argued in (2). In other words, ideas of quantum mechanics for a particle with rest mass could ultimately follow from Fermat’s principle, we argue. A free particle with rest mass is linked with constant= -Et+px and if one wishes to have an eigenfunction of momentum then: -id/dx exp(ipx) = p exp(ipx). For a constant potential, just like with light, p would change, but the form exp(ipx) would remain. Thus it seems that even in average situations (like refraction) one still wants to have a periodic type spatial function W(x) such that -d/dx d/dx W(x) /W(x) = <pp> such that <pp>/2m + V(x) = E for a particle with rest mass. This is the Schrodinger equation and it is found mathematically that only certain En are allowed if one wishes to have specific boundary conditions like W(x) →0 at +/- infinite for say an oscillator problem. Thus we argue that the fundamental idea of taking a derivative of an invariant spatial variable in Fermat’s principle which automatically yields conservation of momentum is a basic idea found in quantum mechanics even for a particle with rest mass, namely that d/dx is associated with momentum. Mathematically d/dx is a translation operator and so the link with Fermat’s principle shows how translation is linked with a spatially invariant spatial variable e.g. x. Furthermore Fermat’s principle shows that even complicated statistical scattering problems such as refraction (in which the photon scatters off of different molecules in different media) can be reduced to a periodic average problem with p→pn. This suggests that for a particle with a rest mass in a potential V(x) one should also have a periodic spatial variable such that d/dx d/dx is linked to average momentum, we argue.
In a previous note (1) we argued that the translational generator d/dx being associated with momentum (px) already appears in Fermat’s principle for both reflection and refraction. We suggested that d/dx is linked to an invariant spatial variable which in turn is associated with momentum conservation in the same direction. This may be written as: d/dx (px)x = (px). Alternatively (px)x = -i ln(exp(i px x) ) with exp(i px x) being a directional probability distribution (i.e. values for p and -p differ) and ln(exp(i px x)) being of the form of information in information theory. One may then ask: How does one link average momentum in the case of a bound state with a potential V(x)? In general, such a bound system is solved using the time-independent Schrodinger equation i.e. conservation of energy using d/dx twice and a p(rms) not a p average obtain from -id/dx ln(W) = p(x) where W(x) is the wavefunction and pave= W dW/dx. After all, the form exp(i (px) x) may be obtained from considerations of d/dx and p, not energy conservation (even though E=pc for light). We argue that exp(ipx) is a directional probability. If both p and -p are present then: p exp(ipx) -p exp(-ipx) is not zero because the spatial probability distinguishes between the two. Only by multiplying exp(ipx) and exp(-ipx) does one remove motion. If p and -p are present, however, as in an OR situation, i.e. exp(ipx) + exp(-ipx), then it is as if the particle is bound or localized and P(p,x)+P(-p,x) = C cos(px). We consider the two components of exp(ipx) in: Integral dx cos(px) sin(px). Why does this equal 0 from a momentum perspective? sin(px) = 1/p d/dx cos(px). Thus the integral is: Integral dx { P(p)+P(-p)} momentum(P(p)+P(-p)) } or the average momentum at x. Overall the particle is bound and so there can be no momentum globally so the integral must be zero and hence cos(px) and sin(px) must be orthogonal over a period even though they represent the same p value. This same idea holds for a linear combination of {P(p,x)+P(-p,x)} sums for different p’s. (We consider overall positive or negative parity.) Thus: Integral dx W dW/dx = 0 holds for even-odd reasons, but also for the physical overall zero momentum. Even with a potential present: -i WW d/dx ln(W) = Pave(x). In other words, bound state quantum mechanics is linked to W dW/dx integrating to 0 even though there are positive and negative pockets at various x values. This follows from the orthogonality of cos(px) and sin(px) over a wavelength from a free particle which conserves momentum. This result holds for any V(x) even though V(x) fixes the coefficients of the linear combination of Sum over p a(p) {P(p,x)+P(-p,x)} using energy considerations. Zero average momentum follows from momentum conservation found in exp(i (px) x). In fact we have argued that the unusual probability wave form exp(i (px) x) exists to ensure constant momentum (momentum conservation) for a free particle. We argue that considering average momentum equal to 0 overall leads to ideas of zero point energy, orthogonal energy states and a crest/trough pattern without considering the time-independent Schrodinger equation which is based on energy conservation. Thus we argue that ideas related to d/dx and momentum carry over from Fermat’s principle into a bound quantum state and its average momentum which is often not considered.
It was argued in (1) that Fermat’s least time principle for light (in a two dimensional (x,y) reflection or refraction scenario) minimizes time by actually varying it in terms of a spatially invariant variable i.e. x (i.e. taking d/dx) with the flat face of the mirror/medium along the x axis at y=0. This is equivalent to conserving momentum in the x direction. Thus d/dx is associated with (px) which may be associated with the Lorentz invariant -Et+p dot r. In other words, d/dx emerges as a momentum operator (up to a factor) and d/dt as an energy operator, yet both are generators in time and space linked with invariance of constant values i.e. conservation. Furthermore an eigenvalue equation: -id/dx exp(ipx) = p exp(ipx) and id/dt exp(-iEt) = E exp(-iEt) automatically (by the nature of eigenvalue) leads to orthogonal eigenfunctions linked to conservation of energy and momentum. In special relativity rest mass mo represents a constant in time which may exist at any point x. In fact, it is often thought of as a point, for example a center of mass point. The actual object with mo, however, may be large with moving parts all within a “container”. What then does x,t in -Et+px mean for constant E,p? It seems that x,t are center-of-mass coordinates for the overall system. We argued in previous notes that 1/p is proportional to a wavelength so for p→0, mo may be at any x point with the same probability. Within the system with moving parts, however, there is motion and an x scale for each moving piece. For a single particle bound system, there is a single energy at each x which is like mo and a potential V(x) and KE(x). In (2) we argued that given the free particle relationship -EE+pp= -momo (c=1) one may retain a constant E value by replacing E with E-V(x) and pp with <pp> (and not ). How do the probabilities appear? They are linked to -Et+px with frequency hbar/E and wavelength hbar/p and are specifically manifested by exp(-iEt) and exp(ipx). For and <pp> we use spatial probabilities i.e. W(x) = Sum over p a(p)exp(ipx). Each exp(ipx) is linked to the operator -id/dx and invariance, but pp is linked with the operator -d/dx d/dx and one does not use an eigenfunction of this operator. Thus the notion of conservation of momentum is retained in the exp(ipx)s and their linear combination W(x) yields a function such that an average kinetic energy exists at each x point i.e KE(x)=KEclassical = -1/2m d/dx dW/dx / W. This is combined with a V(x) to ensure constant E at each x point i.e. create a kind of mo piece linked with id/dt. We argue that E being constant at each x is a statistical equilibrium type of result. Each exp(ipx) is associated with an average motion of x=p/m t and given that exp(ipx) and exp(-ipx) are included, there is also the notion of which is positive and negative in different regions and integrates to 0. prms(x) from <pp> becomes the classical p(x). Thus bound state quantum mechanics seems to be based on a free particle treatment which follows from the special relativistic Lorentz invariants -EE+pp=-momo and -Et+px = constant, directly linked to conservation ideas. Conservation of momentum of an internal bound particle is maintained through exp(ipx), while <pp> by itself is not linked with conservation as there is no eigenstate, but is combined with a V(x) to create a constant En linked with the entire system (and P=0) which then becomes associated with invariance in time and the generator id/dt. In this note, we try to argue that the notion of W*(x)W(x) (where W(x)=wavefunction= Sum over p a(p)exp(ipx)) is spatial density and P(p)=a*(p)a(p) based on spatial invariance and conservation of momentum for a single particle bound state even though there seems to be no spatial invariance or notion of conservation of momentum in W(x) which is simply a solution of the time-independent Schrodinger equation. One may move from a W(x) scenario (i.e. Schrodinger equation) to a W*(x)W(x) picture, by trying to “pick out” invariant (p only terms) using the orthogonality of the exp(ipx)s. Thus W*(x)W(x) (spatial density) emerges naturally in this approach.
In a classical bound state, conservation of energy: p(x)p(x)/2m + V(x) = E holds for all x. Examined in a statistical sense, one may consider each constant dx to have a probability density proportional to dt(x) where v(x)=velocity =dx/ dt(x). Thus the particle is followed in time. In quantum mechanics, even for a free particle exp(ipx), there exists a wavelength hbar/p. Within this region one cannot follow the particle in time. In fact, there are two probability distributions cos(px) and the shifted sin(px). The two dimensional nature of exp(ipx) is required to describe the direction of motion. There exists a mathematical abstraction or average which holds for any x within the wavelength and yields the classical x= p/m t. Thus there is a duality, but within a wavelength one cannot follow the particle exactly using x=p/m t. Rather there is no time and one thinks of probabilities to be at one x or another. The operator -id/dx yields the abstracted or average momentum at any x and (-id/dx)(-id/dx) average pp if one divides by probability. In other words: (-id/dx) exp(ipx)/ exp(ipx) = p and (-id/dx) (-id/dx) exp(ipx) / exp(ipx) = pp. These then apply to any x point. For a bound state, exp(ipx) is replaced with W(x)= Sum over p a(p)exp(ipx). Thus p and -p are included together. From a statistical point of view this is fine because a quantum bound state is described in terms of energy conservation (time-independent Schrodinger equation). Unlike the classical case, the probabilities of p and -p differ, i.e. exp(ipx) and exp(-ipx). The sum exp(ipx)+exp(-ipx) yields cos(px) which is the resolution probability distribution. Taking -d/dx d/dx of W(x) and then dividing by W(x) yields the average kinetic energy at any x within the wavelength. In other words this is the average kinetic energy which matches the classical result for any energy level. As a result, prms(x)prms(x)/2m + V(x) = En holds for any quantum level, but prms(x) is an rms average value which bypasses wave structure. That is why the exp(ipx) for p,-p and -id/dx must be used to calculate average quantities within a wavelength. Many different p,-p combinations are needed so that at each x within a wavelength the correct average kinetic energy value appears. Now consider a high energy level and choose a fixed dx length to represent a tiny region in the system, a cutoff for which there is no further resolution. p(x) is constant within this region. If hbar/ prms(x) <= dx, the wave structure of quantum mechanics does not play a role. (Note that for an oscillator, W(x) is a set of crests and troughs with different heights and widths.) As a result, there is no need for d/dx acting on W(x) to describe average kinetic energy within dx. The average kinetic energy is prms(x)prms(x)/2m and dx is treated like a point. Only if exp(ipx) is used does -id/dx yield the proper classical result. (An alternative scenario is that d/dx acting on a function divided by the function yields the same result as acting on exp(ipx) divided by exp(ipx) e.g. -d/dx d/dx cos(px)/cos(px) = -d/dx d/dx exp(ipx) / exp(ipx). If one has the quantum kinetic energy expression -1/2m d/dx dW/dx / W ((1)) and considers one momentum, namely prms(x), and uses W(x)= cos( prms(x) x), then with prms(x) constant at x, ((1)) yields prms(x) prms(x)/ 2m. Thus d/dx has not broken down even though it does not describe motion within dx which is now a point. A problem arises, however, if one considers Wn(x)= Wn1(x)Wn2(x) as this yields an equation containing dW1/dx dW2/dx/ (W1W2). Then the above procedure yields: dW1/dx/W1 (prms(x) sin(prms(x) x) / cos(prms(x) x). This does not match the classical value. Furthermore, dW2/dx/ W2 contains +prms(x) and -prms(x) with different weights while a classical scenario is -pP(-p,x) + pP(p,x) = 0 because P(-p,x)=P(p,x) while quantum mechanically it doesn’t. We analyze this situation and conclude that d/dx breaks down/ does not apply in the classical limit because it is used to calculate average values within a wavelength only. It only “works” if the effect of the derivative(s) on a function divided by the function is the same as the effect on exp(ipx) divided by exp(ipx). If a wavelength fits within dx, there is no need to calculate values at any lower x level and d/dx is not needed. In fact, there is a single prms(x) value in dx and the average time links different dx’s so that one follows the particle (on average in the sense that one ignores structure within each x) by x(t). As a practical consequence, using -id/dx together with the wavefunction W(x) can lead to problems in the classical limit because d/dx does apply. In short, using -id/dx operating on Wn(x) divided by Wn(x) leads to issues unless it produces the same effect as acting on exp(ipx) divided by exp(ipx).
In a previous note (1) we argued that both the relativistic A= -mo t sqrt(1-vv) and nonrelativistic A=t .5mvv classical free particle actions may be written with v=x/t and x and t varied independently to yield: dA/dx partial = p and dA/dt partial = -E. Thus A is equivalent to -Et + px for a free particle with x and t being treated as independent. We argue in this note that such an assumption is equivalent to a statistical picture for a free particle even at the classical level because x and t are treated as independent. Thus a constant p may occur at any x because no time is linked to x. Given that all space points are the same for constant p one expects a constant probability at all x points. Knowing p exactly means not knowing anything about which x one has. Secondly, one may create an ensemble of free particle p classical actions for a situation containing a potential V(x). Given that x and t are independent, there cannot be any notion of single p accelerating, although a vrms from an ensemble may. At a given x (x known exactly) one does not know which p is present. Thus the ideas of uncertainty of p and x follow immediately from this picture. If one considers a single particle and an ensemble of free particles in the x-t independent scenario, then each free particle action p is linked to A=-Et+px where E=pp/2m. This ensemble would arguably be stirred up by a potential V(x). If one wishes to consider an equilibrium scenario for which there is an overall En for the bound system one would expect an overall Ent for the system and for -pp/2m t pieces to disappear. Thus we postulate that pp/2m +{p part} G(p’s k’s) = En for all p where {p part}G(p’s, k’s) is an unknown function and V(x)=Sum over k Vk exp(ikx). Next one may consider the spatial situation for which V(x) is the classical potential for a single particle. The kinetic energy follows from an ensemble of free particles with the overall -Et removed. From classical considerations one would expect to have a KE(x). Thus one expects: KE(x) = {Sum over p a(p) f(px) } / {Sum over a(p)f(px) } with f(px), an orthonormal basis set, because the action contains px. (It is orthonormal because one wishes the identity of p to exist if not at a point, then over all of x.) We try to argue in this note that f(px)=exp(ipx), the momentum eigenstate which follows from A=-Et+px. We find G(p’s, k’s) which implies entanglement or mixing of a given p state with other p and k states from the potential V(x)=Sum over k Vk exp(ikx). Thus one expects an entropy linked to this momentum entanglement. Given the form exp(ipx), finding a set of ap’s which satisfy pp/2m + G(p’s, k’s)=En leads to a spatial distribution W(x)=Sum over p a(p)exp(ipx) so there should be an entropy contribution from this spatial part as well. In classical statistical mechanics one has both momentum and spatial contributions to entropy. We argue for S=Sp+Sx, but note that even though px has x, p symmetry as does the Heisenberg uncertainty principle var x var p >= hbar/2, Sp and Sx may not be symmetric in terms of important parameters such as n, energy level. In other words, Sp the entanglement entropy with V(x) may be the driver entropy sensitive to energy level changes while Sx the resulting spatial density may not be as sensitive. Furthermore the overall entropy Sp+Sx gives information about uncertainties in x and p as is already known in the literature where bound state Sp+Sx bounds are given.