In Part I of this note, we argued that d/dX Action (where Action = Integral dt L, L being the Lagrangian) for a particle with constant velocity is p whether the classical L=.5mvv or the relativistic L=-mo sqrt(1-vv) (c=1) is used. Thus, a fluctuation or error of delta X leads to a change of (delta X) p in the Action. We tried to argue that this led to the form exp(ipx). In this note, we revisit this scenario. We argue that if one wishes to have a constant error due to a fluctuation/error in X in the Action regardless of p, then delta x = 1/p is a characteristic length. At first one may expect a relative error to be of interest, but we argue that a potential may knock a particle from one p value to another. In such a case, it may be useful to have the error, and not the relative error, remain constant. Thus, one already has a wavelength, but this suggests a periodic function to describe the error of the action throughout space, and there are many possibilities. A periodic function may have positive and negative values corresponding to positive and negative X fluctuations in a regular way. We ultimately argue that exp(ipx) is ideal and is also linked to spatial density through exp(-ipx) exp(ipx)=1. Thus, we argue that two ideas are involved. First, the idea of an error/fluctuation in the Action is used to develop the idea of a wavelength. Secondly, one tries to find a periodic function associated with the wavelength which is also linked to spatial density. Finally, one tries to generalize these ideas to the nonrelativistic case of potential present. This seems to present a statistical picture.
In Part I of this note, we argued that d/dX Action (where Action = Integral dt L, L being the Lagrangian) for a particle with constant velocity is p whether the classical L=.5mvv or the relativistic L=-mo sqrt(1-vv) (c=1) is used. Thus, a fluctuation or error of delta X leads to a change of (delta X) p in the Action. We tried to argue that this led to the form exp(ipx). In this note, we revisit this scenario. We argue that if one wishes to have a constant error due to a fluctuation/error in X in the Action regardless of p, then delta x = 1/p is a characteristic length. At first one may expect a relative error to be of interest, but we argue that a potential may knock a particle from one p value to another. In such a case, it may be useful to have the error, and not the relative error, remain constant. Thus, one already has a wavelength, but this suggests a periodic function to describe the error of the action throughout space, and there are many possibilities. A periodic function may have positive and negative values corresponding to positive and negative X fluctuations in a regular way. We ultimately argue that exp(ipx) is ideal and is also linked to spatial density through exp(-ipx) exp(ipx)=1. Thus, we argue that two ideas are involved. First, the idea of an error/fluctuation in the Action is used to develop the idea of a wavelength. Secondly, one tries to find a periodic function associated with the wavelength which is also linked to spatial density. Finally, one tries to generalize these ideas to the nonrelativistic case of potential present. This seems to present a statistical picture.
The classical action: Integral (0,T) dt1 L(v(t1), x(t1),t1) may be varied to find a stationary solution which yields Newton’s second law, a standard procedure in classical mechanics. Is it possible to obtain quantum mechanical motion from this same classical action? We argue that one may write the action for a relativistic/nonrelativistic particle moving at constant speed in terms of X and T i.e. v=velocity=constant=X/T. Varying with respect to X and T yields: d/dT Action = -E and d/dX Action = p. Thus, changing X by delta X yields a change in the action of p delta X which holds for all X because of the constant velocity. If one envisions a periodic function such as exp(i Action), this behaves as exp(ipx) in terms of -id/dX. Thus, there is an intrinsic wavelength related to 1/p. This follows directly from small changes of the Action. Alternatively, one may create a differential equation such as the Klein-Gordon or Schrodinger equation from classical conservation of energy-momentum using id/dT exp(iAction) = E exp(i Action) and -id/dX exp(iAction) = p exp(i Action). We note that exp(ipx) which describes the fluctuational behaviour of the classical Action may have positive and negative values showing that fluctuations may be both positive and negative These negative and positive fluctuations are, however, periodic so there is some systematic behaviour occurring, it seems.. We argue that exp(ipx) is statistical as its modulus is 1, indicating that a free particle has equal probability to be at any x point. For a system with a potential V(x), one may create an ensemble describing fluctuations of all possible p values i.e. W(x)=wavefunciton= Sum over p a(p) exp(ipx). The fluctuations are regular positive and negative, but there is different wavelength behaviour for each p i.e. 1/p so there is interference to yield an average fluctuation pattern i.e. the wavefunction.
The classical Lagrangian L leads to Newton’s second law which is equivalent to an energy-momentum conservation equation. The classical action: A = Integral dt1 L(t1) for a constant velocity v=X/T may be varied with respect X and T i.e dA/dT = -E and dA/dX= p. Thus, one may create a differential equation in (dA/dT)(dA/dt) and dA/dx dA/dx for the EE=pp+momo equation. Alternatively, one may create other differential equations. Quantum mechanical differential equations, however, are quite different with exp(i Action) being a solution. Even this may be written as the solution of a non-Schrodinger equation i.e. id/dt exp(iAction) = -1/2m (d/dx exp(iAction)) (d/dx exp(iAction)) for a free particle. For a constant velocity, exp(iAction) is equivalent to exp(ipx-iEt). We argue that this solution is statistical because it separates x and t (as independent variables) and holds for all x and t. Thus, quantum mechanics seems to be a statistical approach using free solutions exp(ipx-iEt) to create an ensemble as in classical statistical mechanics. A difference, however, is that exp(ipx-iEt) “interferes” even if one only takes the real part cos(px-Et). We consider the case of a particle in a box with infinite potential walls to argue that the periodicity is warranted i.e. positive and negative values (interference). Even though the modulus is 1, exp(ipx-iEt) may be operated on by various generators e.g. d/dx the generator of translations. This is analogous to operations on vectors in linear algebra. As noted above, quantum equations are not only energy-momentum conservation equations. They yield a wavefunction W(x) which is an unusual momentum distribution W(x)=Sum over p a(p) exp(ipx). exp(-iEt) holds overall because average E is the same at each x. Thus, one needs a W(x) which leads to a momentum-energy conservation equation, but also satisfies certain boundary conditions which become linked to spatial density= W*W. This leads to discrete energy levels and also cases of humps and troughs in W(x). We attempt to investigate some of these ideas in this note.
In (1), the notion of a stochastic classical action is introduced and a probability is linked to exp( constant * action). The objective of (1) is not to derive quantum mechanics, but to derive stochastic dynamics based on maximum entropy in action space. We have argued in a number of previous notes (2) that one may use the classical action to create a quantum free particle probability. This classical action, however, is not chosen a priori, but rather arises due to other considerations. We suggest that stochasticity is present in Newtonian mechanics at some time scale because dt is finite in dp/dt= Force (Newton’s second law) and so postulate that force delivery at some x,t is probabilistic within some dt and dx. A priori, dt and dx are not known, but if there is stochasticity, there should be a free particle probability, i.e. a probability(x,t) where x,t represent an x, t interaction point. Furthermore, the notion of dx and dt should arise naturally. We note that the form exp(i function(x,t,p,E) may be introduced because free particles have no distributed variable (like energy in the Maxwell-Boltzmann case). exp(i function(x,t)) immediately introduces periodicity, but we have argued that stochasticity can only exist within tiny, fixed dx, dt linked to dp/dt=F. In particular, one classical Newtonian scales this stochasticity is not seen. The periodicity should be directly in x and t space, suggesting exp( i A1 t + i A2 x). This expression, however, should be Lorentz invariant, and A1 and A2 should be constants, at least for a fixed mo and v. The reason that the one must have periodicity is that the dx and dt ranges of stochasticity in dp/dt=Fr must appear naturally. At this point, one is led to exp(-iEt+ipx) with -Et+px being a Lorentz invariant which satisfies the above criteria. This Lorentz invariant happens to be the classical action in both the relativistic and nonrelativistic cases, hence giving rise to action as being a stochastic variable (of (1)) because it is a function of the two scholastic variables x and t. Alternatively, one may choose the exp(iA1x+iA2t) due to its conservation properties. One may then argue that A1 and A2 could also be conserved for x,t held constant, i.e. A1 and A2 could be linked to Newtonian energy and momentum in elastic collisions. This allows one to solve for the special relativistic g(v) =1/sqrt(1-vv/cc). Given the idea of a conserved quantity form for a probability, one may argue that exp(i Action) satisfies a maximization of Shannon’s entropy subject to a conserved constraint (which is again a link with (1).) The question then becomes: Why should the Lorentz invariant A= -Et+px be the classical action.One may note that x,t usually refer to trajectory values and so for a free particle, one has x=vt. This means that one may factor out t, i.e. A= t( -E+pv), but this is automatically the form of the Lagrangian with E being the Hamiltonian. This holds relativistically and non-relativistically with the form: dL/dv partial = p. The remaining question is: Why should the classical action be Lorentz invariant? We consider E(v)-p(v) v = -L such that dE/dv - v dp/dv -p = -dL/dv partial. This means that dE/dv - v dpd/v = 0 ((1)). A Lorentz transformation, however, changes mo in a rest frame into p=mov/sqrt(1-vv/cc) and E=mocc/sqrt(1-vv/cc). One may see that this solution may be obtained from ((1)) showing a direct link between the classical action approach and special relativity. The connection may be carried further as Lt = -Et+px is a Lorentz invariant, but also the classical action (in both the relativistic and nonrelativistic cases). We argue that the use of the action in (1) as a stochastic variable is directly related to special relativity. In summary, from finite dt in dp/dt=F, t and x are stochastic variables within some naturally appearing dx and dt. One may write a linear combination of x and t and to create a stochastic Lorent invariant variable which should appear in exp(i Lorentz invariant) and create the dx and dt units. It may then be shown that this Lorentz invariant in E,p,x,t should be equivalent to the classical action.
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.
In a previous note (1) we argued that the free particle classical action A (relativistic or nonrelativistic) may be written in terms of v=x/t and then varied in x and t independently. One finds that dA/dx partial = p and dA/dt partial = -E implying A= -Et+px with x and t independent. This is the opposite of the classical mechanical idea of x(t). Creating an eigenvalue equation yields -id/dx partial exp(iA) = pexp(iA) and id/dt partial exp(iA) = Eexp(iA). exp(iA) plays the role of an unusual type of probability we argue which describes two dimensional periodicity (cos(px) and sin(px) ). We have argued that the quantum particle is the medium for a two dimensional wavelike motion of an energy-momentum wave. A wave type structure is needed to describe dynamics (like a classical wave), but unlike a classical wave two dimensions are required because there is no preference of any x point for a free particle. Both cos(px) and sin(px) both treat x in a preferential manner as their values vary with x. For a classical wave this is fine because the creation point of the wave imposes boundary conditions, but there is no such boundary condition imposed on a free particle. exp(ipx-iEt0 must somehow then link to P(x)=constant. One may note that exp(-iA)exp(iA) = 1 for the free particle. This may be argued to be linked to spatial invariance i.e. P(x)=constant where P(x) is not exp(ipx). Here we argue that exp(-iA)exp(iA) represents a product of two unusual probabilities , one for motion in a certain direction and the other for the time reversed scenario. If the quantum particle is the medium of an energy-momentum wave exp(-iEt+ipx), then the product of the regular and time reversed unusual probabilities cancel or undo each other. In other words one does not see any change in space as long as one has a noninteracting exp(-iEt+ipx) . Given that -id/dx is a linear operator this product yields a momentum of 0. We consider this idea and its implications on a momentum distribution Sum over p a(p)exp(ipx) which may represent a bound state or other states. In particular, it seems that P(x)=W*(x,t)W(x,t) which yields a classical type of probability (P(x) real, positive and normalized to 1) is linked to a product (merging) of forward and time reversed situations so that any spatial density which remain due to the quantum particle acting as a medium for an energy-momentum wave (as well as the time reversed part) would represent an actual change in spatial density. exp(ipx) already contains two periodic spatial functions (cos(px) and sin(px)), but does not manifest it unless there is an interaction.
We try to argue that the idea of probability associated with motion may have classical beginnings in that the center-of-mass (cm) of a piece of mass moving with constant velocity v is present within a dx at some time, but moves out of dx. Flux is constant, but at one time the particle cm leaves dx and arrives at the adjacent dx. Constant flux suggests one does not need to consider time. In such a case, one might associate a periodic function showing probability being removed from a region dx (at x) (i.e. a negative distribution) followed by it being added (a positive distribution) to a region x+dx. This suggests a negative-positive cycle, but this labels certain x regions as negative and others as positive which is inconsistent with all x regions having the same weight based on a constant v. Given that time is removed, it seems this cycling may have nothing to do with velocity v or perhaps anything else for that matter. As a result, one may postulate an Aexp(ibx), where b is a constant, for motion. We will argue for a phase linear in x. AAexp(-ibx)exp(ipx) yields a constant AA, which is flux, and treats all x points the same. At this point, one cannot go farther unless one considers other issues.We consider two such issues. First we ask what happens if the particle is viewed from a frame moving at constant v? The velocity of the particle will not be the same, but we have argued that exp(ibx’) may be the form, yielding no information about b when (x,t) changes to (x’,t’). If on the other hand, the phase cannot change if the particle is viewed from the moving frame, one requires a Lorentz invariant. This suggests that the phase may actually be -Et+px. (We also show that it is not necessary to use special relativity to obtain the form exp(ipx). Rather one may consider ideas of pressure balance associated with continuity of exp(ibx).) As a second consideration, we examine a one dimensional reflection-refraction problem at a point xo for which a probability equation 1= Probability(reflect)+Probability(refract) exists. If exp(ipx) for constant E describes motion, this motion changes with changing p, yet one wishes probability to be a continuous function at xo. This suggests different weight coefficients for different exp(ipx) which seem to be associated with P(reflect), and P(refract). If x=0, one has an equation in the weights, with A+B (x<xo) matching C (x>xo) at xo. If one stops at this point, exp(ipx) does not seem to have any impact. One may note, however, that in general xo is not zero and that one wishes at the very least to have the derivatives match. This immediately links the A,B,C values to the p (momentum) discontinuity. Thus using special relativity to create an exp(ipx) has practical implications because it allows one to deal with p discontinuities. We also show though that special relativity, i.e the use of -Et+px, may be bypassed and the form Aexp(ipx) obtained as a result of its first derivative yielding p which is needed in a pressure balance equation. This then suggests A is the square root of flux.