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 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 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 a series of notes (1), we argued that the time-independent Schrodinger equation may be considered as a statistical conservation of average energy equation. We further argued that V(x)=Sum over k Vk exp(ikx) delivers stochastic potential hits which create a momentum distribution W(x)= Sum over p a(p) exp(ipx) at each x. The form exp(ipx) describes a free particle and a statistical distribution is made of such free particles. The form exp(ipx) has a modulus of 1 indicating that P(x)=constant. d/dx exp(ipx) = ip suggesting the change in conditional probability with space is related to classical momentum. Given a momentum distribution, each p at x occurs at a different time. Thus, one does not follow a classical particle in time x(t), nevertheless an average energy conservation equation applies. Taking the time-independent Schrodinger equation E W = -1/2m d/dx d/dx W + V(x) W and writing EW = i d/dt exp(-iEt), one obtains a time-dependent equation. We argue that if the time-independent case represents a classical conservation of average energy, the time-dependent equation should be linked to Newton’s equation of motion. This immediately leads to a problem because we argue one does not want x(t) for a statistical system. As a result, x in the time-dependent Schrodinger equation, which represented spatial position before, now represents a spatial variable that is independent of time. We consider the oscillator case and note that an amplitude is such a candidate where x(t)=XA(t). Thus, the time-dependent Schrodinger equation ultimately becomes an equation in A(t) with the amplitude cancelling, at least in the free particle and oscillator case. From such a consideration, one may work “backwards” and find a function for which the time-dependent Schrodinger equation (equating powers of X) becomes Newton’s second law. From this, one may show that exp(i Action) where Action = Integral dt1 (0,t) L where L is the classical Lagrangian is this solution with d/dX applying to X which is an amplitude, not x(t).
The Klein-Gordon equation is often created by taking the deterministic relativistic invariant EE = pp+ momo (c=1), replacing E with E-V(x), E=id/dt partial and p with -id/dx partial and having the derivatives act on a function W(x,t). This seems to suggest that the starting point is deterministic classical mechanics (special relativistic form) E,p equation with stochasticity somehow introduced by having average E and p follow from the action of the operators id/dt partial and -id/dx partial on a stochastic function W(x,t). It also makes it somewhat difficult to understand, from first principles, why one would create a linear equation in E,p (compatible with EE=pp+momo), i.e. the Dirac equation. We suggest that the entire approach to Klein-Gordon equation is stochastic, with the EE=pp+momo equation appearing as an invariant expression of average values which describes motion as seen from different frames moving with constant average velocity v. In such a case, one is not limited to the form EE=pp+momo and there is nothing unusual about trying to create an equation linear in E and p. In particular, we argue that the free particle form exp(-iEt) exp(ipx) follows from two separate maximizations of Shannon’s entropy subject to complex constraints (for periodicity). As a result, free motion with a constant E and p is already stochastic as seen from a photon moving in a region with an index of refraction n or a particle moving in a region with constant potential. Changing to a potential V(x) leads to the notion of an average kinetic energy because a constant p is associated with a distribution in x i.e. exp(ipx). One may think in terms of these average values in order to consider frames moving with constant v. This leads to both -Et+px -EE+pp=momo being invariants. If a V(x) is present, one may replace E with E-V(x), but then needs a statistical average Sum over p a(p)exp(ipx) to describe pp.. The time portion may still be exp(-iEt) if one has a bound system characterized by a single average energy. This yields the Klein-Gordon equation.There is, however, no particular reason why one could not use an equation linear in E and p (consistent with -EE+pp=momo), because the latter follows form the an average v (deterministic view) of the stochastic exp(-iEt+ipx) two-dimensional probability.
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.
In previous notes (1) we argued that the free particle action A (either nonrelativistic or relativistic) may have v (velocity) written as x/t and x and t varied independently to yield: dA/dx partial = p and dA/t partial = -E with A=-Et+px, the Lorentz invariant classical Action = t Lagrangian. We further argued that one may write eigenvalue equations i.e. -idexp(iA)/dx partial = p exp(iA) and idexp(iA)/dt partial = Eexp(iA) and call exp(iA) a type of probability. We noted that if a potential is present one may use P(p/x)= a(p)exp(ipx)/W(x) [W(x)=wavefunction = Sum over p a(p)exp(ipx)] and establish a conservation of energy equation KE(x) + V(x) = E such that there exists the same constant E at each x. We described this in two ways. First E represents binding energy so at each x one has the same binding energy i.e. each x point maps to the same E as part of the mass of an overall moving system. Secondly, we argued that E was a parameter indicating that equilibrium existed at each x point, much like temperature T. In this note we wish to consider a different idea suggested in (2), namely that if one has a quantum state: W(x)= Sum over n a(n,t) Wn(x) this may be written as: <i |H| j> a*(i,t)a(j,t) = <W H W>.. If Hamilton’s equations for complex variables are applied to these then: i d/dt partial a*(i,t) = d/da(i,t) partial { <W H W> } ((A)) (2) argues this is the time-dependent Schrodinger equation. In other words, probability weights of the Schrodinger equations (which are complex) obey Hamilton’s classical equation. In this note, we point out that the formalism dA/dx partial = p already introduces operators into classical physics, so it follows that H is an operator which may act on states. We consider the idea of ((A)) as applied to a free quantum particle moving. Next, we apply ((A)) to a quantum bound state and argue that a classical Hamilton’s equation applied to d/dt a(p) partial may describe equilibrium. Finally ((A)) may be applied to a sum of eigenstates, but this is already shown in (2).