A wave equation already exists in classical physics (for example for a vibrating string) with a solution of the form exp(ikx-iwt) where k=2*3.14/wavelength and w=2*3.14 frequency. K and w are not simply linked to momentum and energy, but the wave is physically visible. The wavelike function exp(ikx-iEt) also appears in solutions for the photon, the relativistic Klein Gordon equation (and the Dirac too) as well as in the Schrodinger equation. This solution separates x and k and E and t so -id/dx exp(ikx-iEt) = k exp(ikx-iEt) and id/dt exp(ikx-iEt) = E exp(ikx-iEt). If a classical equation exists linking energy and momentum (like EE=pp+momo or E=.5mvv), one may convert it into a differential equation with independent fluctuations in x and t, but linked through the energy relationship. This retains the classical information, but introduces a wave solution. For the Schrodinger equation with a potential, exp(-iEt + i Integral (0,x) dx1 v(x1)m is a solution which is equivalent to exp(i Integral (0,t) L dt1) where L is the Lagrangian. In this case, -id/dx exp(i action) = p(x) exp(i action) and id/dt exp(i action) = E exp(i action). This also holds for the relativistic Lagrangian -mo sqrt(1-vv). In the case of an electromagnetic potential v dot A, one may write Integral dt1 v dot A = Integral (0,x) dx1 dot A and so A behaves exactly like p(x). For the case of light, exp(ikx-iEt) follows as a solution of a wave equation for either the electric or magnetic fields (which follows from Maxwell’s equations). The equation: d/dt (.5eoEl El + .5/uo BB) + grad dot El x B = 0 (for no source) also follows from Maxwell’s equations. It is an equation relating energy density to momentum flux and also allows for an exp(ikx-iEt) solution if El and B have the same magnitude (c=1) and are perpendicular to each other and the motion of the photon.
A wave equation already exists in classical physics (for example for a vibrating string) with a solution of the form exp(ikx-iwt) where k=23.14/wavelength and w=23.14 frequency. K and w are not simply linked to momentum and energy, but the wave is physically visible. The wavelike function exp(ikx-iEt) also appears in solutions for the photon, the relativistic Klein Gordon equation (and the Dirac too) as well as in the Schrodinger equation. This solution separates x and k and E and t so -id/dx exp(ikx-iEt) = k exp(ikx-iEt) and id/dt exp(ikx-iEt) = E exp(ikx-iEt). If a classical equation exists linking energy and momentum (like EE=pp+momo or E=.5mvv), one may convert it into a differential equation with independent fluctuations in x and t, but linked through the energy relationship. This retains the classical information, but introduces a wave solution. For the Schrodinger equation with a potential, exp(-iEt + i Integral (0,x) dx1 v(x1)m is a solution which is equivalent to exp(i Integral (0,t) L dt1) where L is the Lagrangian. In this case, -id/dx exp(i action) = p(x) exp(i action) and id/dt exp(i action) = E exp(i action). This also holds for the relativistic Lagrangian -mo sqrt(1-vv). In the case of an electromagnetic potential v dot A, one may write Integral dt1 v dot A = Integral (0,x) dx1 dot A and so A behaves exactly like p(x). For the case of light, exp(ikx-iEt) follows as a solution of a wave equation for either the electric or magnetic fields (which follows from Maxwell’s equations). The equation: d/dt (.5eoEl El + .5/uo BB) + grad dot El x B = 0 (for no source) also follows from Maxwell’s equations. It is an equation relating energy density to momentum flux and also allows for an exp(ikx-iEt) solution if El and B have the same magnitude (c=1) and are perpendicular to each other and the motion of the photon.
In Part I we argued that two flow equations for a free particle, namely: d/dx (partial) A(x,t) = p ((1a)) and d/dt (partial) A(x,t) = -E ((1a)) may be applied to both particles with rest mass and those without. In fact we argue that there are at least three scenarios associated with ((1a)) and ((1b)). The first is classical mechanics for a particle with rest mass mo and constant velocity. Using ((1a)) and ((1b)) with p=E(v)v this scenario yields special relativity. A second solution is to write ((1a)) and ((1b)) as eigenvalue equations: -id/dx (partial) exp(iA) = p exp(iA) and id/dt (partial) exp(iA)= E exp(iA) where A=-Et+px. This yields a quantum mechanical solution for a particle with rest mo if one assumes p=Ev. In such a case one may use the special relativistic result EE = pp + momo and establish the Klein-Gordon equation. A third solution pertains to objects with zero rest mass. As argued in Part I, these do not follow p=E(v)v, but rather the classical wave result: v=E/p. In this note we examine massless particles, namely waves on a string, photons and sound, in more detail. For the zero mass case, dA=0 so there is motion in such a way as to keep A constant. If one considers periodic motion and uses ((1a)) and ((1b)), a solution which is real and periodic is: d/dt d/dt cos(A) = vv d/dx d/dx cos(A) ((2)) A second order partial differential equation is needed as one insists the solution be real. This differs from the quantum mechanical solution of exp(-iEt+ipx) which is complex for a reason. The modulus must be linked to a probability of 1 for a free particle. The zero mass solution is one of a smeared-out physical wave. We note that ((1a)) and ((1b)) deal strictly with E energy and p momentum, but cos(A)=cos(-Et+px) is associated with a period proportional to 1/E and a wavelength proportional to 1/p. This raises two questions. First, what physical entity is periodic? Secondly, how does the energy appear in the problem as energy and not just 2*3.14 frequency of the physical entity? We try to consider this issue for a wave on a string, photon and sound wave.
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.
We take as our starting point the free particle classical action which may be written as A= -Et+px if t and x are treated as independent variables. It is possible to find two relationships linking v (velocity) to x and t from dA=0. The first is the classical wave result vp = E while the second is p=Ev where E=mo/sqrt(1-vv) (c=1). Furthermore one may write dA/dx partial = p ((1a)) and dA/dt partial = -E. ((1b)) This may be generalized to two eigenvalue differential equations, but this immediately begs the question: Why do this? We argue that Lorentz invariant form -Et+px represents scaling of both t and x. For example, p scales x and we argue there should be physical relevance to this. This physical relevance results in the wavelength proportional to 1/p. In other words, -Et+px represents more than velocity. The treatment of the scaling for a quantum wave and classical wave differ, we argue, in that the former keeps E,t and p,x separate while a classical wave does not. In other words pv=E applies to a classical wave which is described by cos(-Et+px). In other words, one takes ((1a)) and ((1b)) which separate E,t and p,x into two independent equations and takes account of scaling by combining the two (to include both x and t) i.e. creating the classical wave equation: 1/vv d/dt d/dt partial W(x,t) = d/dx d/dx partial W(x,t). We argue that quantum mechanics on the other hand keeps ((1a)) and ((1b)) separate. The velocity (relativistic) p=Ev is not a wave velocity. A solution to ((1a)) is exp(ipx) and to ((1b)) exp(-iEt). These are usually combined for a free particle to give exp(-iEt+ipx) and a wave is described with a phase velocity pv(phase)=E. We prefer to keep exp(-iEt) and exp(ipx) separate . For example, for two slit interference one does not make use of phase velocity or exp(-iEt). For a bound state one uses p=Ev, but a single free particle is represented by exp(ipx) in W(x)=wavefunction = Sum over p a(p)exp(ipx). exp(-iEn t) appears after one has found En, using the exp(ipx)’s in the sense that one does not include an exp(- i pp/2m t) for each exp(ipx). We argue that by separating E,t and p,x, the latter leads to a scheme for which there is “no time”. It is true that the particle moves at a speed x/t, but p=m1v1=m2v2 etc so p does not uniquely define velocity, rather it defines an impulse hit 2p = Integral F dt . Thus the scaling px which leads to an x, p only wave exp(ipx) is linked to an interaction. If the particle is moving freely and not interacting, then one has classical determinism i.e. v=constant = x/t. As soon as an interaction occurs, one changes from variables x,t which are linked to x, P conditional (x/p) and there is uncertainty in the position of x within a wavelength. The phase in exp(ipx) is only dependent on x not on the time the particle reaches x. exp(ipx) thus represents a probability and one must consider a sum of different possible outcomes (as in a two slit experiment) in order to describe an interaction. This is very different from the local interaction of classical mechanics. Nevertheless for a bound state one makes use of the two formalisms i.e. KE(x)= Sum over p pp/2m a(p)exp(ipx) / W(x) = KEclassical(x). We try to discuss some of these ideas in this note.
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 one may obtain either the Schrodinger or Klein-Gordon equation from the classical free particle Action = Integral dt L = T L(v) by writing v=X/T and using: p= dAction/dx partial and -E = dAction/dt partial. This may be used together with an energy-momentum relationship E=pp/2m (nonrelativistic) or EE=pp + mm (relativistic) to create an eigenfunction equation by using exp(i Action) as a solution. This is equivalent to exp(-iEt+ipx). In (2), we argued that if one generalizes momentum p to Ev where E(v) is energy and uses nonrelativsitic Lagrangian equations: p=dL/dv and H=E= pv-L, one may find the relativistic Lagrangian for a free particle which in turn yields p= d/dx (TL) partial and -E= d/dt (TL) partial i.e. the form of energy and momentum for a free particle in special relativity. In this brief note, we argue that the Action/Lagrangian formalism for a free particle seems to follow directly from a flow(flux) type equation namely: d/dx Action partial = p = Ev (= mo v nonrelativistic limit). If energy is flowing out of a region given by the usual Gaussian flux equation d/dx function (i.e. grad dot vector) represented by d/dx Action partial, then one is losing energy E(v) in this region. Thus d/dt Action partial = -E. As a result, one should analyze the problem in special relativity i.e. use p=E(v) v (but no explicit form for E(v)) and then take the nonrelativistic limit to get the nonrelativistic approximation. As a result, assuming a flow problem with a function Action(x,t) leads to the Lagrangian formalism. From this one may obtain the forms of special relativity, classical equations of motion and quantum mechanics for a free particle (relativistic or nonrelativistic).
The classical action (relativistic or nonrelativistic) 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 = -Et+px is a solution. This expression also appears in classical wave forms and in the electromagnetic description of a photon (from Maxwell’s laws). -Et+px is a Lorentz invariant and so is directly linked to special relativity and different frames moving with constant velocity. In (1) we argued that x and t should be treated as independent in A. This suggests that p is a scaling in space and E in time. Thus there should be physical lengths proportional to 1/p and physical time intervals proportional to 1/E. Special relativity considers different frames (moving with constant velocity) and 4-vectors so it seems to be quite different from a scenario with t and x separated and scaling of x by p. We argue that although -Et+px is a Lorentz invariant and as such may be used to find E(v) and p(v), quantum mechanics is usually performed in a special frame, namey one in which V(x) or V(x,v) the potential is at rest. (For electromagnetism one treats (A, phi) as a 4-vector, but this is not the norm.) As a result, energy En found in a rest frame of the potential also represents “rest energy” so En is binding energy which is added to rest mass energy. In other words motion in the rest frame of a potential creates rest energy = rest mass. We argue that V(x) may be written as a sum over k Vk exp(ikx) and so no time related terms appear, only momentum linked ones. Thus momentum is the key feature for the interaction, even though velocity and time are still in the picture (just sometimes not directly in the interaction scheme). For example, a two slit interference or single slit diffraction does not depend on time or speed although these are certainly in the problem. For a bound state v and t are also in the problem as one calculates average kinetic energy using pp/2m = .5 p v which is rate dependent, but there is no acceleration which is a time-dependent phenomenon. Rather one deals with an ensemble of exp(ipx)’s because these interact with the potential in a wavelike manner to create the effect of acceleration. exp(ipx) a momentum wave linked to the scaling of x by p in px is not only involved in an interaction with V(x), but represents extra information from the classical mechanical picture. exp(ipx) for a known rest mass mo means one knows v=x/t=dx/dt, yet there is an extra piece of information namely the phase px which is involved in even “free” interactions such as two idendical particle with different phases arriving at the same x at the same t or the average kinetic energy in a bound state. Velocity v is converted into a momentum expression through p/mo in the norelativistic case and so exp(ipx) governs the whole scenario. Acceleration then becomes a mathematical abstraction of the motion of vrms(x) for any quantum bound energy level with the real physics associated with exp(ipx) being ignored.