In a previous note (1) we argued that two flux/flow equations d/dx partial A(x,t)=p ((1a)) and ddt partial A(x,t) = -E ((1b)) lead to the Lorentz invariant solution A(x,t)= -Et+px. These have two stationary solutions (dA=0), namely dE/dv= dp/dv with x/t=v and p=E(v)v (rest mass nonzero) and dx/dt = E/p (rest mass i.e. a classical wave). The first solution leads to relativistic/nonrelativistic quantum mechanics if ((1a)) and ((1b)) are written as eigenvalue equations. The second represents classical waves and their eigenfunction equation is the classical wave equation d/dt d/dt cos(A) = 1/vv d/dx docs(A)/dx. A periodic solution, we argue, implies action-reaction suggesting force or tension in the system. This seems to be the case for a classical wave on a string with fixed ends or for a sound wave in a gas with pressure. Light treated as a classical wave, however, is more complicated because there is no medium with tension, but rather one has periodically varying electric and magnetic fields. An electric field, however, is considered a force field and so there is an analogy with a periodically changing pressure. A classical wave requires a real periodic solution. A quantum mechanical free particle solution (nonzero rest mass) does not require a real periodic solution because there is no medium or obvious real entity changing periodically, but one may establish a complex solution using eigenvalue equations:id/dt partial exp(i A) = E exp(iA) and -id/dx partial exp(iA) = p exp(iA). At first a complex solution seems unphysical, but one may note that for a fixed t (for example), exp(ipx) is 1 at x=0 and again at pL=2*3.14. Thus there is a special length in the problem which yields the classical mechanical result d/dx partial A = p namely the wavelength 2.*3.14/p. In other words the quantum solution describes the classical solution at each x= n * wavelength. Within each wavelength region one has an exp(ipx) solution which consists of two out of phase periodic functions such that the modulus is 1. We try to argue that like in a classical wave there should be some impedance to a disturbance (like tension of a string or pressure of a gas). This impedance, however, is not represented by one physical entity (e.g. pressure in a sound wave) as one has a complex solution. A complex solution exp(ipx) is not alarming because light treated as a classical wave has two perpendicular entities, namely the electric and magnetic fields, that are both periodic. One may write a classical wave equation for each, but one may write a combined quantum-like wave equation for a photon: i d/dt partial Wi + eijk (-id/dxj) Wk = 0 (where eikj is the Levi-Civita symbol) which has a vector wavefunction Wi= Electric field (i) (x,t) + i Magnetic field (i) (x,t). The only impedance in the zero rest mass problem seems to be the rest mass (inertia) which resists motion. To be more general, it seems that momentum overall should be an impedance to motion. The entire momentum (E(v)v) is associated with an impact and a change in the momentum of a quantum particle (with rest mass) may be resisted by the entire momentum. In classical physics constant velocity motion seems to be a simple translation from one x point to another. The exp(ipx) solution to ((1a)) and ((1b)) shows that this interpretation only holds for n *wavelength points. Within each wavelength region out of phase two dimensional periodic motion occurs which suggests a kind of action-reaction. This behaviour occurs for various objects such as an electron, nucleon, nucleus, atom or even fairly large molecules etc. It has been argued that this quantum motion may apply for even objects in the macroscopic world e.g. mirrors. Impedance for nonzero rest mass quantum objects is velocity dependent as one may many different velocities associated with a nonzero rest mass momentum while classical waves have one velocity (for a given tension scenario). E.g. light always as a velocity of c because it has only one tension scenario i.e. electric and magnetic fields. Changing wavelengths and frequencies are linked to changing energy and momentum, but not velocity. We try to investigate some of these ideas.
In a previous note (1) we argued that two flux/flow equations d/dx partial A(x,t)=p ((1a)) and ddt partial A(x,t) = -E ((1b)) lead to the Lorentz invariant solution A(x,t)= -Et+px. These have two stationary solutions (dA=0), namely dE/dv= dp/dv with x/t=v and p=E(v)v (rest mass nonzero) and dx/dt = E/p (rest mass i.e. a classical wave). The first solution leads to relativistic/nonrelativistic quantum mechanics if ((1a)) and ((1b)) are written as eigenvalue equations. The second represents classical waves and their eigenfunction equation is the classical wave equation d/dt d/dt cos(A) = 1/vv d/dx docs(A)/dx. A periodic solution, we argue, implies action-reaction suggesting force or tension in the system. This seems to be the case for a classical wave on a string with fixed ends or for a sound wave in a gas with pressure. Light treated as a classical wave, however, is more complicated because there is no medium with tension, but rather one has periodically varying electric and magnetic fields. An electric field, however, is considered a force field and so there is an analogy with a periodically changing pressure. A classical wave requires a real periodic solution. A quantum mechanical free particle solution (nonzero rest mass) does not require a real periodic solution because there is no medium or obvious real entity changing periodically, but one may establish a complex solution using eigenvalue equations:id/dt partial exp(i A) = E exp(iA) and -id/dx partial exp(iA) = p exp(iA). At first a complex solution seems unphysical, but one may note that for a fixed t (for example), exp(ipx) is 1 at x=0 and again at pL=2*3.14. Thus there is a special length in the problem which yields the classical mechanical result d/dx partial A = p namely the wavelength 2.*3.14/p. In other words the quantum solution describes the classical solution at each x= n * wavelength. Within each wavelength region one has an exp(ipx) solution which consists of two out of phase periodic functions such that the modulus is 1. We try to argue that like in a classical wave there should be some impedance to a disturbance (like tension of a string or pressure of a gas). This impedance, however, is not represented by one physical entity (e.g. pressure in a sound wave) as one has a complex solution. A complex solution exp(ipx) is not alarming because light treated as a classical wave has two perpendicular entities, namely the electric and magnetic fields, that are both periodic. One may write a classical wave equation for each, but one may write a combined quantum-like wave equation for a photon: i d/dt partial Wi + eijk (-id/dxj) Wk = 0 (where eikj is the Levi-Civita symbol) which has a vector wavefunction Wi= Electric field (i) (x,t) + i Magnetic field (i) (x,t). The only impedance in the zero rest mass problem seems to be the rest mass (inertia) which resists motion. To be more general, it seems that momentum overall should be an impedance to motion. The entire momentum (E(v)v) is associated with an impact and a change in the momentum of a quantum particle (with rest mass) may be resisted by the entire momentum. In classical physics constant velocity motion seems to be a simple translation from one x point to another. The exp(ipx) solution to ((1a)) and ((1b)) shows that this interpretation only holds for n *wavelength points. Within each wavelength region out of phase two dimensional periodic motion occurs which suggests a kind of action-reaction. This behaviour occurs for various objects such as an electron, nucleon, nucleus, atom or even fairly large molecules etc. It has been argued that this quantum motion may apply for even objects in the macroscopic world e.g. mirrors. Impedance for nonzero rest mass quantum objects is velocity dependent as one may many different velocities associated with a nonzero rest mass momentum while classical waves have one velocity (for a given tension scenario). E.g. light always as a velocity of c because it has only one tension scenario i.e. electric and magnetic fields. Changing wavelengths and frequencies are linked to changing energy and momentum, but not velocity. We try to investigate some of these ideas.
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.
In a previous note (1), we argued that both free particle quantum and classical mechanics follow from an energy flow/flux scheme based on: d/dx (partial) A(x,t) = p ((1a)) and d/dt (partial) A(x,t) = -E ((1b)) A(x,t) turns out to be the classical action tL(v) (L=Lagrangian) with x/t=v and the above hold for both the relativistic and nonrelativistic cases. We further argued that ((1a)) and ((1b)) lead to two sets of solutions, one classical and the other quantum. The classical solution assumes dt→0 and dx→0 i.e. flows may be followed at each point in space (and time). In (2) we argued that ((1a)) and ((1b)) are consistent with p=dL/dv (partial) and H=E= pv - L. From these together with p=Ev one may find: E=mo/sqrt(1-vv) (c=1). For the nonrelativistic case, E= mocc + .5mvv. In this classical analysis one may resolve each t and x, thus if one has a function V(x) (potential) representing energy, the nonrelativistic result is: (E-V(x)) = .5mvv. The quantum solution to ((1a)) and ((1b)) is very different. It establishes eigenvalue equations i.e. id/dt (partial) exp(iA) = E exp(iA) and -id/dx (partial) exp(iA) = p exp(iA) for the free particle. The importance of this is that new physical mechanisms seem to come into play. A= -Et+px so exp(-iEt+ipx) suggests time and spatial regions proportional to 1/E and 1/p in which there is some ambiguity i.e. there are two dimensions and there seems to be some kind of probability for the particle to be in one dimension or the other at x (and similar results for t) i.e. cos(px) and sin(px). As a result, one may not have dx→0 and dt→0. Furthermore, it is argued that a two dimensional associated probability scheme must accompany energy flow. At first this may seem strange, but if one considers a photon which has an energy flow Ev, one has electric and magnetic fields perpendicular to each other and perpendicular to the motion which undergo periodic motion linked to cos(wt) and sin(wt). Thus, the quantum particle periodic probability mimics an already known result from Maxwell’s equations. In fact, we have argued that for fermions there is additional angular motion i.e. spin linked with the recognition of 3-space. Thus we argue that quantum mechanics is involved with periodic type probability motions which are needed to resolve dx and dt (linked to exp(-iEt+ipx) and periodic probability motion (spin) and are also needed for fermions to resolve 3-space as two p vectors may have point in different directions (i.e. spin). Quantum mechanics therefore seems to be a very different theory from classical mechanics, but both appear to be based on energy flow/flux. As a result, the average of a quantum bound state should be linked to a classical system we argue in this note as one only deals with energy flow/flux.
In classical physics it is well known that a stationary solution of the classical action Integral dt L ,where L is the Lagrangian, leads to a differential equation whose solution x(t) describes classical mechanical motion. In previous notes we have argued that two free particle flow/flux equations d/dx partial A(x,t) = p and d/dt partial A(x,t) = -E lead to A(x,t) being the classical free particle action with a solution: A(x,t)= -Et+px. In this note, we consider two stationary solutions i.e. dA=0 which differ from the classical approach. One is for a particle with rest mass for which we assume p=E(v)v. The stationary behaviour leads to the form (Lorentz matrix) of special relativity and also to relativistic/nonrelativistic quantum mechanics for a free particle. The second dA=0 solution leads to a zero rest mass particle for which the Lorentz transformation of the first dA=0 solution also holds because (x,t) transforms in the same manner for both zero and nonzero rest masses. We argue that dA=0 i.e. the action is stationary because A=-Et+px is a Lorentz invariant i.e. a dot product.
In previous notes (1)(2), we argued one may write two flow/flux equations for a free particle using an unknown function A(x,t) i.e. d/dx (partial) A = E(v) v ((1a)) and d/dt (partial) A = -E(v) ((1b)) where E(v) is energy which is a function of rest mass and velocity v=constant. A(x,t), but after taking derivatives, x/t is set to v. As shown in (2), one may write A=tL(v) and find E(v)=mo/sqrt(1-vv) (c=1) (and p=Ev) i.e. the result of special relativity. Thus special relativity seems to follow directly from a flow/flux equation where motion v increases energy. Alternatively, one may obtain both relativistic/nonrelativistic quantum mechanics for a free particle, by converting ((1a)) and ((1b)) into differential equations i.e. id/dt exp(iA) = Eexp(iA) and -id/dx exp(iA) = pexp(iA) with A=-Et+px yielding the same result. In both cases, physical motion v has physical implications. For special relativity, energy changes from mo to mo/sqrt(1-vv) and there are effects on clocks and lengths. For quantum mechanics there is time and space resolution linked to dx→1/p and dt→1/E. In this note we try to investigate if there is a connection between the two. In particular we try to see why -Et+px within the free particle wavefunction exp(-iEt+px) is a Lorentz invariant.
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).
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.