It is well known that uniform classical motion may be described by a constant velocity ie. one which does not change in time. A different constant number, however, is needed for each velocity so uniform motion in general is not being represented by a single number in this scenario. In (1) we noted how a single number such as energy E could represent systems linked to different physical theories. For example, a classical particle may move according to .5mvv + V(x) = E where v is velocity. On the one hand a single number E describes each point (x,t), but at each point there is different detailed motion such as potential energy, velocity etc. Similarly in a Maxwell-Boltzmann gas exp(- ( .5mvv + V(x))/T) is proportional to probability. For a given E1 = .5mvv + V(x) there are many v,x points which have the same E1 and thus probability. Finally for a quantum bound state there is a single En (energy) even though there is a potential energy V(x) and average kinetic energy -1/2m d/dx dW/dx / W (W=wavefunction) at each x. In this note we ask whether a single number may apply to all uniform motion with this constant being equal then to different details in each system. The reason we ask this question is that we think a given constant velocity, say 25 m/s, does not characterize uniform motion. The problem we argue is one of resolution of x and t. For example, 25 m/s may characterize very erratic motion from x=0 to x=25 m in a period of 1 sec. This simply means the average speed is 25 m/s, not that the motion is uniform. To bypass this argument one may suggest perfect x, t resolution for all speeds. In such a case erratic motion may only exist in an extremely tiny dx or dt which approaches 0. This suggests there exists a spacetime (x,t) which may analyze all uniform motion with infinitesimal precision i.e is all-seeing. It is not clear that there exists any physical mechanism for this. We argue instead that velocity is a property of a moving object, not of spacetime. An object undergoes force over distance and receives a velocity which is one of its properties. In such a case infinitesimal resolution of t and x may not apply to all velocities. In fact if v velocity is a property of the object, x,t resolution may also be associated properties. Thus details of a particular uniform motion are not simply v (which yields p and E=pp/2m nonrelativistically), but also x and t resolution. Is it possible that these details may combine to create a single constant representing all uniform motion? We argue that this is possible and that this constant is in fact the free particle action -Et+px which holds both relativistically and nonrelativistically. For t proportional to 1/E and p proportional to 1/p (same proportionality constant) one has a single constant, Action=0, describing all uniform motion. We however, arrive at the form -Et+px without using notions of action or Lagrangian.
It is well known that uniform classical motion may be described by a constant velocity ie. one which does not change in time. A different constant number, however, is needed for each velocity so uniform motion in general is not being represented by a single number in this scenario. In (1) we noted how a single number such as energy E could represent systems linked to different physical theories. For example, a classical particle may move according to .5mvv + V(x) = E where v is velocity. On the one hand a single number E describes each point (x,t), but at each point there is different detailed motion such as potential energy, velocity etc. Similarly in a Maxwell-Boltzmann gas exp(- ( .5mvv + V(x))/T) is proportional to probability. For a given E1 = .5mvv + V(x) there are many v,x points which have the same E1 and thus probability. Finally for a quantum bound state there is a single En (energy) even though there is a potential energy V(x) and average kinetic energy -1/2m d/dx dW/dx / W (W=wavefunction) at each x. In this note we ask whether a single number may apply to all uniform motion with this constant being equal then to different details in each system. The reason we ask this question is that we think a given constant velocity, say 25 m/s, does not characterize uniform motion. The problem we argue is one of resolution of x and t. For example, 25 m/s may characterize very erratic motion from x=0 to x=25 m in a period of 1 sec. This simply means the average speed is 25 m/s, not that the motion is uniform. To bypass this argument one may suggest perfect x, t resolution for all speeds. In such a case erratic motion may only exist in an extremely tiny dx or dt which approaches 0. This suggests there exists a spacetime (x,t) which may analyze all uniform motion with infinitesimal precision i.e is all-seeing. It is not clear that there exists any physical mechanism for this. We argue instead that velocity is a property of a moving object, not of spacetime. An object undergoes force over distance and receives a velocity which is one of its properties. In such a case infinitesimal resolution of t and x may not apply to all velocities. In fact if v velocity is a property of the object, x,t resolution may also be associated properties. Thus details of a particular uniform motion are not simply v (which yields p and E=pp/2m nonrelativistically), but also x and t resolution. Is it possible that these details may combine to create a single constant representing all uniform motion? We argue that this is possible and that this constant is in fact the free particle action -Et+px which holds both relativistically and nonrelativistically. For t proportional to 1/E and p proportional to 1/p (same proportionality constant) one has a single constant, Action=0, describing all uniform motion. We however, arrive at the form -Et+px without using notions of action or Lagrangian.
Newtonian mechanics focuses on dx and dt units which tend to zero and so presents, it seems, a smoothed over or averaged type of motion. This motion in principle extends to dx and dt values which are almost zero. There is, however, a priori no reason not to expect that such a smooth averaging is at some length scale based on another kind of motion. The question is: What should that “other” kind of motion be? If it is deterministic in x and t, one may reapply the idea of this motion breaking down at yet some smaller scale. Thus it seems one should consider decoupling of x and t because one cannot follow a particle from one x point to another in time within some small length scale as this would be deterministic. The opposite approach seems to be a probabilistic one with t and x decoupled. The external ruler and clock which exist in the Newtonian picture, however, do not disappear simply because a length scale is small. Thus having x positions based on probabilities at some length scale still uses the notion of the probability itself being smoothed over because it contains x which may take on any value. This is not necessarily a problem because the probabilistic behaviour must still appear to be Newtonian on average and so x and t based on the external clock/ruler with infinite resolution need to “appear” again. (For example, an electron has a wavelength and exhibits interference from two slits, but while moving towards the slits, distance = velocity* time with distance and time measured by an external clock. It is a mathematical generalization, however, to expect that d=vt actually fully describes motion (i.e. other than on average) within a wavelength length. If it did there would be no interference pattern. The idea of a smoothed over (or averaged motion) based on probabilistic motion also appears in the classical world, but for many particles. For example, a Maxwell-Boltzmann gas with no potential has uniform density. If one considers a hypothetical box containing many molecules, it is like a Newtonian stationary object with a balance of pressure. In the case of a potential (1-D) one may consider a dx unit (with many molecules) and balance Force(x) density(x) with the difference of pressure to find the density distribution proportional to exp(- V(x)/T). These ideas are completely Newtonian, but they are smoothed over average notions. At a microscopic level molecules undergo probabilistic collisions linked to T (temperature). Thus the microscopic picture is not static - it is only the equilibrium average or big picture which is static. Even in the case of no potential, focusing on a tiny region of space smaller than the interparticle average separation and arguing that density is constant seems to be an average argument which bypasses much of the reality. If x and t may then become probabilistic and -Et+px is a Lorentz invariant, then why should there not be stochasticity linked with E and p? In fact we argue they are == as shown by Maxwell’s solution of a photon from his electromagnetic equations. E and p (energy and momentum) are average values. Their density is given by: .5C1( Electric field dot Electric field + C2 Magnetic field dot Magnetic field) and the Poynting vector for momentum. Both the Electric field and Magnetic field fluctuate as cos(-Et+px) with E and p being average values. At first this was ascribed to wave motion in a medium, but in 1905 Einstein’s photoelectric arguments against energy being spread out seem to suggest a breakdown of deterministic physics describing a wave in a medium. Instead a photon is probabilistic and has a wavefunction of Electric + i Magnetic field. (2)
A classical bound system may be said to have a spatial density proportional to the amount of time dt a particle spends in a constant dx region i.e. C/v(x) (1) We stress that in this scenario there is a well defined and constant dx so that “dt” may be linked to velocity v(x). We also note that there is no spatial representation of action-reaction here even though this exists in Newton’s theory. In a classical Maxwell-Boltzmann gas there are many particles which each act dynamically according to .5mvv+V(x) = E (where v and E are particular to the particle when it is not colliding). At the same time there is a spatial density which “represents” macroscopic static chunks of matter which are balanced through pressure differences (P=spatial density(x) RT) and -dV/dx density where V(x) is the potential. From the pressure equation, T is proportional to energy which in turn is proportional to RMS speed squared. RMS speed may then be linked to a dt if a fixed dx is understood. In the classical statistical case, however, spatial density is directly linked to the number of particles in a region. Time enters in the pressure expression through a term proportional to the average kinetic energy i.e. pv where p is like an impulse and v a flux. What happens in a single particle quantum bound state? In a sense it is like a classical bound state as the equation: prms(x)prms(x)/2m + V(x) = En holds for any energy level. Can one then take vrms(x), choose a fixed dx and argue that spatial density should be equal to dt or C/vrms(x)? We argue that one cannot due to the lack of defined “dx” quantity. Why is dx problematic to define? We argue that it is because quantum mechanics is based on action-reaction x-probaiblity distributions linked to impulse. For example, exp(ipx) for a free particle captures only p or impulse (not velocity) and cos(px) and sin(px) are its action-reaction regions.These lead ultimately to a density is given by the well-known W*(x)W(x) where W(x)=Sum over p a(p)exp(ipx), although kinetic energy with pp/2m = .5pv is used to find W(x) as well. . W*(x)W(x) has crests and troughs similar to cos(px) in exp(ipx), except that these now have varying heights and widths. We try to argue that even though each x point maps to a different KEave(x), the hump physically represents action-reaction in the system (just as the humps in cos(px) do except for a single p) while KE(x) represents an average of p’s. This action-reaction is linked with impulses. In general, if one takes d/dx of the time-independent Schrodinger equation one finds an average impulse term, -dV/dx and KE(x) dW/dx / W where dW/dx / W is the average impulse. At peaks, dW/dx =0 and so an average force through time*impulse term balances -dV/dx=Newtonian force. These peaks are also associated with W*(x at peak)W(x at peak) which is the spatial density. We interpret this as representing the average time the single particle spends near x at peak. This value, however, follows from ideas of impulse and not from speed as there is no well defined dx even though there is a well defined vrms(x). Thus the ratio of W*(x)W(x) at two peaks is not the inverse of the ratio of vrms(x) except apparently when the energy level n becomes very high. We investigate these ideas.
We try to consider free particle and single bound state quantum mechanics from the viewpoint of classical physics. We first assume that a free particle moves as x=vt. Secondly, we assume that dx and dt cannot be 0 as in Newtonian mechanics, but must be finite. This begs the question: How does one measure dx and dt? If one can measure a particle at a given x and t, then dx→0, dt→0. Thus, finite dx, dt means that a given p=constant and pp/2m=kinetic energy (nonrelativistic) interact with a force over a finite dx. In other words, one does not know where the interaction occurs within dx. As long as the free particle moves without interacting, then x=vt and there is no interest in any dx or interactions. For example, one may measure very large x and t ranges. Even if there are dx and dt effects at the start and end points, these are small compared to the large x and t values. As a result, one has a F(x,p) probability within dx, which repeats itself for the next dx (i.e. is periodic), where F is a probability for a p impulse hit at x within dx. At the same time, the particle moves with x=vt it seems within dx. In general, one might then think that F(x,p) and x=vt are decoupled. In previous notes, we have argued that one way to couple F(x,p) and x=vt is through special relativity, i.e. A = -Et+px. X and t values used would presumably satisfy x=vt, but for any x,t one has x+hbar/p and t+hbar/E as solutions yielding the same A value and so might consider dx= hbar/p and dt=hbar/E. The question we ask here is whether one may arrive at this conclusion without using special relativity. The answer seems to be that one can if one suggests that F(x,p) must contain information making it consistent with x=vt, i.e. with every x and t point having the same weight. Clearly, however, each x,t do not have the same weight within dx and dt, but are rather governed by F(x,p). This means, however, that F(x,-p) also represents a probability distribution and so if one has F(x,p)F(x,-p) there is no momentum at a given x point. In such a case, the distribution within dx must disappear and one must have a uniform distribution. This holds even if F(x,p) has the same dx value for any p. Thus, F(x,p)F(x,-p) = constant and must be periodic in x. This implies that F(x,p) = 1/F(x,-p) for constant=1. Given that F(x,p) =0 at the initial and final points of dx, then F(x,-p) = infinite at these points and that is not allowable. Thus, one is forced to use complex variables and try exp(i g(p) x). If g(p) is a function of p (i.e. not a constant), then conservation of momentum implies that exp(i g(p1) x) exp(i g(p2) x) = exp(i g(p1+p2) x), suggesting exp(ipx). What, however, if g(p) = constant? In such a case, one has issues creating an average kinetic energy: ( Sum over p a(p) exp(i constant x) pp/2m) / (Sum over p a(p) exp(i constant x) ((1)) This leads to a kinetic energy which does not depend on x and cannot be used with V(x) in a conservation of energy equation. Thus, we choose exp(ipx). Note: there is no reason to choose a(p,x) because the physics of finite dx and dt is restricted to exp(i constant x). Finally, we try to explain why the same dx=hbar/p and dt=hbar/E together with x=vt follow from special relativity.
Newtonian mechanics often employs a potential V(x) such that .5m v(x)v(x) + V(x) =E ((1)) where v(x) is velocity. Mathematically V(x) is defined at each x, but in reality there should be a tiny cutoff i.e. finite dx. Newtonian mechanics makes use of the idea of small dx lengths which tend to 0 mathematically. To first order v(x) is constant within a dx range, but the overall mathematics is continuous instead of discrete. Thus there is a continuous equation ((1)) for conservation of energy. A photon may be described by an eclectic field varying as cos(-Et+px). In terms of wavelengths exp(ipx) is often used as a mathematical convenience and noncontinous behaviour is not an issue. For example in refraction, momentum = hbar/wavelength abruptly changes and one straight line path is joined to another with a different slope. exp(ipx) is associated with repeatable wavelength patterns with a single wavelength representing a spatial distribution/ (kind of probability) region. A particle with rest mass may also be described by exp(ipx) which suggests a characteristic length hbar/p. Within this length (i.e at each x point which exist within hbar/p) the particle has p and pp/2m. Thus for a square well potential there is conservation of energy at each x point even though there is a wavelength in the problem. This is fine for a constant momentum, but what about a potential V(x) which implies p must change at each dx region? exp(ipx) is an eigenfunction of -id/dx which is linked to spatial invariance. It manifests itself physically by interference (e.g. two slits) which again involves non-smooth motion (i.e. one compares rays with completely different slopes). The characteristic length hbar/p is fixed and associated with a single p, yet momentum is supposed to change from one x point to the next. hbar/p does not fit within a single dx and this is a problem. Quantum mechanics remedies this issue by allowing for a superposition of exp(ipx)s i.e. W(x) = wavefunction = Sum over p a(p) exp(ipx) such that energy conservation is forced at each x. Thus one combines the idea of wavelength (which is physical) with energy conservation which is also physical. A consequence of this, however, is that -id/W/dx / W leads to a strange average momentum which is not the square root of -1/2m d/dx dW/dx/ W as well as discrete energies. A mathematical benefit is that one has a continuous differential equation, namely the time-independent Schrodinger equation. The question we ask is: What happens in the so-called classical limit? In the nonrelativstic case this leads to Newtonian mechanics in which there is no wavelength. How does the wavelength disappear so to speak? We suggest (as has been suggested often in the literature and textbooks) that the wavelengths in W(x) may become extremely tiny so as to not be detected easily in classical length scales. Mathematically the time-independent Schrodinger equation holds and there is a smooth W(x) at all points, but we suggest that this smoothness may not be necessary because it is not found in refracting light. If the wavelength is tiny compared to small dx regions in the classical limit, then many wavelengths of a given p fit into dx. Thus the idea of spatial invariance approximately holds within dx and there is no issue of the wavelength spilling out of the dx region as it does when it is comparable to the size of the system as in quantum bound problems. Thus within dx a single p with many (or at least several) wavelengths holds and there is a single p. The kinetic energy is pp/2m and the potential V(x) where x is in the center of dx. One may think of p changing abruptly (nonsmoothly) at the boundary of one dx with another such that conservation of energy holds. Thus one has p wavelengths which do not spill out of dx and give an approximate sense of spatial invariance together with energy conservation. The only loss is that of a smooth wavefunction, but it is not clear that such smoothness is a requirement. V(x) is physically associated with a limiting dx length (at which the concept of V(x) does not hold, but rather there are stochastic hits) and if one expands this length i.e. chooses a much bigger dx in keeping with classical scales, it seems that there should not be a problem. It as if V(x) delivers an impulse hit at each dx junction (like a series of different media in refraction).. This approach leads to the decoherence of wavelengths in the classical limit.
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).
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.