A quantum particle is often described as exhibiting dual particle-wave nature. A particle seems to be an object which obeys Newton’s laws/special relativity with a wave also being a phenomenon existing in Newtonian mechanics (special relativity). In other words the particle and wave are governed by x-t external variables which have infinite resolution. It was originally thought that x and t in addition to being external were absolute, but special relativity changed this idea. We have suggested in previous notes that a free quantum particle with energy E and momentum p creates its own space time i.e time intervals are proportional to 1/E and space to 1/p. This means no external clock is needed and different internal rulers and clocks exist for different E,p. It also suggests finite time and space intervals. In fact, an absolute is introduced namely A=0 = -Et+px with t and x being independent. This suggests that t→b/E and x→b/p. A is the free particle Lagrangian (both relativistic and nonrelativistic) with x and t being considered independent. Thus we suggest that the external x ruler and t clock of Newtonian mechanics represent an average of the internal picture and are also described by A, but with x/t=v and not x and t being treated as independent. In (1) we showed that having discrete x spacing with a running time (i.e. Netwon’s time variable) and insisting on spatial invariance leads to dA/dx partial = p. Thus instead of considering particle wave duality we suggest an underlying wave structure which creates its own spacetime intervals (1/E, 1/p), but may be mapped in certain situations to the external running x and t of Newton’s theory. For example, consider an electron (a quantum particle) with a fixed p moving over a distance much greater than its wavelength. It may be mapped to Newton’s external running x,t values with x=vt, but this is an average mapping into continuous x and t. In reality the situation is more complicated because there may be a Bohm-Aharonov effect occurring. Thus the underlying wave picture is in fact physically important and linked to interactions. Thus a self-ruler is not simply a measuring device - interactions may occur unlike in the case of the external ruler. Given that p defines a kind of spatial invariance approximated by a periodic function exp(ipx) (eigenfunction of -id/dx) and that Newton’s running time is not needed in a quantum bound state equilibrium, one may wonder what one should do in the case of a particle bound in a V(x) potential. In Newtonian mechanics V(x) suggests non-spatial invariance and v(x) velocity changes with x as well in contrast with exp(ipx) which is periodically invariant. In fact Newtonian mechanics (and Lagrangian/Hamiltonian as well) focus on d/dt i.e x(t). We have argued above that time intervals 1/E are independent of 1/p spatial ones. It may be noticed, however, that different p values lead to different wavelengths and so even though there is a kind of spatial invariance for each, when compared to each other this invariance is broken and superpositions of exp(ipx) may create local x probability densities. At the same they create spatial momentum densities because -d/dx (d/dx) 1/2m yields a kind of average kinetic energy when normalized. This seems to describe the x-p uncertainty relationship. In order to create spatial resolution (other than the built-in one in the wavelength) one needs to superpose different momenta, each of which are spatially invariant according to their periodic scheme exp(ipx). Thus one may create an average 1/2m <pp> and obtain a vrms. This may be described in terms of Newtonian external running x and t variables which have infinite resolution, but this vrms does not map to a physical state i.e to exp(i (mvrms) x) localized at x. Rather the various exp(ipx)’s which create vrms are the physical entities as may be seen by knocking a particle out of a bound state. The physical exp(ipx) momenta states have spatial invariance and so cannot be associated with a specific position in space. Only their superposition creates the notion of specific position in space, but then there is a distribution of p. This seems to be the basis for the x-p uncertainty principle. Thus we suggest that the particle of particle-wave duality is created through a mapping of a physical wave scenario to a continuous (infinite resolution) x-t scheme which in reality is an abstraction because each p,E free particle state has it sown internal clock-ruler i.e. 1/E and 1/p time and space units. Given that 1/E and 1/p have finite values the idea of dt and dx tending to 0 only makes sense as an abstraction which only holds in certain situations.
A quantum particle is often described as exhibiting dual particle-wave nature. A particle seems to be an object which obeys Newton’s laws/special relativity with a wave also being a phenomenon existing in Newtonian mechanics (special relativity). In other words the particle and wave are governed by x-t external variables which have infinite resolution. It was originally thought that x and t in addition to being external were absolute, but special relativity changed this idea. We have suggested in previous notes that a free quantum particle with energy E and momentum p creates its own space time i.e time intervals are proportional to 1/E and space to 1/p. This means no external clock is needed and different internal rulers and clocks exist for different E,p. It also suggests finite time and space intervals. In fact, an absolute is introduced namely A=0 = -Et+px with t and x being independent. This suggests that t→b/E and x→b/p. A is the free particle Lagrangian (both relativistic and nonrelativistic) with x and t being considered independent. Thus we suggest that the external x ruler and t clock of Newtonian mechanics represent an average of the internal picture and are also described by A, but with x/t=v and not x and t being treated as independent. In (1) we showed that having discrete x spacing with a running time (i.e. Netwon’s time variable) and insisting on spatial invariance leads to dA/dx partial = p. Thus instead of considering particle wave duality we suggest an underlying wave structure which creates its own spacetime intervals (1/E, 1/p), but may be mapped in certain situations to the external running x and t of Newton’s theory. For example, consider an electron (a quantum particle) with a fixed p moving over a distance much greater than its wavelength. It may be mapped to Newton’s external running x,t values with x=vt, but this is an average mapping into continuous x and t. In reality the situation is more complicated because there may be a Bohm-Aharonov effect occurring. Thus the underlying wave picture is in fact physically important and linked to interactions. Thus a self-ruler is not simply a measuring device - interactions may occur unlike in the case of the external ruler. Given that p defines a kind of spatial invariance approximated by a periodic function exp(ipx) (eigenfunction of -id/dx) and that Newton’s running time is not needed in a quantum bound state equilibrium, one may wonder what one should do in the case of a particle bound in a V(x) potential. In Newtonian mechanics V(x) suggests non-spatial invariance and v(x) velocity changes with x as well in contrast with exp(ipx) which is periodically invariant. In fact Newtonian mechanics (and Lagrangian/Hamiltonian as well) focus on d/dt i.e x(t). We have argued above that time intervals 1/E are independent of 1/p spatial ones. It may be noticed, however, that different p values lead to different wavelengths and so even though there is a kind of spatial invariance for each, when compared to each other this invariance is broken and superpositions of exp(ipx) may create local x probability densities. At the same they create spatial momentum densities because -d/dx (d/dx) 1/2m yields a kind of average kinetic energy when normalized. This seems to describe the x-p uncertainty relationship. In order to create spatial resolution (other than the built-in one in the wavelength) one needs to superpose different momenta, each of which are spatially invariant according to their periodic scheme exp(ipx). Thus one may create an average 1/2m <pp> and obtain a vrms. This may be described in terms of Newtonian external running x and t variables which have infinite resolution, but this vrms does not map to a physical state i.e to exp(i (mvrms) x) localized at x. Rather the various exp(ipx)’s which create vrms are the physical entities as may be seen by knocking a particle out of a bound state. The physical exp(ipx) momenta states have spatial invariance and so cannot be associated with a specific position in space. Only their superposition creates the notion of specific position in space, but then there is a distribution of p. This seems to be the basis for the x-p uncertainty principle. Thus we suggest that the particle of particle-wave duality is created through a mapping of a physical wave scenario to a continuous (infinite resolution) x-t scheme which in reality is an abstraction because each p,E free particle state has it sown internal clock-ruler i.e. 1/E and 1/p time and space units. Given that 1/E and 1/p have finite values the idea of dt and dx tending to 0 only makes sense as an abstraction which only holds in certain situations.
Quantum mechanics is often compared with classical mechanics, but it seems that the momentum/spatial treatment of bound quantum mechanics provides for a particle-wave duality with both features having an experimental basis. In bound state quantum mechanics, one has a density pattern often similar to a two slit pattern i.e. with humps, yet at the same time one may knock a particle out of the bound state and obtain a probability distribution a(p)a(p) of momentum states. Describing a particle simply by its momentum even in the bound state seems to be classical as does using pp/2m to describe the kinetic energy of a plane wave in the decomposition of W(x), the wavefunction. Wave aspects are linked to classical particle features like momentum and do not change the particle energy pp/2m. The wave, we argue, may be related to some type of periodic cyclic forward backward motion which creates a spatial probability needed to estables resolution on the order of a wavelength proportional to 1/p. It is often not seen unless there is a potential or other probability waves which may interfere with it In this note, we consider the scenario that momentum (i.e. mov in the nonrelativistic case) is directly linked to a kind of forward-backward cyclic motion of a quantum particle regardless of its spin. The fact that spin exists suggests some kind of internal motion. In the case of a photon, it may be directly linked to angular motion of energy, although for spin ½ matters are more complicated. This motion again does not change classical energy pp/2m unless spin interacts with a magnetic field etc. We argue that a bound state is stochastic and so various free particle momentum states exist without being restricted to an x position. This is the particle scenario. The internal cycling motion, however, gives rise to a resolution of the order of 1/p suggesting a spatial distribution to account for such a resolution. Furthermore, the internal cycling (i.e. forward and backwards motion) even for a particle traveling to the right requires two dimensions to describe overall forwards and backwards motion. The spatial density of the cycling motion may be associated with internal changes in velocity, but overall there is a particle with mass mo moving with v (nonrelativistic case) and kinetic energy pp/2m. This holds for a free particle, but also for a particle in an infinite potential well with a single average |p|. An issue occurs, however, if one wishes to form an average of different p values at the same point x. In such a case the spatial probability density exp(i pj x) differs for each p at x and so the average [p1 exp(i p1 xo) + p2 exp(i p2 xo) ] / [ exp(i p1 xo) + exp(i p2 xo)] becomes relevant. In other words, one takes the average momentum, say p1, and links it to the spatial probability i.e. multiplies two averages and then divides by spatial probability to find a momentum flow at a specific point within a wavelength region. For the particle aspect, such a resolution is non-issue. The particle is the same at each x point in space.
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)
Newtonian mechanics (from the 1600s) defines a quantity called momentum p in terms of mv where m is mass and v=dx/dt with dx and dt tending to zero. Newton’s first law is equivalent to conservation of momentum in the absence of a force. The second demonstrates how p changes under a force. If one has equal action/reaction then overall momentum is conserved. As a consequence the idea of a “wave” associated with p does not arise. The main focus is on the idea of force. Fermat’s Least Time Principle for light (1600s), however, is based on minimizing the overall time traveled by light which reflects or refracts. Young’s interference experiments did not occur until the very early 1800s and Newton considered light to be a particle at this time. In fact light bounces off a foil creating an impact like a particle. Fermat’s principle does not directly make use of momentum or its conservation. It is focused on minimizing time. To do so, however, one must take a derivative with respect to a certain variable and set it equal to 0. This variable is the spatially invariant one in a problem. For example consider reflection (stationary or moving mirror) or refraction in two dimensions x,y. Let the mirror or medium face lie along the x axis at y=0. There exists a constraint on y and given a fixed starting point one cannot change y. This leaves x to be varied because it is invariant. Thus the extremization of time is really a statement of invariance of a spatial variable. In other words, Fermat’s principle is linked to the physical variables t and x while Newton’s focuses on p (momentum). This invariance in turn gives rise to conservation of momentum in the direction of the invariant variable i.e. (px). There is still the idea of Newton’s change in momentum as (py) abruptly changes in refraction as it hits the interface of media. The point we wish to make is that from Fermat’s least time principle one sees that conservation of momentum in the absence of a force is completely intertwined with invariance in space. The two cannot be separated in this approach and there is nothing unphysical about t or x. In fact, one may create a function A= -Et + (px)x + (py)y such that dA/dx partial = px = conserved quantity. In this function (px) and x are on the same footing. One does not have one without the other. This idea is carried over into special relativity as -Et+p dot r is a Lorentz invariant. Given that p momentum is physical (it manifests itself in impulse hits) and p and x are inseparable in (px)x we suggest there should exist a physical manifestation in x i.e. in space linked to the maintenance of conservation of momentum in this direction even in the absence of a force. This we argue leads to the physical idea of a wavelength which follows from A=-Et+px with A=0 in all constant velocity frames leading to a frequency proportional to 1/E and wavelength to 1/p. We have discussed these ideas in previous notes, but the main point we wish to make here is that equivalence of spatial invariance in the x direction in Fermat’s principle with conservation of momentum in the same direction suggests that if p is physical, the spatial invariance must also be manifested physically. It is not simply a mathematical convenience to link t to x to px, we argue. Furthermore this wavelength should be linked to establishing conservation of momentum as it does in quantum mechanics where the orthogonality of exp(ipx) i.e. Integral exp(-i p1 x) exp(ip x) dx automatically ensures p1=p without imposing momentum conservation a priori as in Newton’s first law. Both Newtonian mechanics and Fermat’s principle suggest that p may change abruptly due to a force or an immediate change in medium. The idea of a physical wavelength, however, suggests there may be interactions completely different from an abrupt change in p due to an impulse. In particular there may be interference or Bohm-Aharonov interactions. Newtonian Mechanics
In special relativity, energy and momentum are part of a 4-vector, but are also properties of a particle. X and t also form a 4-vector, but are not properties of a particle, but rather are measured relative to an external ruler or clock. Shifting the origin of the ruler/clock changes their value, but does not change the value of velocity and hence momentum.In (1), we called x and t running variables. Special relativity deals with observations based on measurers in different frames moving at constant velocities. The ruler or clocks used by these people, however, may be shifted so there is no absolute property of time or x in this case either. Shifting a ruler or clock does not change the value of velocity, it is a property of the particle (assuming there are equally space gradations and clock intervals). In order to make x and t properties of a particle, we argue (as in (1)) that one may use A= -Et+px, a Lorentz invariant. For -EE+pp, (c=1), another Lorentz invariant, there is a link between all frames with rest mass squared (c=1). Within a specific frame (E,p exact numbers), -Et+px creates a link between special degenerate (in terms of A) x and t points, i.e. x+ n constant/p and t + n constant/E. These keep A constant due to the minus sign. Thus E and p create an internal clock and ruler for a particle as we have argued in (1). This means that n constant/p are special “x” points (and the same with n*constant /E and time) and suggests a probability distribution in x with a period of constant/p. Thus E and p interacting with x,t allow the latter to be properties of a particle, but at the cost of introducing probability. Given the notion of probability, this suggests that entropy should come into play. One may ask: Is entropy maximized subject to a constraint? If so, what constraint should one use? We argue for a constraint of ipx associated with the period constant /p, the “i” being associated with periodicity. In such a case, the Lorentz invariant -Et+px leads to exp(ipx) as a dynamic probability without the use of an eigenvalue equation.
Addendum Nov. 27, 2022. In Newtonian mechanics, one may slice the x axis into equally spaced dx with a particle being in one dx at time t1 and in the next at t2. In this note we suggest describing directional constant motion with a probability density P(p,x). If the particle moves out of x we suggest that;. P(p,x) dP(p,x)/dx=0 or P(p,x) must be at least two dimensional i.e. P(p,x) = F(p,x) + iG(p,x) such that F dF/dx + G dG/dx =0. This is equivalent to: FF + GG = constant for any x which shows all x points being the same even though P(p,x) is a function of x which distinguishes direction. Finally, with P(p,x)P(-p,x) = constant because this represents no motion and so no distinguishing density is needed one has P(p,x) = exp(ipx). In Part I we argued that directional motion should be associated with a different spatial probability densities. dtime(x)= dx/velocity(x), however, taken as proportional to a density in the literature (1) does not distinguish. We suggested that Newton’s first law is equivalent to Fermat’s least time principle which for reflection from a fixed mirror is equivalent to d/dx (px) x = (px). (px)x, however, may be written as -i ln[exp(i (px) x) ] which has the form of constant multiplied by ln probability in information theory. Thus we suggested in Part I that exp(ipx) is a kind of probability density which distinguishes direction in motion. We also noted that exp(-ipx) exp(ipx) = 1 which is linked to the dt(x) density for constant motion and that Integral exp(-i p1 x) exp(ipx) dx = constant delta (p1, p). In this note we wish to examine in more detail the notion of x, t and wavelength. exp(ipx) depends on x and p=mo dx/dt or (p=mo v / sqrt(1-vv/cc) with v=dx/dt). X and t are continuous variables linked to the idea of invariance in space for constant motion. exp(ipx) and exp(-iEt) on the other hand define built-in particle space and time intervals for a given p,E. Thus exp(ipx) defines constant motion in space without the notion of time (i.e. t and x are independent within a built-in ruler gradation proportional to 1/p (two humps-one positive one negative). This leads to a contradiction if the particle is tiny compared to 1/p. On the one hand the particle has smooth motion v=x/t, but on the other a “created” particle length unit must be defined suggesting a probability density over the length proportional to 1/p. To allow for the two, one must assume a constant average motion over the length 1/p as measured by an external ruler and clock and at the same time abandon the notion of following the particle in x,t (infinite resolution) within the length proportional to 1/p even though its probability distribution exp(ipx) is well defined at each x. Thus the identity of the particle is defined by the probability exp(ipx) within 1/p regions even though a direct hit with another particle within this region involves p. Furthermore given that the spatial probability defines the particle it is not enough to state that for two particles one has p1 and p2, the spatial probabilities must distinguish by being orthogonal i.e. having a 0 dot product: Integral dx exp(-i p1 x) exp(ip2 x). We argue in the note that not only is particle identity described by exp(ipx) and exp(-iEt), but also conservation of energy and momentum in various problems. In Part I we suggested considering the product of P(p,x) and P(-p,x) in order to obtain a constant value linked to spatial variance throughout x even though exp(ipx) and exp(-ipx) are not spatially invariant. We first note that exp(ipx) and exp(-ipx) are the time reversal of each other and then consider exp(ipx) AND exp(-ipx) together within a little region of the order of 1/p. exp(ipx)exp(-ipx)= 1, but each moves with its own density profile expx(ipx) and exp(-ipx). Thus constant density ignoring the direction of motion seems to be identical to the product of the time reversed density states. Even though constant motion is only a translation when viewed on average, we suggest that directional motion must be linked with skewed density and be based on internal distances and clocks. This contradicts the idea of a smooth translation and leads to an unusual probability description within hbrar/p wavelength units. The catch seems to be, however, that one needs to introduce two dimensions in order to demonstrate the skewness linked with directional motion. By creating a spatial wavelength, the region within is one in which one may not follow the particle in x and t i.e. x and t are independent as in the classical action: A = -Et+px. dA/dx partial = p, but one cannot write this unless x and t are independent and only represent constant v on average i.e. x/t. Finally the wavelength and interval in time, which are physical, are directly linked to ensuring conservation of energy and momentum in the direction of an invariant variable. We argue this by consider a two dimensional reflection problem i.e. Fermat’s principle.