We argue that in classical mechanics there exist at least two types of measurements of a constant moving particle. (This may be extended to an accelerating one.) First there is the x(t) description which is sometimes considered a complete description of the mechanical case. This implies one may measure position and time of the particle successively and thus know about its behaviour in space and time. We argue that there is a second measurement which is not one of space or time, but rather of impulse. A constant moving particle may strike a foil at any place in its path at any time (if one has an ensemble of such classical particles) and the foil will “recognize” that it has been hit by a momentum p. In quantum mechanics we argue that there are also two types of measurement for a free particle. The first is the identical momentum impulse measurement i.e. the particle strikes a foil which recognizes it has momentum p. There seems to be no quantum feature to this. Such a measurement has already been performed classically for photons. This is the “particle” like measurement which has nothing to do with wave features. A. Einstein in 1905 in his work on the photoelectric effect already noted that particle type behaviour of a photon. The second type of measurement deals with spatial features which are very different from x(t). A quantum free particle is represented by exp(ipx) which contains “p” which may be measured by an impulse. We argue, however, that spatial features, unlike those for a classical wave in which the medium and wave disturbance are separated, require interference i.e. the summing of exp(ipx) and exp(i p1 x) or exp(i p1 x) or exp(i p (x+b)) or exp(i p1 (x+b1)). Thus exp(ipx) behaves like a complex probability. Mathematically it has a modulus of 1 which is said to be linked to equal probability for it to be at any x. Why should this be relevant? We argue that it is because of the impulse measurement which indicates that a p hit may occur at any x. We further argue that exp(ipx) behaves as part of a projection operator because it is orthogonal to other exp(ip1x)’s. It is this orthogonality which seems to allow for one to build a probability medium, but requires W*(x)W(x)=P(x). In such a case, a*(p)a(p) terms create the medium from which probability may be added and removed by cross terms exp(ip1 x) exp(ip2 x). Orthogonality ensures that overall what is added and removed cancel. Thus there are two levels of probability, one at W(x) linked to OR situations created by interactions which reveal the spatial nature of a quantum free particle and the other W*(x)W(x).
We argue that in classical mechanics there exist at least two types of measurements of a constant moving particle. (This may be extended to an accelerating one.) First there is the x(t) description which is sometimes considered a complete description of the mechanical case. This implies one may measure position and time of the particle successively and thus know about its behaviour in space and time. We argue that there is a second measurement which is not one of space or time, but rather of impulse. A constant moving particle may strike a foil at any place in its path at any time (if one has an ensemble of such classical particles) and the foil will “recognize” that it has been hit by a momentum p. In quantum mechanics we argue that there are also two types of measurement for a free particle. The first is the identical momentum impulse measurement i.e. the particle strikes a foil which recognizes it has momentum p. There seems to be no quantum feature to this. Such a measurement has already been performed classically for photons. This is the “particle” like measurement which has nothing to do with wave features. A. Einstein in 1905 in his work on the photoelectric effect already noted that particle type behaviour of a photon. The second type of measurement deals with spatial features which are very different from x(t). A quantum free particle is represented by exp(ipx) which contains “p” which may be measured by an impulse. We argue, however, that spatial features, unlike those for a classical wave in which the medium and wave disturbance are separated, require interference i.e. the summing of exp(ipx) and exp(i p1 x) or exp(i p1 x) or exp(i p (x+b)) or exp(i p1 (x+b1)). Thus exp(ipx) behaves like a complex probability. Mathematically it has a modulus of 1 which is said to be linked to equal probability for it to be at any x. Why should this be relevant? We argue that it is because of the impulse measurement which indicates that a p hit may occur at any x. We further argue that exp(ipx) behaves as part of a projection operator because it is orthogonal to other exp(ip1x)’s. It is this orthogonality which seems to allow for one to build a probability medium, but requires W*(x)W(x)=P(x). In such a case, a*(p)a(p) terms create the medium from which probability may be added and removed by cross terms exp(ip1 x) exp(ip2 x). Orthogonality ensures that overall what is added and removed cancel. Thus there are two levels of probability, one at W(x) linked to OR situations created by interactions which reveal the spatial nature of a quantum free particle and the other W*(x)W(x).
In classical physics, a particle traveling to the right is the mirror image of one traveling to the left. In terms of a spatial density (the time spent in each constant dx), the value is the same, as is the energy. A similar notion of such a spatial invariance is in the classical conservation of energy equation for a bound particle: p(x)p(x)/2m + V(x) = E. Quantum mechanics seems to be different because exp(ipx) and exp(-ipx) are distinctly different spatial probabilities, i.e. they are dynamic. It is not just that case that a particle moves to the right for exp(ipx) and to the left for exp(-ipx). The impulse hit probabilities (represented by exp(ipx) and exp(-ipx)) are distinctly different. If one wishes to create an OR situation and write: W(x) = Sum over p a(p)exp(ipx) and use this in a conservation of energy equation, then simply including only positive or negative p values in W(x) leads to a complex average kinetic energy -1/2m d/dx dW/dx / W which cannot be used with V(x) and E. Furthermore, W_ (only negative p) and W+ (only positive p) values do not yield the same average kinetic energy. Fortunately, given that this is a statistical average for a localized object (somewhat analogous to a gas), one may combine W+ + W- and obtain a real kinetic energy. We have argued in previous notes that there exists a quantum-classical duality. For example, for a free particle one has two independent notions. One is x=vt which describes the center-of-mass of the particle. We take this to represent a state description which treats v and -v as mirror images. exp(ipx), on the other hand, represents probabilistic impulse hits and does not define the state by itself, we argue, only its interactions. Interactions are physical and define what happens in an interacting system. One may, however, measure a system, i.e interact with exp(ipx), and this should define a measurement state ( a second state ensemble average). For example, exp(ipx)exp(-ipx) = 1 which is equivalent to the constant dt in each dx associated with x=vt. Thus, the two states coincide, but exp(ipx)exp(-ipx) means that one may place the 2-slit apparatus at any x point and is not about the center-of-mass, as x=vt is. In some cases, the two states are quite different. W*(x)W(x) represents an interaction state, but vrms(x) from .5mvrms(x)vrms(x) = -1/2m d/dx dW/dx / W represents a center-of-mass state description. (Note: in these cases a state means an average state, i.e. an average over many ensemble entries.) If one takes W*(x)W(x) and integrates over x, then interference terms vanish and one obtains a classical counting of distinct p value states with their weight a*(p)a(p).This, we argue, leads to the notion of orthogonality of probabilities. In classical probability, there is no such notion, but there is the notion of AND (multiply) probabilities. Given exp(ipx) and exp(-ipx), if one multiplies one removes p, i.e. the interactions and only sees the state left. W*(x)W(x) yields a set of peaks which represent a pattern of probabilistic hits in space which define a state which one would measure in space. The interference pattern, however, must be one such that if one integrates over x, one only obtains a classical sum of probabilities a*(p)a(p)’s as mentioned above. This leads to the notion of orthogonal exp(ipx)’s (which follows mathematically anyway). If one considers different bound En states, then these two are orthogonal because W*n1(x)Wn2(x) mean n1 and n2 are associated with different energies and integrating over x cannot show a result which looks like a state result, where W*n(x)Wn(x) shows a state result. In order to achieve this orthogonality (linked directly to the notion of a state) one must have that Wn(2) usually has 0 points and is positive and negative with the possible exception of a ground state. In other words, if Wn1*(x)Wn2(x) integrated over x is not 0, one violates conservation of energy. As a result, we argue an average probability for a bound state must statistically include +p and -p contributions and must create a function which has positive and negative peaks (except possibly for the ground state), i.e. W(x) must have 0 points. This W(x) is linked to a measurement state W*(x)W(x), but this is strictly considering exp(ipx)s. We argue that there are two pieces of information for a free particle, x=vt and exp(ipx). There must also be an average state linked to x=vt and this given by KEave(x) = .5m vrms(x)vrms(x) + V(x) = E which shows that the particle may exist and have average kinetic energy even at W(x)=0 points. When measuring a system, one interacts with the exp(ipx)s and so the W*(x)W(x) state is the one which is traditionally presented, but whether one measures the center-of-mass motion or not, one also has the notion of the x=vt, i.e. vrms(x) state (average) as well, we argue.
Quantum mechanics is based on two different probability schemes, namely W(x) =wavefunction and W*(x)W*(x)=P(x), with one being linked to the other. Why should there be two probability schemes? In (0) we argued that there are two spatial behaviour schemes because an impulse p (momentum) is automatically associated with two spatial distributions cos(px) and sin(px) in exp(ipx), yet a particle with constant p moves as x=p/m t. Thus if one is only concerned with motion, each x point has the same weight, and this is physically measurable as a photon or electron travels a certain distance in a certain time in free space. On the other hand, an impulse interaction is associated with exp(ipx) is a completely different spatial picture, and this too is measurable as in a 2-slit experiment. Thus it seems the two schemes are measured differently. We argue that this is an important issue because if there are two x-probability schemes there should be different entropies associated with each. Ttraditionally (1), one sees two quantum entropies: - W*(x)W(x) ln {W*(x)W(x)} and - a*(p)a(p) ln{ a*(p)a(p) } where W(x) = Sum over p a(p)exp(ipx). These are associated with the x=p/m t scheme or x-dependence based on motion or average motion. W*(x)W(x) represents spatial density which should be measured presumably by some sort of scattering, but we ask whether some information is lost in this approach? In particular, we note that a wavefunction may be obtained for a bound state. In the high energy limit, the envelope over the density W*(x)W(x) is said (2) to match C/v(x) from classical physics where .5mv(x)v(x) + V(x) = E. C/v(x), however, is proportional to delta t (with delta x being constant) and has nothing to do with the strength of an impulse hit. In the classical case, there is a single v(x) and so a single p(x) in each delta x thus no impulse entropy within a delta x i.e. at a point x. In quantum mechanics, however, we argue that a constant p automatically has an x probability distribution exp(ipx) associated with it and there might be an entropy linked to this.This entropy is associated with a probability to receive a p impulse at x and is so not linked to interference while the W*(x)W(x) is. In (3) we introduced the impulse or wavelength entropy (for a free particle |cos(px)|/C ln{| cos(px)|/C},, but here we wish to investigate its properties more closely.
Momentum and velocity are two well-known physical observables in classical physics (and quantum physics as well because quantum particles have impulse and move through space in a certain amount of time based on an external clock and ruler). They are, however, measured in different ways. Velocity describes motion through x in some time t, while p is measured through impulses. Thus one does not need a clock to measure momentum. It may be measured in terms of the damage done to a foil, compression of a spring etc. Thus p=m1v1=m2v2 etc., i.e. a numerical number associated with momentum does not automatically yield the velocity unless one knows rest mass. In many situations rest mass is known (i.e. one deals with an electron etc) and so there seems to be a tendency to consider momentum and velocity to be almost the same thing when they are distinctly different. In fact, we argue that the notion of wavefunction W(x) and density P(x)=W*(x)W(x) are associated with momentum and velocity respectively. In fact, for a free particle these two measurables are linked with completely different spatial probabilities. Constant velocity impies P(x)=constant while exp(ipx) describes the two-dimensional x-probability associated with p which is based on a constant mo and v. In special relativity for a free particle (i.e. p constant, v constant), there are two well known Lorentz invariants: EE = pp + momo (c=1) and A = -Et + px. One may note that v does not appear explicitly even though one may write E and p in terms of v i.e. E = mo/sqrt(1-vv) and p=Ev. The first invariant describes E in terms of rest mass and p (measured through impulse). The second involves p,E and x,t with x and t appearing as independent even though ultimately x(t) and v=dx/dt=constant = x/t if x and t both start at 0. This independence is crucial because it means one may consider fluctuations which keep A constant separately. For example, if t→ t+ hbar/E and x→ x+hbar/p, then A remains constant. This leads to the idea of the quantum wavelength and a spatial probability independent of t, namely exp(ipx). One average, however, x=vt, but how does this appear in Lorentz invariants? There is no explicit v. We argue that the Lorentz invariant A = -Et + px must convert E from a function of p to one in v and that this occurs through a Legendre transformation. Thus the form of the invariant is itself a Legendre transformation with E being transformed into L, the Lagrangian. There is, however, a second important feature because p=dL/dv (partial) and given that p is constant, dp/dt=0 so d/dt dL/dv partial =0 which is in the form of an extremization. We argue that this is not an accident. dL/dv=p follows from E being constant, so, in other words, it follows from the px term. This same term leads to exp(ipx) which means that one does not have smooth x=vt motion within a wavelength. Thus, we argue that when one transforms to the second measurable v, the notion of this fluctuation behaviour must also be manifested. In other words, v=constant or dv/dt is not enough. The extremization behaviour of the Lagrangian, we argue, is thus not a mathematical convenience, but gives an indication that even though the extreme path x=vt is followed on average, other paths are sampled. We argue that this idea is incorporated into special relativity, in fact within a single Lorentz invariant A = -Et + px.
The wavefunction for a quantum free particle is exp(-iEt + ipx) In this case, E and p are linked through either E=pp/2m (nonrelativistic case) or EE =pp + momo (relativistic case c=1). We argue that there are different interactions involved with E and p. P (momentum) is energy flow and is involved in impulse type interactions with a potential usually V(x). For V(x,t) matters are more complicated, but will also be discussed. The impulse hits are associated with exp(ipx) and lead to interfering plane waves and an overall crest-trough spatial scenario. There is a kinetic energy (nonrelativistic case) associated with each p, but a new overall energy En (n= eigenlevel) is created which yields the free form exp(-iEnt). This energy is linked to the cycling of exp(ipx)’s. It is also part of the new mass of the bound state system i.e. mo+En. This new system could be considered as an overall free particle and obtain a new overall momentum. (The initial bound state has 0 overall momentum.) Thus, time and space parts of the wavefunction behave in very different manners due to the nature of the interaction even though there is a link between 1/E and cycling of exp(ipx) terms. We call this case an impulse or space based type of interaction. It is also possible to have a time based interaction i.e. the absorption of a photon, phonon etc. In such a case, there is momentum conservation and so the whole system receives a new momentum, but this is usually ignored. The main effect is for the system to move from one bound state E(n1) to another E(n2). This has consequences on the spatial density and cycling of exp(ipx) terms in the wavefunction, but is not an impulse interaction in space. In the case of a system: W(x,t)= Sum over n an(En) exp(-iEnt) Wn(x), one has time based interactions which knock the system from one bound state to another (as in heat bath interaction). Again there are links between each En and the spatial density, but there is an overall spatial density(x,t) and average energy(x,t) which depends on the interference of exp(-iEnt) terms. (This disappears if one integrates over all x.) This is similar to the diffraction/interference spatial pattern, but in time. In the impulse scenario for a(p)=a(-p) one has cross terms in the spatial density of cos( (pi-pj)x) which create the crest/trough pattern seen. For the heat bath time type interaction, there exist cos( (Ei-Ej)t) type terms if an(En)=a*n(En) which are linked to the energy Ei-Ej of the intermediary photon/phonon etc and its time period. Thus in both cases, the quantum crest/trough pattern (in space or time) seems to be linked to the momentum or energy transfer. The energy transfer is known to be linked to periodic objects (photons/phonons) and it seems the momentum transfer pi-pj also has a similar periodicity, but in space. It is almost as if these “entities” are creating the unusual quantum crest/trough patterns.
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.
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.