In quantum mechanics, a free particle is represented by exp(ipx). It has often been noted that exp(-ipx) exp(ipx) =1 yields equal probability for any point x. In classical mechanics, one might argue that a constant real number suffices. The form exp(ipx) is also identical to the classical electromagnetic expression associated with light (i.e. a photon). In this note, we wish to begin with the idea of an ensemble of momenta at x using a real probability f(p,x) and create a conservation of energy equation i.e. KEave(x) + V(x) = E. This equation may be differentiated to yield a force equation with d f(p,x)/dx. If one argues that should appear in such an equation, one has a link between d f(p,x)/dx and . One finds that f(p,x), which we assume to be the same for p and -p so that pf(p,x)-pf(p,x)=0 as in classical statistical mechanics, may be generalized to f(p,x) + i g(p,x) with g(p,x) linked to d f(p,x)/dx and . In other words, quantum mechanics includes two components, one associated with V(x) and <pp/2m at x> and another with F(x) and . The Bohm Aharonov effect seems to justify such an approach because one may have V(x) non-zero and F(x) zero and still have physical effects, thus a potential V(x) and F(x) may both have separate physical consequences. Using cos(px) and isin(px) seems to track these two separate consequences through the separate consequences of and <pp/2m at x>. For a classical particle, p becomes pp by squaring so the two do not need to be treated with two separate “dimensions’ i.e. cos(px) and isin(px). In this note, we try to obtain the form exp(ipx) from the more general exp( i Theta(x,p)) without resorting to the form exp(ipx) of a photon in classical electromagnetism and also consider an interpretation.
In quantum mechanics, a free particle is represented by exp(ipx). It has often been noted that exp(-ipx) exp(ipx) =1 yields equal probability for any point x. In classical mechanics, one might argue that a constant real number suffices. The form exp(ipx) is also identical to the classical electromagnetic expression associated with light (i.e. a photon). In this note, we wish to begin with the idea of an ensemble of momenta at x using a real probability f(p,x) and create a conservation of energy equation i.e. KEave(x) + V(x) = E. This equation may be differentiated to yield a force equation with d f(p,x)/dx. If one argues that should appear in such an equation, one has a link between d f(p,x)/dx and . One finds that f(p,x), which we assume to be the same for p and -p so that pf(p,x)-pf(p,x)=0 as in classical statistical mechanics, may be generalized to f(p,x) + i g(p,x) with g(p,x) linked to d f(p,x)/dx and . In other words, quantum mechanics includes two components, one associated with V(x) and <pp/2m at x> and another with F(x) and . The Bohm Aharonov effect seems to justify such an approach because one may have V(x) non-zero and F(x) zero and still have physical effects, thus a potential V(x) and F(x) may both have separate physical consequences. Using cos(px) and isin(px) seems to track these two separate consequences through the separate consequences of and <pp/2m at x>. For a classical particle, p becomes pp by squaring so the two do not need to be treated with two separate “dimensions’ i.e. cos(px) and isin(px). In this note, we try to obtain the form exp(ipx) from the more general exp( i Theta(x,p)) without resorting to the form exp(ipx) of a photon in classical electromagnetism and also consider an interpretation.
Quantum mechanics utilizes a wavefunction W(x)=Sum over p a(p)exp(ipx) (in time independent cases) with exp(ipx) being what we call a free particle complex probability. In such a case, W(x) is an OR scenario of these various probabilities. We have argued in previous notes that exp(ipx) is the probability for a p impulse hit at x, even if the center-of-mass of the particle is not at x. This means that impulse hits may occur probabilistically in space as in the 2-slit example which requires adding two probabilities as an OR case for each slit. The point then seems to be that W(x) (or even its momentum frozen form W*(x)W(x) gives a profile in space of the number of interactions (various p impulse hits) at each x. Classically, the kinetic energy (nonrelativistic) is pp/2m. In a classical bound problem this kinetic energy is considered to exist (in calculations) at the center-of-mass, but forces are also said to be rendered at this point so pp/2m, the center-of-mass and the force hit point all coincide. In quantum mechanics, a p impulse hit and the center-of-mass do not coincide. If p is said to hit at a probabilistic x, then it seems that pp/2m should also be seen to exist at this same hit point. A physical interaction point verifies in a sense the location of pp/2m in a probabilistic manner, i.e. a(p) exp(ipx)/W(x) is the probability used to calculate KEaverage and this is based on impulse hit locations. By normalizing using W(x), one removes the relative number of average hits at each x and obtains a pure energy value at x. (This may then be used in a KEave(x)+V(x)=E equation.) On average W(x) has zero points, which means that if one considers W*(x)W(x) to represent the motion-frozen-out spatial distribution of hits, then no hits are rendered at certain x points (on average). One might argue then that the particle cannot exist at x, but the center-of-mass can exist anywhere. One might next argue that there is no kinetic energy at x, but that cannot be true because 1/2m d/dx d/dx W has the same 0 points as W and the two divide. Thus, there is average kinetic energy at all bound points and this represents an average based on impulse hits (exp(ipx)). Classically, if one has many particles in a potential, one may write density(x)V(x) to find the potential energy of the particles. This implies that the particles interact with V(x), e.g. neutrinos usually do not interact. Density(x) in a classical sense means that the center-of-mass and force hit point are both at x. In quantum mechanics, exp(ipx) represents the probability for a force hit point, W(x) represents an average and W*(x)W(x) represents a motion-frozen average hit point. W*(x)W(x)V(x) seems to represent the average potential energy of the particle at x because W*(x)W(x) is linked with interaction. Even if a particle is not present, however, there is energy V(x) in space. It seems that if W*(x)W(x)=0 at some x, then there is no average particle interaction at this point, but there is still average kinetic energy Sum over p a(p)pp/2m exp(ipx) / W(x) based on the probability for average probability hits. The key idea is that one normalizes probability by dividing by W(x) in the average kinetic energy calculation. W*(x)W(x) is normalized not at x, but integrating over dx and setting this value to 1. One has relative hit at x values represented by W*(x)W(x). For exp(-ipx)exp(ipx)=1, each dx has the same average hit value (with one not being able to distinguish details in hbar/p, i.e. this is a motion frozen result. The question then becomes: Why would one write: W*(x)W(x) KEave(x)? One might argue that this term occurs if one takes the quantum mechanical equation: KEave(x) +V(x) = En and multiplies by W*(x)W(x). One may ask: Does V(x)W*(x)W(x) have physical meaning? V(x) is the potential energy at x. The average interaction probability with V(x) at x should be W(x)V(x) and not W*(x)W(x)V(x). The latter is a classical mechanical idealization. If W*(x)W(x) is 3 at x1 and 2 at x2, then it is as if the particle interacts 1.5 times more at x1 and so if V(x1)=V(x2), then this approach assumes 1.5 the interaction energy at x1. KEave(x), however, is simply free particle energy (average) at x and one does not need to take this average value and multiply it by the relative number of interactions at x. W(x) has zero points meaning that there is no interaction at such a point on average. KEave(x) on the other hand is the average energy at x based on the particle kinetic energy presence pp/2m being at the point of an impulse hit. This does not mean that an interaction is occurring with V(x) at this point in this formalism. Even a free particle has an exp(ipx) probability when it is not interacting. The sum Sum over p a(p)exp(ipx) means that all of the interactions linked with x have occurred and one is making an OR calculation. KEave(x) is free particle kinetic energy at x and V(x) is potential energy (not interaction potential energy with the particle). Multiplying by W*(x)W(x) is a classical minded approach to creating a classical equation: density(x) (KEave(x)+V(x)) = E density(x) ((1)) Here KEave(x) would represent the average kinetic energy of a single particle and density(x) would be the average number of particles at x. density(x)KE(x) has a clear meaning. W*(x)W(x) KEave(x) on the other hand is different. KEave(x) is actually the classical kinetic energy value at x and so it is not clear why one would multiply it by W*(x)W(x). W*(x)W(x) represents the probability for an impulse hit at x in a p-frozen out picture. The notion of exp(ipx) (i.e. probability for a p hit at x) is already buried within the quantum calculation of KEave(x) (which yields the classical value). In order to find the physical meaning of W*(x)W(x) multiplied by KEave(x) or V(x) it seems that one has broken dx into equally sized segments which correspond to different detection hits of the particle. This is then somehow equated with the presence of the particle per unit time in dx which is then somehow rendered as the average energy at x multiplied by the detected presence at x through a steady rate of measurement/time. Thus, multiplying by W*(x)W(x) is not multiplying by a physical density.
Newton’s second law dp/dt = Force seems to be written in terms of a cause F(x) and effect p(x), because momentum is written in terms of a change i.e. d/dt. His third law, however,describes two causes i.e equal and opposite causes (forces) leading to two effects. We argue that a constant momentum, the effect of a force, may also act as a “force” if divided by a constant time i.e. change in p / dt(constant) = Force. A conservative external force may be written in terms of the derivative of a potential i.e. -dV/dx, but we suggest that a quantum free particle is linked with two distributions cos(px), sin(px) which describe an internal force/acceleration for even a constant p. The two combine in two dimensions i.e. exp(ipx) to create a type of probability which has a constant modulus i..e treats each x point equally. Each x point is treated equally because there is no external force -dV/dx, but we argue that there is still an internal one linked with probability and spatial distributions i.e cos(px),sin(px). If an impulse is applied to a quantum free particle to change it from p1 to p2, the change in energy is completely described in terms of momentum i..e p2p2/2m - p1p2/2m. There is no need to consider space, but at the same time there is a change in the two distributions cos(px) and sin(px) due to the change in p. Thus a constant p is linked with two spatial distributions which have physical consequences i.e. two slit interference etc. There is an internal distribution in space. If a quantum particle is placed in a potential V(x), there is now both an external potential, but also an internal one associated with each exp(ipx) if one assumes an ensemble of exp(ipx)’s is created i.e. W(x)=wavefunction = Sum over p a(p)exp(ipx). The point we make is that both the external and internal potentials are linked with x distributions, thus applying d/dx effects both, but Newton’s conservation of energy KE(x) + V(x) = E deals with changes in kinetic energy only linked to the external potential V(x). Formally, one may write: -1/2m d/dx dW/dx / W + V(x) + E, but we argue that external and internal potentials “interact” and we try to describe the situation considering V(x)W(x) to demonstrate the interplay of internal and external forces from the point of view of the internal force (probability) exp(ipx) and statistical mechanical loss of information i.e. equilibrium.
In (1), it is suggested that a Bohmian quantum potential for a wavefunction A(x,t) exp(i S(x,t)) is given by ( -1/2m d/dx d/dx A)/A. Here we wish to describe how the notion of such a potential might arise based on viewing the free particle wavefunction as a complex momentum conserving probability. In a number of previous notes, we argued that one may derive the free particle wavefunction exp(ipx) by seeking a free particle probability which conserves energy and momentum in Newtonian 2-body (or more) elastic scattering. We argued that for a given initial e1,e2 (energies) and p1,p2 (momentum vectors), any ei,ej and pi, pj (vectors) which conserve energy and momentum have equal likelihood of occurring as final outcomes after the collision. This leads to the exp(i C1 E) and exp(i C2 p) (p along the x-axis) as a solution, but for p=m1v1=m2v2, the same p satisfies a conservation of momentum equation. In a boosted frame , the p’s are different and so do not satisfy the same momentum conserving equation which is a problem, so one requires a Lorentz invariant argument, i.e. exp(-iEt+ipx). If one only considers momentum, then one has exp(ipx). In other words, a particle with momentum p and kinetic energy pp/2m (nonrelativistic) is linked with a momentum conserving probability exp(ipx) which is impulse-centric. Even if one considers kinetic energy pp/2m, in the probabilistic view of exp(ipx), one cannot separate energy from impulse hits as one does in Newtonian mechanics. For example, pp/2m + V(x) = E is strictly about energy. A quantum scenario cannot be so. It must combine both ideas and it this notion of including a picture of impulse hits which gives rise to a kinetic energy term for a bound state (i.e. the Bohmian quantum potential -1/2m d/dx d/dx A), which is a statistical average, but at the same time has the appearance of a “potential”. In this view, the potential V(x) may be written as Sum over k Vk exp(ikx) with exp(ikx) also representing momentum impulse hits, while at the same time Vk represents energy.
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.
A classical particle is considered to be a point described by x(t) i.e. x and t are both a set of points. In a one dimensional problem, however, dx= v(x) dt so if all dx tend to zero, then dt’s are small, but don’t tend to zero in the same way. In other words dx may be considered small, but of a finite size. To have a particle within dx, it should have a probability distribution that goes to zero at the endpoints of dx i.e. one could have a repeated set of these for a constant velocity. This, however, contradicts P(x)= C1 dt = C1 dx/v(x). Thus it seems first that there are two kinds of probability distributions and that the set of distributions fitting into each dx is incomplete because if this set is described by a function F(x), this function should have the same magnitude at F(x+dx) for a free particle. This suggests a phase function linear in x i.e. exp(i C1 x). P(x) = C1dx/v(x) is based on dynamics and no dynamics appears in exp(i C1 x). It seems that dynamics should occur leading to the possibility of exp(i p x). (This , however, is ad hoc.) We call this a dynamic probability as opposed to P(x). In quantum mechanics it is called a wavefunction. If one considers the classical free action (relativistic or nonrelativistic) with x and t independent, then A= -Et + px (1) and one may create eigenvalue equations -id/dx partial exp(iA) = p exp(iA) and id/dt partial exp(iA) = E exp(iA) so exp(ipx) also appears more formally. In other words a fixed resolution periodic distribution requires the same shifted distribution (i.e. cos(px) and sin(px)) kept as separate dimensions i.e. “i” in order to create a dynamic probability which is a phase function which may be linked to P(x). This phase function exp(ipx), however, consists of two classical-like waves showing that in each dimension probability is “carved” out of troughs and added to crests. The two dimensions allow for the dynamic probability to be consistent with P(x)=constant for a free particle.
In Part II of this note, we argued that if one describes a bound particle in a potential in a statistical manner, like classical statistical mechanics, one is led to a complex, periodic space invariant probability, exp(ipx), for a free particle. We showed that exp(ipx) represents a two vector solution to the vector operated on by a rotation matrix (using cos(px), -sin(px)) set equal to a translation operator 1+delta X d/dx operating on the same vector. In this note, we examine the electromagnetic scenario. The idea of a “wave” describing a photon first emerged with Maxwell and his equations in the 1800s, with interference/diffraction and the speed of light both appearing. It was later found (Einstein and the photoelectric effect) that electromagnetic energy is not spread out all over space like a wave, but is concentrated. Furthermore, if one uses probabilistic ideas to obtain exp(ipx) for a quantum particle, like an electron, which undergoes two slit interference, then why would not two slit interference for photons also be considered in terms of probability? In this note, we examine these ideas.