In classical mechanics a constant velocity may be defined by dx/dt where x is in the direction of motion. Thus two dimensions are required and the concept of positive/negative x. Quantum mechanics of a free particle involves a v=x/t, but also independent periodic fluctuations of x and t i.e. exp(-iEt) and exp(ipx) may be considered separately. One, however, cannot lose information we argue so two dimensions are still required to describe motion only using x as well as a concept of negative and positive. (The notion of “only using x” applies to a bound problem.) If the two dimensions only depend on x, then there is no a priori dimension which may be strictly positive such as t. Both dimensions should allow for positive and negative values so that somehow overall the direction of velocity may be defined. Quantum mechanics is concerned with probabilities, but classical probabilities do not introduce negative values. As argued in previous notes, it is possible to have two layers of probabilities P(x) (classical) and P(p/x) linked to exp(ipx) (quantum). The second result contains the two required dimensions and the negative/positive quality. As argued in (1) a quantum free particle has a velocity v=x/t, but the classical action = -Et+px is unchanged if t=1/E and x=1/p. Thus we argue that there may be periodic fluctuations about the classical motion v=x/t. In other words there are uncoupled t and x fluctuations which are linked to p (i.e. v) and are systematic and periodic. These define spatial density. v=x/t defines velocity and has two dimensions and the notion of positive and negative x. We argue that the fluctuation scheme must also define velocity and so requires two dimensions and a notion of positive/negative. As a result the mathematical result of a wavefunction component cos(px) or sin(px) taking on a negative value accounts for information which cannot be lost. For a particle with a single p in a box with infinite walls, a positive or negative hump represents the same probability (sin(px)sin(px) C), but one does not account for the direction of motion. Even using one dimension cos(px) as in a bound state problem, the information of negative and positive values must be retained (in cos(px) over x) because the negative/positive property is used to define overall velocity (even if it is 0 on average for a particle in a box). Thus the concept of interference arises because one cannot lose information in the quantum probabilistic scheme. As noted before it seems one defines new “density” and kinetic energy dimension exp(ipx) and pp/2m exp(ipx) etc.
In classical mechanics a constant velocity may be defined by dx/dt where x is in the direction of motion. Thus two dimensions are required and the concept of positive/negative x. Quantum mechanics of a free particle involves a v=x/t, but also independent periodic fluctuations of x and t i.e. exp(-iEt) and exp(ipx) may be considered separately. One, however, cannot lose information we argue so two dimensions are still required to describe motion only using x as well as a concept of negative and positive. (The notion of “only using x” applies to a bound problem.) If the two dimensions only depend on x, then there is no a priori dimension which may be strictly positive such as t. Both dimensions should allow for positive and negative values so that somehow overall the direction of velocity may be defined. Quantum mechanics is concerned with probabilities, but classical probabilities do not introduce negative values. As argued in previous notes, it is possible to have two layers of probabilities P(x) (classical) and P(p/x) linked to exp(ipx) (quantum). The second result contains the two required dimensions and the negative/positive quality. As argued in (1) a quantum free particle has a velocity v=x/t, but the classical action = -Et+px is unchanged if t=1/E and x=1/p. Thus we argue that there may be periodic fluctuations about the classical motion v=x/t. In other words there are uncoupled t and x fluctuations which are linked to p (i.e. v) and are systematic and periodic. These define spatial density. v=x/t defines velocity and has two dimensions and the notion of positive and negative x. We argue that the fluctuation scheme must also define velocity and so requires two dimensions and a notion of positive/negative. As a result the mathematical result of a wavefunction component cos(px) or sin(px) taking on a negative value accounts for information which cannot be lost. For a particle with a single p in a box with infinite walls, a positive or negative hump represents the same probability (sin(px)sin(px) C), but one does not account for the direction of motion. Even using one dimension cos(px) as in a bound state problem, the information of negative and positive values must be retained (in cos(px) over x) because the negative/positive property is used to define overall velocity (even if it is 0 on average for a particle in a box). Thus the concept of interference arises because one cannot lose information in the quantum probabilistic scheme. As noted before it seems one defines new “density” and kinetic energy dimension exp(ipx) and pp/2m exp(ipx) etc.
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.
The classical Lagrangian L leads to Newton’s second law which is equivalent to an energy-momentum conservation equation. The classical action: A = Integral dt1 L(t1) for a constant velocity v=X/T may be varied with respect X and T i.e dA/dT = -E and dA/dX= p. Thus, one may create a differential equation in (dA/dT)(dA/dt) and dA/dx dA/dx for the EE=pp+momo equation. Alternatively, one may create other differential equations. Quantum mechanical differential equations, however, are quite different with exp(i Action) being a solution. Even this may be written as the solution of a non-Schrodinger equation i.e. id/dt exp(iAction) = -1/2m (d/dx exp(iAction)) (d/dx exp(iAction)) for a free particle. For a constant velocity, exp(iAction) is equivalent to exp(ipx-iEt). We argue that this solution is statistical because it separates x and t (as independent variables) and holds for all x and t. Thus, quantum mechanics seems to be a statistical approach using free solutions exp(ipx-iEt) to create an ensemble as in classical statistical mechanics. A difference, however, is that exp(ipx-iEt) “interferes” even if one only takes the real part cos(px-Et). We consider the case of a particle in a box with infinite potential walls to argue that the periodicity is warranted i.e. positive and negative values (interference). Even though the modulus is 1, exp(ipx-iEt) may be operated on by various generators e.g. d/dx the generator of translations. This is analogous to operations on vectors in linear algebra. As noted above, quantum equations are not only energy-momentum conservation equations. They yield a wavefunction W(x) which is an unusual momentum distribution W(x)=Sum over p a(p) exp(ipx). exp(-iEt) holds overall because average E is the same at each x. Thus, one needs a W(x) which leads to a momentum-energy conservation equation, but also satisfies certain boundary conditions which become linked to spatial density= W*W. This leads to discrete energy levels and also cases of humps and troughs in W(x). We attempt to investigate some of these ideas in this note.
Classical mechanics was developed in the 1600s and is considered a complete theory in that it matches observables in the macroscopic world. In the 1920s, however, it was found that different mechanical rules seem to apply to the electron in an atom. In this note we try to argue that classical mechanics is a subset of quantum mechanics i.e. it is based on partial information or the dropping of information. In (1) we noted that a classical spatial density Cdx/v(x,) where v(x) is velocit, is sometimes presented in the literature (2). This expression is proportional to dt associated with a fixed length dx so the density is a relative one. If one considers two bound classical systems (a particle in a box) with different constant speeds (in the interior) then dt is different for each, but Cdx/v(x) does not distinguish. Thus we argue that information has been lost. The higher v particle not only has a higher impulse, but an increased frequency and we argued in (1) that quantum mechanics takes note of this extra frequency by increasing the number of crests associated with higher constant velocities. In this note, we point out that quantum mechanics itself is linked to the dropping of information. If one considers free particle wavefunctions W(x) for different momenta (i.e velocities if mo is the same), one has exp(i p1 x), exp(i p2 x) etc. These clearly have different information. They describe direction, but more importantly they have an associated wavelength proportional to 1/p. P(x)=W*(x)W(x)=constant and so this information, which is physical, is lost . P(x) is related to the question: What is the probability for a particle to be at x? If one considers cos(px) as a nonlocal spatial distribution which shows motion i.e. negative and positive parts of cos(px), then sin(px) is a spatially shifted distribution which helps to indicate direction, but also shows that all x points are the same. One has a peak of cos(px) at x=0 and a lower value at x=delta, but due to the motion, the peak may later be at delta just like a classical wave. This is what W*(x)W(x)=constant captures. It shows that even though cos(px) maps to x in a certain way with sin(px) in an extra dimension, overall the free particle has no preference for one x point versus another. There are two points we wish to make. First, classical mechanics seems to be based on P(x) i.e. it drops information about a wavelength. Space is broken into tiny dx units and the particle has a constant velocity in each even if it is accelerating. This is the Newtonian picture and leads to dp/dt = Force or pp/2m + V(x) = E. This information, however, is contained completely in quantum mechanics because for a bound state with V(x) one has: -1/2m d/dx dW/dx / W + V(x) = En. -1/2m d/dx dW/dx / W is exactly KE(x) classical. Quantum mechanics, however, displays more information than the classical picture. For example, only certain En are allowed and W*(x)W(x) leads to a spatial density which does not appear in classical mechanics. This leads to a second point. exp(ipx) contains information about a wavelength which is lost if one only considers P(x) for a free particle. If there is an interaction (two electrons shifted in space, a V(x) or a single or double slits etc) exp(ipx) acts like a probability and adds in OR or interaction situations. In such a case the information regarding the wavelength becomes physically crucial. In other words for an interaction this information does not drop out as it does in the classical scenario of pp/2m + V(x) = E. Thus we argue that classical mechanics is a subset of quantum mechanics. Wavelengths are tiny in the macroscopic world and so are simply not observed, allowing the subset scenario of pp/2m + V(x)=E (nonrelativistic) to be an excellent approximation. Indeed for every energy level of a quantum bound state this relationship also holds. It is the spatial density which is altered.
In classical physics, a potential V(x) acts on a particle at a location x regardless of its velocity which is taken to be constant within dx (even though the particle accelerates). Thus writing potential energy V(x) at a given x is like taking a snapshot of the particle when it is at x. The snapshot is static and does not indicate the dynamics of the problem as the particle may be moving either to the right or left, which is irrelevant in terms of the exact potential energy and even kinetic energy at point x. A quantum particle moving with constant velocity is described by a two-dimensional dynamic probability exp(ipx), with cos(px) representing a static density and sin(px) a time-shifted density to capture one directional motion such that exp(ipx)exp(-ipx)=1, we argue. V(x), however, only acts on the static density (linked to cos(px) and not sin(px)). Thus it seems that bound state quantum mechanics needs to remove the dynamic pieces of free particle wavefunctions exp(ipx) through interference. One might argue that for wavelengths of the order of the system length one does not know if one has exp(ipx) or exp(-ipx) at an x point, but for high energy levels, wavelengths become small with respect to the system length and yet interference is still present. Thus it seems that for the calculation of energy one requires a static average snapshot of the bound particle. Physically, we argue that the dynamic piece is still present i.e.. if one knocks out a bound particle and it is found in the p momentum state, it is represented by exp(ipx). Thus the purpose of the bound state wavefunction is to facilitate the calculation of average energy which is associated with a static snapshot type of density and not necessarily represent all of the dynamical information in the system i.e. the two dimensionality of each exp(ipx). As a result, the bound wavefunction does not show motion to the right and then to the left in the high energy limit.
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.