Classical mechanics traces a particle’s movement through space and time through x(t). For example, one may consider the case of a particle moving at constant speed (momentum p) in one direction or a particle bouncing back and forth at constant speed in a box. These problems are described in time and have direction and position associated with time etc. It is possible to define a probability P(x,t) in the classical picture based on delta time, the amount of time a particle spends in a little region of space dx. P(x,t), by itself, loses much relevant information present in the x(t) scenario because a particle moving to the right, or to the left, or bouncing back and forth all have the same probability, 1/L, where L is length. Furthermore, this probability is the same for all p (momentum) values. Thus a probability description, without extra information, is a subset of the relevant information in the problem. As a result, to describe the classical picture fully, one might say that a particle has a probability of 1/L, but also a mass mo and velocity +v (i.e. momentum mov nonrelativistically). For the particle bouncing back and forth in a box with constant p, one needs to give the initial position at time=0 and then a subsequent time (as well as the initial velocity and mo) in order to fully describe the problem, even though probability = 1/L. In the literature (see (1)), there have been many attempts to derive the quantum Schrodinger equation from a classical Fockker-Planck equation or a continuity equation (which follows from the Fokker-Planck equation as shown in (1)). The continuity equation contains a probability P(x,t) and a velocity(x,t) and is generally argued to represent classical physics. It does, if one retains appropriate classical conditions such as the discontinuity of density for a particle in a box (i.e. density=0 outside the box, but 1/L inside) and the notion of differing directions within the box. In fact, one may write density=W*(x,t)W(x,t) where W(x,t) is an ad hoc complex function and still retain compatibility with classical physics. If one does not, however, retain the classical conditions completely, nonclassical physics is introduced, even if density(x,t)=1/L. For example, in quantum mechanics density(x) is considered to be continuous in all x. Thus for a particle moving in a box, density outside the box is 0 so at the box wall it must also be 0. In the classical case, the density is 1 inside the box and 0 outside. There is a discontinuity classically and this cannot be changed and this contradicts the quantum mechanical picture. Thus quantum mechanics does not follow form classical Fokker-Planck or continuity equations which retain classical conditions.. Furthermore, because time is completely removed in the FP-continuity approach as applied to quantum mechanics, one cannot distinguish between forwards and backwards motion. As seen by the velocity field given in (1), v is zero for a bound particle which completely removes the sense of time and violates the classical physics conditions. It is, however, not zero for a particle moving in one direction. Thus there seems to be no issue in describing a Fokker-Planck-continuity equation as describing classical physics. One may even introduce density(x,t)= W*(x,t)W(x,t) provided one retains classical conditions allowing for discontinuity in density and different W(x,t) values at different times i.e. a discontinuity in W(x,t) at a box wall. If these conditions, however, are not met, then one no longer describes classical physics, but a new kind of physics (quantum physics). Thus one cannot argue that quantum mechanics follows from classical statistical Fokker-Planck-continuity equations. It may follow from these if one adds a set of extra conditions. Given these extra conditions, however, one could derive quantum mechanics without starting with classical density.
Classical mechanics traces a particle’s movement through space and time through x(t). For example, one may consider the case of a particle moving at constant speed (momentum p) in one direction or a particle bouncing back and forth at constant speed in a box. These problems are described in time and have direction and position associated with time etc. It is possible to define a probability P(x,t) in the classical picture based on delta time, the amount of time a particle spends in a little region of space dx. P(x,t), by itself, loses much relevant information present in the x(t) scenario because a particle moving to the right, or to the left, or bouncing back and forth all have the same probability, 1/L, where L is length. Furthermore, this probability is the same for all p (momentum) values. Thus a probability description, without extra information, is a subset of the relevant information in the problem. As a result, to describe the classical picture fully, one might say that a particle has a probability of 1/L, but also a mass mo and velocity +v (i.e. momentum mov nonrelativistically). For the particle bouncing back and forth in a box with constant p, one needs to give the initial position at time=0 and then a subsequent time (as well as the initial velocity and mo) in order to fully describe the problem, even though probability = 1/L. In the literature (see (1)), there have been many attempts to derive the quantum Schrodinger equation from a classical Fockker-Planck equation or a continuity equation (which follows from the Fokker-Planck equation as shown in (1)). The continuity equation contains a probability P(x,t) and a velocity(x,t) and is generally argued to represent classical physics. It does, if one retains appropriate classical conditions such as the discontinuity of density for a particle in a box (i.e. density=0 outside the box, but 1/L inside) and the notion of differing directions within the box. In fact, one may write density=W*(x,t)W(x,t) where W(x,t) is an ad hoc complex function and still retain compatibility with classical physics. If one does not, however, retain the classical conditions completely, nonclassical physics is introduced, even if density(x,t)=1/L. For example, in quantum mechanics density(x) is considered to be continuous in all x. Thus for a particle moving in a box, density outside the box is 0 so at the box wall it must also be 0. In the classical case, the density is 1 inside the box and 0 outside. There is a discontinuity classically and this cannot be changed and this contradicts the quantum mechanical picture. Thus quantum mechanics does not follow form classical Fokker-Planck or continuity equations which retain classical conditions.. Furthermore, because time is completely removed in the FP-continuity approach as applied to quantum mechanics, one cannot distinguish between forwards and backwards motion. As seen by the velocity field given in (1), v is zero for a bound particle which completely removes the sense of time and violates the classical physics conditions. It is, however, not zero for a particle moving in one direction. Thus there seems to be no issue in describing a Fokker-Planck-continuity equation as describing classical physics. One may even introduce density(x,t)= W*(x,t)W(x,t) provided one retains classical conditions allowing for discontinuity in density and different W(x,t) values at different times i.e. a discontinuity in W(x,t) at a box wall. If these conditions, however, are not met, then one no longer describes classical physics, but a new kind of physics (quantum physics). Thus one cannot argue that quantum mechanics follows from classical statistical Fokker-Planck-continuity equations. It may follow from these if one adds a set of extra conditions. Given these extra conditions, however, one could derive quantum mechanics without starting with classical density.
In (1) it is shown how the Schrodinger equation can be “forced” in a sense to follow from the Fokker-Planck classical stochastic equation which contains a classical probability density. Other approaches in the literature which are similar use continuity equations and also a classical density. The key to all of these approaches is to assume, in an ad hoc manner it seems, that classical probability is equivalent to W*(x)W(x) where W(x) is a complex function called the wavefunction. In principle, one may write a density as W*(x)W(x). This is mathematically correct even if, for the sake of argument, it is ad hoc. The problem is that for a constant density (i.e. constant momentum and speed), one does not know if W(x)=exp(ipx) or W(x)=exp(-ipx). We consider here the arguments shown in (1). If a free particle has a density(x)=1/L, it may be associated with W(x) = exp(ipx)/sqrt(L). A free particle density, however, is associated with constant p (momentum) and assumes one knows x(t). If one considers the free particle to be trapped in a one-dimensional box with infinite potential walls, it travels to the right with p and then to the left with -p. The density(x) classically is 1/L, where L is the length of the box. The density is constant in time, but the direction of motion is a function of time. Thus, information is lost considering an equation such as the Fokker-Planck equation which uses a density function which does not track the direction of motion in time. In particular, writing density(x) = W*(x)W(x), which is done in (1) and applying this density to a bound particle in a box with infinite walls leads to an equation linear in W(x). Thus exp(ipx) and exp(-ipx) are both possible solutions if V(x)=0. Given that the conservation type equation obtained from the Fokker-Planck (or continuity) equation is linear in W, exp(ipx)+exp(-ipx) is a solution. Thus there is interference which leads to a density which does not match a constant classical density which should apply to a particle moving to the right and then to the left in a bound state because one assumed a classical picture a priori. Simply suggesting that density classical = W*(x)W(x) mathematically does not mean that the a priori classical picture may be dropped. As a result, we argue that there exists a contradiction in approaches which write classical density as W*(x)W(x) and then obtain the Schrodinger equation. As a remedy, we suggest that W(x) does not follow from a theory of classical density=W*(x)W(x), but vice versa. One needs equation which deal with W(x) a priori, and once it is found, one may create a classical density through W*(x)W(x).
In the literature, stochastic approaches employed to derive the Schrodinger equation seem to focus on the idea of a classical spatial density which depends on time and space ie. density(x,t). As a result, d/dt (partial) density appears in various equations such as the continuity and Focker Planck equations. At one point in the derivation the quantum wavefunction is introduced “mathematically” as W(x,t)=sqrt(density(x,t)) exp(i S(x,t)) where S is related to a velocity field through u(x,t)=velocity= d/dx S. From quantum mechanics, -Et becomes a term of S the velocity field. The stochastic/hydrodynamical approaches employ the term du/dt partial = d/dt (partial) d/dx S and then “pull out”” d/dx leaving d/dt (partial) S which yields -E average energy from the -Et term. In this note we argue that for a bound quantum state density(x) so d/dt (partial) density = 0, yet the statistical ideas of quantum mechanics should be present in such a case. Furthermore for the free particle wavefunction exp(ipx-iEt), density=1. In fact, the Schrodinger equation should follow for a free particle even the case of density=1. Stochastic processes seem to be based on equations which consider a change in classical spatial density at a point x in time due to flow in or out or stochastic changes. Quantum processes seem to deal with periodic probability changes from one dimension to another exp(ipx) (2 dimensions) which are somehow related to a physical process of momentum flow. Ultimately one links this process to an energy conservation which does not seem to be the same as a classical stochastic/hydrodynamical approach. In this note we argue that a free particle with a constant momentum p has periodic stochastic processes associated with it through its complex probability change in the region of its wavelength, but that these are linked to creating the momentum p. Furthermore momentum is linked to energy so we argue that the Schrodinger equation is a stochastic equation linking energy and momentum and that time and space are separated i.e. there is no situation of following a change in spatial density in time for a free particle with momentum p. For a bound state, one has a distribution of free particles, but again time is removed from the problem in the sense that it only appears in exp(-iEt) in the wavefunction to again establish a relationship between energy and average momentum and potential. There is again no nonzero d/dt (partial) density term which seems to be key in stochastic equations. Only if one takes a linear combination of energy eigenstates Sum over En an(En) exp(-iEnt) Wn(x) does one obtain the form density(x,t), but we argue that quantum features should already appear in exp(ipx) and the bound state scenario with a zero d/dt (partial) density(x,t). We argue, however, that the Schrodinger equation and fundamental stochastic ideas of quantum mechanics should already be present at the level of exp(ipx).
There seems to be much interest in the literature (1) in associating quantum mechanics (in particular the Schrodinger equation) with a classical joint phase space probability F(x,p,t) such as appears in classical Liouville theory etc. Here, we consider that P(x)dx = dx/L for a particle at rest and analyze the extra physics which appears when one considers a single particle bouncing back and forth in a box, which is what a person in a frame moving with -v would see (albeit with a Lorentz contracted L.) We suggest that the constraint L is important as it is linked with physical forces which appear when the particle moves. In other words, a particle at rest is linked with a static probability, while for motion, one must have a probability which is linked to more than x and t, but to force considerations in x and t, a dynamical probability. (Experiment bears this out as we discuss.) In particular, we argue that in the case of a moving particle, two physical concepts related to force appear, namely pressure which is an average over time and impulse hit which is only linked with space. Thus, two different particles may have the same p, but different v and pressure values. We thus suggest a physical grouping of (p,x) and (pressure, t). We then note that one needs to somehow preserve the notion of P(x)=1/L, i.e motion does not introduce a bias to x, nor to time. In a rest frame, there is no motion and so there is neither p nor pressure, but P(x)=1/L pertains. If one tries to define a probability with two groupings, one for pressure, t and the other for p,x, then if one multiples a p1 case by a p2 case, the p,x grouping should yield p1+p2, x. Furthermore, a p,x grouping should imply mixed terms, e.g. px etc, otherwise one does not really have an association between the two variables. This suggests a pf(x) form linked to the p-x grouping. Given the notion of special relativity, it seems that one may replace pressure with the notion of energy and create a Lorentz invariant probability. This would then lead to the free particle quantum wavefunction (probability) exp(-iEt+ipx), but we argue that the roots are in classical physics and a desire to physically describe the physical information present for a particle bouncing back and forth in a box.
In (1), a variational approach is applied to Fisher’s information (with constraints) to obtain a differential equation in P(x)=probability to be at x. It is suggested the equation may be simplified using P(x)=W(x)W(x) where W(x) is real. The result is the nonrelativistic Schrodinger equation with mass=1. As a first observation, we note that Fisher’s information, which is a purely statistical quantity, should not necessarily yield a nonrelativistic equation which is an approximation of relativistic results. In particular, we argue that Fisher’s information is used to find <pp> not necessarily 1/2m <pp>, but this is a minor point. A more important issue, we argue, is the meaning of P(x) in a bound system. A classical bound system is sometimes said to have a P(x)=C/v(x) i.e. P(x)dx is proportional to dt, the amount of time a particle spends in each equally sized dx. In the high energy limit, the envelope of a quantum bound probability P(x)=W(x)W(x) (W(x)=wavefunction) is said to roughly match C/v(x), yet this density does not follow from the extremization of Fisher’s information (together with two general constraints which yield E and V(x)=potential). This begs the question: What is P(x)? If one considers a particle moving at constant speed (i.e constant momentum) then P(x)=constant because each dx point carries the same probability. Classically a particle placed in a box with infinite potential walls would have a P(x) =constant and p and -p motion would be separated in time. Thus if P(x) is to have meaning in a box, one must conclude that the probabilities associated with p and -p are not constant, but rather functions of x. This then leads to the notion of a complex new kind of probability (which maps to P(x)=constant) for a free particle i.e. W(x) =cos(px) + i sin(px) so that the modulus W*(x)W(x)=P(x)= 1. These ideas seem to go beyond the notion of Fisher information. What then is the role of Fisher information? We argue it is related to two very specific moments of the distribution cos(px) or rather W(x) = Sum over p a(p)cos(px) with a(p)=a(-p), namely and <pp>. In classical physics there exists p and pp at each x point (but one is the square of the other), but with distributions <p does not equal to <pp> in general. Both and <pp> are linked to physical concepts which exist in Newtonian mechanics, namely impulse and work. With the presence of cos(px) one has a combination of Newtonian mechanics and a statistical distribution. We argue that the role of Fisher’s information is to allow one to transform from to <pp>. In particular, -idW/dx appears in Fisher’s information (and is linked to ), but the specific role of a variation is to transform it into <pp> which appears in a conservation of energy equation. We argue that this is why Fisher’s information may be used to derive the Schrodinger equation.
Classical mechanics contains a well-known dichotomy, namely that of work: Integral Fdx = change in KE and impulse, Integral Fdt = change in p (momentum). The former view is concerned with time and velocity, while the latter is not as p=m1v1=m2v2, i.e. the same impulse may occur for two different velocities. When solving classical mechanical problems, however, the time view (conservation of energy equation, Newton’s second law) is utilized and the impulse picture, usually ignored. We suggest that in quantum mechanics, one must retain both dichotomies as they are linked to each other as they are in classical physics. In fact, we argue that in quantum mechanics interactions are based on the impulse picture, which in some problems, such as one dimensional photon reflection-refraction, make it the only scenario which solves the problem. (Actually the impulse scenario makes use of flux values and so is intertwined with the time picture.) In the literature (1) etc, the continuity equation: d/dt partial density = d/dx { velocity(x) density(x,t) } seems to be a common starting point for deriving the Schrodinger equation. We argue that this approach only considers the time-work-kinetic energy scenario a priori and ignores, a priori, the impulse approach to interactions. The impulse picture is a physical description of interactions and so both scenarios should be introduced together a priori. What is usually done is to consider only the time picture and then hypothesize on mathematical grounds that: density(x,t) = W*(x,t)W(x,t) instead of considering the wavefunction form as the representation of the impulse picture which governs interactions. In fact, the time-independent Schrodinger equation is a combination of both scenarios. The work-kinetic energy picture must appear because V(x) is used, but actual interactions occur via impulses and their unusual associated exp(ipx) probability.
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.