It is well known that a Lagrangian (usually kinetic energy - potential) may be varied to produce Newton’s second law of motion. In (1) it is noted that kinetic energy is proportional to a Fisher’s information of the form b Integral dt dx/dt dx/dt (b=constant) and so in a sense Fisher’s information may be varied together with another piece i.e. -V(x). In (2), it is noted that Fisher’s information, this time with a probability P(x), i.e. b Integral dx P(x) {d/dx ln(P) } {d/dx lnP) } may be varied (with constraints) to yield the Schrodinger equation. (One may note that the two definitions of Fisher’s information are the same if P(x)=x(t)x(t) in the first case.) Here we consider the notion of impulse and pressure which depends on p. We argue that a free quantum particle with constant p undergoes x=p/m t on average, but is associated with a probability distribution exp(ipx) which contains uncertainty in x (within a wavelength) and no time. P, however, represents an impulse which the particle may impart. In the case of potential V(x) one may consider a linear (interfering) sum of exp(ipx)’s i.e. W(x)= wavefunction = Sum over p a(p)exp(ipx). This leads to an average impulse of -idW/dx / W. Classically pressure is created by multiplying impulse by velocity (flux). If time does not exist within a wavelength then there is no time within it. There is, however, still an external clock and p ave in a sense should correspond to an m v(ave) nonrelativistically. Thus WW (bound system) is in a sense like a pressure. One may then suggest that for an equilibrium scenario one extremizes this pressure with respect to constraints (EWW and V(x)W). The variation is with respect to W(x), the wavefunction. Given that a bound particle maps to an rms momentum such that KE(x) = prms(x)prms(x)/2m where prms(x) is the classical value, then at the classical level prms(x) vrms(x) should be pressure and should also satisfy an extremization equation. In fact, the extremization equation using WW should be very closely related to it. One is in fact the d/dx of the other. The classical notion of pressure, proportional to vv, may be extremized subject to constraints. This leads to Newton’s second law which contains d/dt explicitly. Changing to “x” concepts yields v(x)d/dx which may be integrated to yield an energy conservation equation matching the quantum one. Thus we try to think of extremization of Fisher’s information i.e. EPI (extreme physical information) in terms of pressure, in both the classical and quantum cases.
It is well known that a Lagrangian (usually kinetic energy - potential) may be varied to produce Newton’s second law of motion. In (1) it is noted that kinetic energy is proportional to a Fisher’s information of the form b Integral dt dx/dt dx/dt (b=constant) and so in a sense Fisher’s information may be varied together with another piece i.e. -V(x). In (2), it is noted that Fisher’s information, this time with a probability P(x), i.e. b Integral dx P(x) {d/dx ln(P) } {d/dx lnP) } may be varied (with constraints) to yield the Schrodinger equation. (One may note that the two definitions of Fisher’s information are the same if P(x)=x(t)x(t) in the first case.) Here we consider the notion of impulse and pressure which depends on p. We argue that a free quantum particle with constant p undergoes x=p/m t on average, but is associated with a probability distribution exp(ipx) which contains uncertainty in x (within a wavelength) and no time. P, however, represents an impulse which the particle may impart. In the case of potential V(x) one may consider a linear (interfering) sum of exp(ipx)’s i.e. W(x)= wavefunction = Sum over p a(p)exp(ipx). This leads to an average impulse of -idW/dx / W. Classically pressure is created by multiplying impulse by velocity (flux). If time does not exist within a wavelength then there is no time within it. There is, however, still an external clock and p ave in a sense should correspond to an m v(ave) nonrelativistically. Thus WW (bound system) is in a sense like a pressure. One may then suggest that for an equilibrium scenario one extremizes this pressure with respect to constraints (EWW and V(x)W). The variation is with respect to W(x), the wavefunction. Given that a bound particle maps to an rms momentum such that KE(x) = prms(x)prms(x)/2m where prms(x) is the classical value, then at the classical level prms(x) vrms(x) should be pressure and should also satisfy an extremization equation. In fact, the extremization equation using WW should be very closely related to it. One is in fact the d/dx of the other. The classical notion of pressure, proportional to vv, may be extremized subject to constraints. This leads to Newton’s second law which contains d/dt explicitly. Changing to “x” concepts yields v(x)d/dx which may be integrated to yield an energy conservation equation matching the quantum one. Thus we try to think of extremization of Fisher’s information i.e. EPI (extreme physical information) in terms of pressure, in both the classical and quantum cases.
For the case of a free particle, it is known that one may extremize a Lagrangian L=.5mvv, where v=dx/dt, to obtain: d/dt dx/dt = 0. In this classical case, one is interested in x and t or x(t). One may think of moments i.e. powers of derivatives of x(t). Given the first moment is equal to a constant, all higher moments are 0. Each moment yields a piece of information. Thus there is one piece of information, namely the constant velocity v=dx/dt. A second derivative d/dt dx/dt =0 demonstrates there is no further information. The form d/dt dx/dt, however, is associated with an extremization of a function, namely L=.5mvv or Integral .5mvv dt. Thus we argue that extremization is really linked to information found in the problem for the free particle case. Next, we consider .5 Integral dt vv = .5 Integral v dx. This is proportional to Integral p dx where p represents a physical impulse. The equation d/dt dx/dt =0 shows there exists only one piece of information, regardless of the value of v. In other words, all v values satisfy this equation (where v=constant). In analogy, if one wishes all p values to satisfy Integral p dx, p = constant, without distinguishing p, then x= constant/p. This, however, is the quantum wavelength. If it is repeated, in space, one desires a periodic function in px which has a modulus of 1, i.e. an eigenfunction of -id/dx namely exp(ipx). This does not follow from a Fisher’s information type variation, but rather from the idea of information and two quantities of interest, namely a kind of probability exp(ipx) and x. The modulus of 1 is equivalent to the single piece of information present i.e. v=constant. To see Fisher’s information enter the quantum picture, one may consider a particle in a box with infinite walls. In such a case, exp(ipx) and exp(-ipx) are present or cos(px) if a(p)=a(-p). There is one piece of information, so d/dx cos(px) = function(px). function(px) is the one piece of extra information like v. Finding a second moment cannot yield any further information i.e. d/dx d/dx cos(px) = constant cos(px). If it did not, there would be extra information in the system, but v=constant is the only information. d/dx d/dx cos(px), however, follows from the extremization of Fisher’ s information subject to the constraint Integral cos(px)cos(xp) CC = 1. Thus we argue that in the free particle case, extremization really represents moments (shown by derivatives) and information. We also consider the case of a potential V(x) present.
Newtonian force is based on a change in momentum in time, but momentum is a vector and so may change in magnitude and direction. In many cases, the change is of one form or the other. For example, for a bound particle the change is in kinetic energy i.e. in magnitude, while for a particle moving at constant speed in a circle or pressure in a Maxwell-Boltzmann gas, the change is in direction. For an MB gas, the probability for p and -p are the same so the average momentum at any point x is 0 and thus chunks of the gas do not move, even though a single particle in the gas moves. The pressure, on the other hand, is the average of 2pv (2p for impulse, v for flux) and so pressure for p and -p do not cancel. A quantum free particle is based on exp(ipx) which does not contain time and is a dynamical probability, This is both an eigenfunction of the momentum operator -id/dx and the kinetic energy operator -1/2m d/dx d/dx, yet as pointed out in Part II, there are two pieces (cos(px),sin(px)) with a spatial density and hence a kind of pressure/force. These, however, combine to give the sense of constant motion in one direction. We argue this is due to a dynamical probability, unlike a static balance of pressures/forces found in an MB gas or Newtonian statics. There is a density with pressure distribution cos(px), but this is continuously changing because the particle moves in one direction with constant p. Thus the two pressure distributions “balance” so to speak to create a constant p. Interference of exp(ipx) and say exp(-ipx) may “bring out” a “static” spatial density i.e. 2cos(px), but behind the scene the dynamical probabilities still exist. It is these which “hold” the cos(px) together which appears strange from a static point of view. For example, a quantum bound state has a static W*(x)W(x) spatial density at the same time that the single particle has an average classical kinetic energy KE(x)=En-V(x). Interference also gives rise to the notion of an impulse-pressure in quantum as one ignores the isin(px) and i sin(-px) terms. Creating a pressure scenario with p and -p leads to two considerations. First, there is an impulse-pressure (defined without flux) at each x point and secondly, this impulse-pressure changes with x. The time-independent Schrodinger equation is formulated in terms of energy (i.e. a scalar): KE(x)+V(x)=En where KE(x)= {-1/2m d/dx dW/dx} \ W and W=Sum over p a(p)exp(ipx). Thus, as in Newtonian mechanics, one thinks of -dV/dx solely accelerating a particle. The quantum particle however, is described by an ensemble W(x) which has pressure. This pressure is due to a combination of the form exp(ipx) and V(x), so V(x) does not simply change root mean square speed. exp(ipx), by itself, creates pressure when interference occurs as in a two slit scenario. We try to examine these ideas in this note.
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.
The extremization of Fisher’s information subject to constraints usually containing V(x), the classical potential, appears in the literature (1), (2). V(x), however, is a concept introduced by Newton and associated with conservation of energy for a conservative potential. We argue the appearance of V(x) in an extremization approach to obtain an “equation” bypasses the already existing conservation of energy equation. If V(x) is used, we argue that one cannot dispense with conservation of energy and that it should be written a priori. If that is the case, what does extremization of Fisher’s information guarantee? In (1) a probability distribution P(x) is set equal to W(x)W(x) its square root. In (2), a complex function is defined: W(x) = sqrt(density(x)) exp(i S/hbar) where S is the classical action. Here density(x)=P(x). In both cases, the square root of the density appears. The problem, we argue, is that for a free particle with density(x)=P(x)=1, there is no Fisher’s information to extremize, yet the free particle is a key part of quantum mechanics. Its probability (square root probability) is the solution of an eigenvalue problem of the translation operator -id/dx and it is consistent with Newtonian mechanics i.e. conservation of energy with E= pp/2m. From this one may create an ensemble W(x) = Sum over p a(p) exp(ipx). The average kinetic energy is -1/2m d/dx/ dW/dx / W which happens to be the extremized value of Fisher’s information for P(x)=W(x)W(x). In other words, we argue that conservation of energy must occur and that one needs a solution for a free particle and this does not follow from Fisher’s information extremization. Extremization of Fisher’s information produces the form d/dx dW/dx which is taken to be linked to <pp> which already follows from -d/dx d/dx exp(ipx). One argument made in the literature (3) is that Newtonian conservation of energy also follows from Fisher’s information extremization i.e. the Lagrangian for a free particle .5m dx/dt dx/dt is of the form of a Fisher’s information with xx = probability (which is unusual). The strategy, it seems, is that one may apply extremization of Fisher’s information directly to a quantum problem by assuming a P(x)=W(x)W(x) and use d/dx instead of P=xx with d/dt. An issue, as noted, is that for a free particle W(x) exists, but extremization of Fisher’ information does not exist. Even if E=pp/2m (nonrelativistically) and this follows from the extremization of Fisher’s information, W(x)=exp(ipx) does not, but it is consistent with E=pp/2m. The reason for this consistency is that it contains p (momentum) which by its very definition must ensure E=pp/2m (nonrelativisitcally). Fisher information extremization, however, yields the bound state solution i.e. may be used to derive the Schrodinger equation. In (1) this is done for a bound state, in (2) for the more general case, but (2) defines W(x) = sqrt(density) exp(iS/hbar) where S= the classical action. For a free particle, the definition of W(x) is exp(-iEt+ipx) which is the solution of the free particle problem without using any equation. Thus the definition is the solution itself for the free particle case in (2). It seems that Fisher’s information, which uses density(x), does not include a scheme for describing motion in one direction in space without time present. exp(ipx) i.e. cos(px) and sin(px) distributions in two dimensions allow for this. This motion is associated with the translation operator d/dx which we argue is key in solving for the square-root probability of a free particle in keeping with E=pp/2m, which (3) argues follows from the extremization of Fisher’s information. Thus, even if one accepts the argument that Fisher’s information is responsible for the energy conservation equation, using Fisher’s information = Integral dt dx/dt dx/dt constant, does that allow one to immediately switch to a problem with P(x)=W(x)W(x) and the derivative d/dx and apply Fisher’s extremization there, we ask? We argue that there is an implicit assumption that: C1 dx/dt dx/dt = C2 d/dx dW/dx / W where C1 and C2 are constants This implies that for a free particle which treats each x point the same i.e. P(x)=constant, W(x)W(x) cannot be P(x). It also implies that d/dx dW/dx = pp W ((1)) where p=mov nonrelativistically. ((1)), however, is the solution to the free particle problem without performing any Fisher information extremization i.e. exp(ipx) is a solution. This solution may be used to create an ensemble W(x)=Sum over p a(p)exp(ipx) which may be used in a conservation of average energy equation. Thus we argue that Fisher extremization used to create the Schrodinger equation is based already on the idea that W(x)=exp(ipx) for a free particle and simply produces the conservation of energy equation which is already known to exist from Newtonian mechanics or Fisher’s information extremization which is argued to be the same as Lagrangian variation in (3).
Note March 9, 2023 At the same time that Fisher's information is linked with variance of an estimator, it is also linked with a classical notion of pressure for a particle in a box with infinite potential walls. Thus minimizing the variance subject to Integral P(x)dx=1 may also be thought of as extremizing pressure with respect to Integral P(x)dx=1. We argue that the cos(px), sin(px) distributions are linked with the impulse p and at the same time the particle moves with an average speed p/m (nonrelativistically). A quantum free particle is represented by the free particle wavefunction exp(ipx). This consists of two periodic distributions (in 2 dimensions) such that their modulus is 1. sin(px) and cos(px) are a possible solution set. sin(px) and cos(px) are, however, linked to a unit radius which rotates in 2 dimensions. Alternatively one may note that d/dx cos(px) = -p sin(px) and d/dx sin(px) = p cos(px). Thus one may determine the form of a free particle “probability” using the notion of periodicity, 2 dimensions and a constant modulus. No concept of extremization is needed. It is interesting to note, however, that d/dx d/dx cos(px) = pp cos(px) and d/dx d/dx sin(px) = pp sin(px). Each distribution sin(px), cos(px) by itself represents a solution of extremization problem, namely the extremization of Fisher’s information defined by I= Integral dx P(x) {d/dx ln(P)} {d/dx ln(P)} where P(x)=W(x)W(x) and W(x)=cos(px) or sin(px). Then I = Integral dx dW/dx dW/dx. Extremizing with respect to the constraint Integral P(x) = 1 yields: d/dx d/dxW = constant W. cos(px), sin(px) and exp(ipx) are solutions of this extremization equation. In order to have a solution which treats all x points the same in a certain sense, one may choose exp(ipx). The point is that even for a free particle, the two distributions involved are linked with extremization of Fisher’s information (with Integral W(x)W(x)dx=.5Length), although one may obtain exp(ipx) with no extremization and no knowledge of Fisher’s information. Why should Fisher’s information be extremized? Fisher’s information is a statistical quantity which appears in the Cramer-Rao bound: var(estimator) >= 1/ I. One may take the inverse of this equation and consider variance(estimator) to be linked to a spread or uncertainty in x. If one wishes to minimize this, one may maximize I subject to constraints. Thus there seems to be a purely statistical approach (i.e the extremization of Fisher’s information) even for a free particle. This extremization approach, however, is linked directly to the fact that one has two probability distributions in a 2-dimensional object with a constant modulus. This does not have the appearance of any extremization, but is in fact equivalent to one for each distribution function. In fact, we argue that if P(x)=W(x)W(x) with Integral P(x)dx=1, then if W(x) is cyclical/periodic i.e. d/dx d/dx W = constant W, then W may be obtained through the extremization of Fisher’s information subject to the constraint Integral P(x)dx =1. Thus periodicity and extremization are linked, but in a somewhat unusual manner.
A classical bound system may be said to have a spatial density proportional to the amount of time dt a particle spends in a constant dx region i.e. C/v(x) (1) We stress that in this scenario there is a well defined and constant dx so that “dt” may be linked to velocity v(x). We also note that there is no spatial representation of action-reaction here even though this exists in Newton’s theory. In a classical Maxwell-Boltzmann gas there are many particles which each act dynamically according to .5mvv+V(x) = E (where v and E are particular to the particle when it is not colliding). At the same time there is a spatial density which “represents” macroscopic static chunks of matter which are balanced through pressure differences (P=spatial density(x) RT) and -dV/dx density where V(x) is the potential. From the pressure equation, T is proportional to energy which in turn is proportional to RMS speed squared. RMS speed may then be linked to a dt if a fixed dx is understood. In the classical statistical case, however, spatial density is directly linked to the number of particles in a region. Time enters in the pressure expression through a term proportional to the average kinetic energy i.e. pv where p is like an impulse and v a flux. What happens in a single particle quantum bound state? In a sense it is like a classical bound state as the equation: prms(x)prms(x)/2m + V(x) = En holds for any energy level. Can one then take vrms(x), choose a fixed dx and argue that spatial density should be equal to dt or C/vrms(x)? We argue that one cannot due to the lack of defined “dx” quantity. Why is dx problematic to define? We argue that it is because quantum mechanics is based on action-reaction x-probaiblity distributions linked to impulse. For example, exp(ipx) for a free particle captures only p or impulse (not velocity) and cos(px) and sin(px) are its action-reaction regions.These lead ultimately to a density is given by the well-known W*(x)W(x) where W(x)=Sum over p a(p)exp(ipx), although kinetic energy with pp/2m = .5pv is used to find W(x) as well. . W*(x)W(x) has crests and troughs similar to cos(px) in exp(ipx), except that these now have varying heights and widths. We try to argue that even though each x point maps to a different KEave(x), the hump physically represents action-reaction in the system (just as the humps in cos(px) do except for a single p) while KE(x) represents an average of p’s. This action-reaction is linked with impulses. In general, if one takes d/dx of the time-independent Schrodinger equation one finds an average impulse term, -dV/dx and KE(x) dW/dx / W where dW/dx / W is the average impulse. At peaks, dW/dx =0 and so an average force through time*impulse term balances -dV/dx=Newtonian force. These peaks are also associated with W*(x at peak)W(x at peak) which is the spatial density. We interpret this as representing the average time the single particle spends near x at peak. This value, however, follows from ideas of impulse and not from speed as there is no well defined dx even though there is a well defined vrms(x). Thus the ratio of W*(x)W(x) at two peaks is not the inverse of the ratio of vrms(x) except apparently when the energy level n becomes very high. We investigate these ideas.