The classical Fisher information, F, in one dimension is given in the literature as: Integral dx d(x) [ d/dx ln( d(x) ) ]2 ((1)) where d(x) is the spatial probability density. For quantum mechanics, (1) uses d(x)=W*(x)W(x) for quantum mechanics, where W(x) is the complex wavefunction. The Cramer Rao inequality is used to show how 1/ F is related to the variance of d(x). For a bound state wavefunction, which is real, one may show Fisher’s information leads to Integral dx v(x)v(x) d(x) which is proportional to the average kinetic energy. Using the correspondence principle, d(x), for high energy levels, should approach C/v(x) where v(x) is the classical velocity. On the other hand, using the definition of F directly with d(x)=C/v(x) leads to a different result. This seems unusual. We try to investigate in this note to see that quantum features exist even in the high energy limit. The Fisher information calculated with W(x)W(x)=C/v(x) seems to capture quantum information, while that with d(x)=C/v(x), information about the envelope function.
The classical Fisher information, F, in one dimension is given in the literature as: Integral dx d(x) [ d/dx ln( d(x) ) ]2 ((1)) where d(x) is the spatial probability density. For quantum mechanics, (1) uses d(x)=W*(x)W(x) for quantum mechanics, where W(x) is the complex wavefunction. The Cramer Rao inequality is used to show how 1/ F is related to the variance of d(x). For a bound state wavefunction, which is real, one may show Fisher’s information leads to Integral dx v(x)v(x) d(x) which is proportional to the average kinetic energy. Using the correspondence principle, d(x), for high energy levels, should approach C/v(x) where v(x) is the classical velocity. On the other hand, using the definition of F directly with d(x)=C/v(x) leads to a different result. This seems unusual. We try to investigate in this note to see that quantum features exist even in the high energy limit. The Fisher information calculated with W(x)W(x)=C/v(x) seems to capture quantum information, while that with d(x)=C/v(x), information about the envelope function.
Linking quantum mechanics to classical mechanics seems to have been an early goal in quantum theory as represented by the correspondence principle. More recently there have been attempts to link the entropy of both a single bound quantum state and a thermal statistical system of quantum states to classical entropies (1),(2). For the former, the entropy is proportional to Integral dx density(x) ln[density(x)] where density(x)= delta t = 1 / v(x) with v(x) being velocity. Thus, density, which is a probability in Shannon’s entropy equation, is a measure of the change in the value of delta t(x) in this classical entropy. Bound state quantum mechanics is said to be time-independent, but by the correspondence principle (1) the probability of the envelope of the wavefunction i.e. Envelope(W(x))* Envelope(W(x)) = delta t(x) so quantum probability (albeit linked to the envelope) is related to classical time intervals which adjust with x to yield the proper ration v(x)=delta x / delta t(x) with delta x being constant. In this note we argue that the expectation value: Integral W(x) d/dx dW/dx = Integral WW KEclassical(x) for a bound state is, in the high energy limit Integral dt KEclassical(x) i.e. a portion of the action, just as in the classical case (i.e. the correspondence principle). This leads to the association of the higher energy limit of the envelope of W times itself with delta t(x).
In the literature (e.g. (1)), the expression - density(x) ln(density(x)) is used as Shannon’s spatial entropy for a quantum system (bound or not). For the bound case, the correspondence principle is said to hold for high energy levels (2). In that case, quantum density = W(x)W(x) (W(x)=real wavefunction) is said to be equivalent to the classical density C/v(x) where v(x) the velocity is given by .5mv(x)v(x) + V(x) = E. Classical density yields a curve which passes through peak regions of quantum density humps. In the classical case, particle motion is completely determinate. C/v(x) as a density (probability) is proportional to dt, the amount of time a particle spends passing through a fixed length dx. If the particle moves quickly, it spends little time, and hence there is less time for a measurement (i.e one may miss the particle due to its speed). We argue in this note that such a scenario does not seem to be in keeping with the ideas of Shannon’s entropy where one has probabilities for certain mutually exclusive events at each ti. For example at each ti, one may toss a coin. If it is not “heads”, this means it is tails. For the classical particle in the correspondence principle, each dx region has its own probability proportional to dt = dx/v(x). Thus, each dx is a region of its own in terms of probability. Thus, if Ci measurements of the presence of the particle are made within dx of xi out of N trials, the probability that a measurement is made in dx at xi is Ci/N and the probability that it is not is 1-Ci/N. Thus, unlike Shannon’s picture, various arrangements do not have the same weight or probability. We investigate this scenario in more detail and argue that it does not follow the approach of Shannon’s entropy. We also try to connect with the quantum situation.
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.
In (1), Shannon’s spatial entropy is computed using - [W*W] ln[W*W], where W(x) is the wavefunction and a classical envelope approximation: W*(x)W(x)=C/v(x), where v(x) is classical velocity, is used (according to the idea of the correspondence principle). A momentum entropy: - Integral dp a(p)a(p) ln[a(p)a(p)] (one dimension) is also calculated. Recently in (2), we argued that one should form Shannon’s entropy using the probability = a(p1)exp(ip1 x) a(p2)exp(ip2 x). This leads to the following if one sums the first term over p1,p2 and integrates the second over x: Entropy= Integral dx -2W x d/dx W - Integral dp a(p)a(-p) ln[a(p)a(-p)] ((1)) one dimension In (1), the spatial entropy is found to behave as ln(n) for high n whereas for ((2)), spatial entropy is always 1. Thus, there is a very large difference between the two. In (1), a momentum entropy -a(p)a(p)ln[a(p)a(p)] is added to spatial entropy and in ((1)) a very similar momentum piece appears. For the case of the oscillator, W(x) and a(p) are both Gaussians and so the overall result for entropy from (1) and from this note both behave as ln(n) when n is large, but the factor of ln(n) differs. Ours is half as large for the case of a quantum oscillator. For our case, all of the entropy comes from the momentum term in the large n limit. In this note, we consider these ideas in more detail.
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).
In bound state quantum mechanics, high energy eigenvalues are said to lead to a density W(x)W(x) where W(x) is the wavefunction, whose envelope approaches the classical value 1/v(x), where v(x) is the classical velocity given by sqrt[2m(E-V(x)]. For the quantum oscillator, the ground state mimics classical statistical mechanics, while the high energy states mimic classical mechanics. For both classical statistical mechanics and classical physics, we argue in this note that one may use a reaction picture to derive probabilities. Traditionally, the classical density 1/v(x) is obtained from dx = v dt or dx/v = dt with “dt” being proportional to probability. In classical statistical mechanics, one has two particle scattering and imposes a balance using P(p1)P(p2)=P(p3)P(p4). In the case of V(x), one may also write a reaction equation (1) showing particles leaving and entering a little box. In this note, it we argue it would be good to have a consistent picture when dealing with densities, namely a reaction picture, with “breakup or decay” balancing “synthesis”. We then try to argue that such a reaction picture may actually also apply to high energy eigenvalue quantum mechanical bound states.