It is well known that a suggestive relation exists that links Schrödinger's equation (SE) to the information-optimizing principle based on Fisher's information measure. The connection entails the existence of a Legendre transform structure underlying the SE. Here, we show that appeal to this structure leads to a first order differential equation for the SE's eigenvalues that, in certain cases, can be used to obtain the eigenvalues without explicitly solving the SE. Complying with the above mentioned equation constitutes a necessary condition to be satisfied by an energy eigenvalue. We show that the general solution is unique.
It is well known that a suggestive relation exists that links Schrödinger's equation (SE) to the information-optimizing principle based on Fisher's information measure. The connection entails the existence of a Legendre transform structure underlying the SE. Here, we show that appeal to this structure leads to a first order differential equation for the SE's eigenvalues that, in certain cases, can be used to obtain the eigenvalues without explicitly solving the SE. Complying with the above mentioned equation constitutes a necessary condition to be satisfied by an energy eigenvalue. We show that the general solution is unique.
A suggestive relation links Fisher' information measure (FIM)Iand Schrödinger equation (SE). The connection is based upon the fact that the constrained minimization of I leads to a SE. This, in turn, is the origin of intriguing relationships between various aspects of SE, on the one hand, and the formalism of statistical mechanics derived from Jaynes's maximum entropy principle (MaxEnt), on the other one. The link entails the existence of a Legendre transform structure underlying the SE, which allows for the emergence of two first-order differential equations that must, respectively, be satisfied by i) the Fisher measure and ii) the SE energy eigenvalues. The complete A) I-solution and B) energy-solution are both obtained bypassing the SE and, furthermore, linked by the Legendre structure.
Schrödinger's equation (SE) and the information-optimizing principle based on Fisher's information measure (FIM) are intimately linked, which entails the existence of a Legendre transform structure underlying the SE. In this comunication we show that the existence of such an structure allows, via the virial theorem, for the formulation of a parameter-free ground state's SE-ansatz for a rather large family of potentials. The parameter-free nature of the ansatz derives from the structural information it incorporates through its Legendre properties.
Abstract We study the scaling properties of Fisher’s information measure (FIM) and show that from these one can straightforwardly deduce significant quantum-mechanical results. Specifically, we investigate the scaling properties of Fisher’s measure I and encounter that, from the concomitant operating rules, several interesting, albeit known, results can be derived. This entails that such results can be regarded as pre-configured by the conjunction of scaling and information theory. The central notion to be arrived at is that scaling entails that I must obey a certain partial differential equation (PDE). These PDE-solutions have properties that enable the application of a Legendre-transform (LT). The conjunction PDE+LT leads one to obtain several quantum results without recourse to the Schrödinger’s equation.
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.
A process exists in the literature (1) of creating an information term based on Fisher’s information using spatial density P and adding it to a classical Lagrangian based on a velocity field i.e. velocity=dS/dx. Maximization of S and P is performed and a differential equation in terms of spatial density and S is obtained. Finally, it is noted that a change of variables: W= sqrt(p) exp(iS/hbar) may simplify the differential equation so that it becomes the Schrodinger equation with W as the wavefunction. In this note, we consider the wavefunction as appearing “naturally” as a normalization constant when using conditional probability and argue that W is the velocity field. Given that the Schrodinger equation includes a term of the form -1/2m d/dx d/dx W (in 1-D) i.e. the first derivative is related to the average velocity and the second to the ensemble average of p*p. This follows because unnormalized conditional probability f(p,x) is the Fourier basis function i.e. exp(ipx). The second derivative term - 1/2m d/dx d/dx W may be obtained from a Lagrangian containing (d/dxW) (d/dx W), thus formally one may use maximization to obtain the Schrodinger equation at a level using W directly. The maximization form, however, is identical to the “classical translational” form (2) of Fisher’s information. Thus, it seems that the Fourier basis exp(ipx) and the maximization of classical translational Fisher’s information are linked. In this note, we wish to investigate this in further detail. We also use translational properties together with conditional probability to show that f(p,x) where P(p/x)= a(p) f(p/x) / [Sum over p a(p) f(p/x)] is exp(ipx). In particular, we obtain an equation taken from the translation of the conservation of energy equation at point x to x+delta x. This links df(p,x)/dx with -dV/dx (the force). If the force is stochastic and yields “k” momentum hits, -dV/dx= Sum over k kVk(x). We argue that df(p,x)/dx also brings out this k term i.e. is an eigenfunction of k i.e. -idf(p,x)/dx =k f(p,x) yields exp(ipx) as a solution. This f(p,x) is automatically an orthonormal basis (multiplied by an appropriate constant) and helps to show why the “momentum operator” is linked to a “translation operator” i.e. d/dx. The bound state Schrodinger equation is considered in this note.