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Journal of Physics A Mathematical and General

Fisher information, kinetic energy and uncertainty relation inequalities

Shunlong Luo

Published Jun 14, 2002• 43 citations
Publisher

Abstract

By interpolating between Fisher information and mechanical kinetic energy, we introduce a general notion of kinetic energy with respect to a parameter of Schrödinger wavefunctions from a statistical inference perspective. Kinetic energy is the sum of Fisher information and an integral of a parametrized analogue of quantum mechanical current density related to phase. A family of integral inequalities concerning kinetic energy and moments are established, among which the Cramér–Rao inequality and the Weyl–Heisenberg inequality, are special cases. In particular, the integral inequalities involving the negative order moments are relevant to the study of electron systems. Moreover, by specifying the parameter to a scale, we obtain a family of inequalities of uncertainty relation type which incorporate the position and momentum observables symmetrically in a single quantity.

DOI
10.1088/0305-4470/35/25/303

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By interpolating between Fisher information and mechanical kinetic energy, we introduce a general notion of kinetic energy with respect to a parameter of Schrödinger wavefunctions from a statistical inference perspective. Kinetic energy is the sum of Fisher information and an integral of a parametrized analogue of quantum mechanical current density related to phase. A family of integral inequalities concerning kinetic energy and moments are established, among which the Cramér–Rao inequality and the Weyl–Heisenberg inequality, are special cases. In particular, the integral inequalities involving the negative order moments are relevant to the study of electron systems. Moreover, by specifying the parameter to a scale, we obtain a family of inequalities of uncertainty relation type which incorporate the position and momentum observables symmetrically in a single quantity.
The inequality , with L being the grand orbital quantum number, and its conjugate relation for (⟨ r 2 ⟩, ⟨ p −2 ⟩) are shown to be fulfilled in the D -dimensional central problem. Their use has allowed us to improve the Fisher-information-based uncertainty relation ( I ρ I γ ⩾ const) and the Cramer–Rao inequalities (⟨ r 2 ⟩ I ρ ⩾ D 2 ; ⟨ p 2 ⟩ I γ ⩾ D 2 ). In addition, the kinetic energy and the radial expectation value ⟨ r 2 ⟩ are shown to be bounded from below by the Fisher information in position and momentum spaces, denoted by I ρ and I γ , respectively.
It is shown that the assumption that nature acts to make optimum estimates of position maximally in error leads to the Schrödinger (energy) wave equation (SWE). In this way, the SWE follows from a simple statement of uncertainty. The approach grows out of probability estimation theory, in particular the principle of minimum Fisher information (or maximum Cramer–Rao bound). The minimized Fisher information turns out to be proportional to the mean kinetic energy of the particle. This approach is an attractive supplement to conventional ways of introducing quantum mechanics to students, since it avoids the use of imperfect physical models (such as vibrating strings) or the immediate need for complex momentum operators, and follows from a plausible assumption as to how nature works.
For a quantum state and a set of observables, we can construct an associated covariance matrix and a natural quantum Fisher information matrix. These two matrices characterize the uncertainty and information content of the observables in the relevant state. An inequality between these two matrices is established. This inequality may be interpreted as a general quantification of the Heisenberg uncertainty principle from a statistical estimation perspective. In particular, it implies a new uncertainty relation which refines the celebrated Schrödinger uncertainty relation.
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.
Rigorous and universal relationships among radial expectation values of any DDD-dimensional quantum-mechanical system are obtained, using R\'enyi-like position-momentum inequalities in an information-theoretical framework. Although the results are expressed in terms of four moments (two in position space and two in the momentum one), especially interesting are the cases that provide expressions of uncertainty in terms of products $\ensuremath{\langle}{r}^{a}\ensuremath{\rangle}{}^{1/a}\ensuremath{\langle}{p}^{b}\ensuremath{\rangle}{}^{1/b}$, widely considered in the literature, including the famous Heisenberg relationship $\ensuremath{\langle}{r}^{2}\ensuremath{\rangle}\ensuremath{\langle}{p}^{2}\ensuremath{\rangle}\ensuremath{\geqslant}{D}^{2}/4$. Improved bounds for these products have recently been provided, but are always restricted to positive orders a,b>0a,b>0a,b>0. The interesting part of this work are the inequalities for negative orders. A study of these relationships is carried out for atomic systems in their ground state. Some results are given in terms of relevant physical quantities, including the kinetic and electron-nucleus attraction energies, the diamagnetic susceptibility, and the height of the peak of the Compton profile, among others.
We introduce a generalized Wigner-Yanase skew information and then derive the trace inequality related to the uncertainty relation. This inequality is a non-trivial generalization of the uncertainty relation derived by S.Luo for the quantum uncertainty quantity excluding the classical mixture. And we introduce a generalized Fisher information and then derive a generalized Cramer-Rao inequality. We also give an example for our generalized Fisher information and then derive the uncertainty relation for two observables.