Abstract
Fisher’s information measure plays a very important role in diverse areas of theoretical physics. The associated measures and , as functionals of quantum probability distributions defined in, respectively, coordinate and momentum spaces, are the protagonists of our present considerations. The product has been conjectured to exhibit a nontrivial lower bound in Hall (2000). More explicitly, this conjecture says that for any pure state of a particle in one dimension . We show here that such is not the case. This is illustrated, in particular, for pure states that are solutions to the free-particle Schrödinger equation. In fact, we construct a family of counterexamples to the conjecture, corresponding to time-dependent solutions of the free-particle Schrödinger equation. We also conjecture that any normalizable time-dependent solution of this equation verifies for .
1. Introduction
A very important information measure, with manifold physical applications, was conceived by R. A. Fisher in the 1920s—for detailed discussions see [1–4]. Recent developments show that Fisher’s information has a fundamental role in quantum mechanics [5–19]. In particular, it allows for the formulation of new quantum uncertainty principles [20–24]. It is usually abbreviated as and can be thought of as a measure of the expected error in a measurement [1].
A particular instance of great relevance is that of translational families [1]. These are distribution functions whose form remains invariant under displacements of a shift parameter . Thus, they are shift invariant distributions (in a Mach sense, there is no absolute origin for ). The measure exhibits Galilean invariance [1]. Given a probability density , with and a family of parameters, the concomitant Fisher matrix is [25] where is the volume element in . In particular, for , one defines translational families , with elements , where represents the partial derivative with respect to the coordinate . The trace of this matrix, given by , is a good uncertainty indicator for probability distributions associated with quantum wave functions [26]. If is a normalized wave function in coordinate space (-dimensions) and is its momentum-counterpart, the corresponding probability densities are, respectively, and , with associated Fisher measures which allow one to study uncertainty relations via the product [26].
For instance, one can demonstrate that if (or ) is real, then [5], with equality for coherent states of the harmonic oscillator (HO) [26]. For general, mixed states it is clear that the product does not possess a nontrivial lower bound (e.g., one can use thermal HO states, represented by Gaussian distributions in both coordinates and momenta, in the high temperature limit). In the case of pure states, though, the existence of such a lower bound for was an open question. Hall conjectured that the relation might hold in general for pure states in one dimension [26, page 3]. We will next present a couple of counterexamples that show this conjecture to be incorrect. Our examples give rise to a new conjecture: for a bounded wave function one has when .
2. Counterexamples
Our first example is taken from the considerations (in a different context) of [5]. A free-particle’s one-dimensional wave packet (unit mass) evolves according to Schrödinger’s equation Setting the initial conditions with , , and , that correspond to a Gaussian packet, one finds the solution where . The associated probability densities are The product obeys the relation for . Also, one has for .
We pass now to another free-particle solution, given by the first partial derivative of with respect to : (see (4)), correctly normalized. It is easy to see that is a solution by deriving both members of (4); that is, The new solution is with . The two corresponding densities are The product is , verifying for , and when .
In general, one can show that the whole family of solutions of (4) given by successive derivatives of , that is, the set , verifies that when , with the pertinent normalization constants. Thus, the family yields infinite counterexamples to Hall’s conjecture. To see this, one needs first to rewrite the Fisher measure in wave function’s terms, so that (2) becomes equivalent to or, in one dimension, . Further, . Thus, can be expressed in terms of and . In one dimension one has In general, for , the Fisher’s measure associated with the distribution becomes
We show now that the integrand tends to zero for . Thus, in such a limit. Actually, we will show that for . The th derivative of is proportional to the th derivative of a Gaussian distribution, given by where and is the Hermite polynomial of degree in the variable . The time-dependent parameters , , and vanish for . What is the behavior of ? Let us see what happens with , the th solution in momentum space, corresponding to the Fourier transform of . We have Demanding normalization leads to Thus, Remember that with , and One finds the following limits for the absolute values of the wave functions: Equation (20) indicates that all functions vanish for . Accordingly, from (13) we find for for all . Further, (21) shows that does not vanish in this limit. In fact, does not depend on .
So as to understand what happens with let us see an expression analogous to (13) in momentum space: Expanding the integrand using (15) we have Introducing this into (22), and remembering that both and are independent of , we find Since (see (16)), one has and thus becomes bounded. Thus, there exists such that for all . We conclude that for for the whole family of solutions .
3. Conclusions
We conclude by reiterating that we have found an infinite number of counterexamples to the conjecture , for pure states, put forward in [26]. On the basis of these results, we conjecture that, for any normalizable wave function , the corresponding time-dependent solution of the free-particle Schrödinger equation satisfies for .
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.