Minimum uncertainty for antisymmetric wave functions

We study how the entropic uncertainty relation for position and momentum conjugate variables is minimized in the subspace of one-dimensional antisymmetric wave functions. Based partially on numerical evidence and partially on analytical results, a conjecture is presented for the sharp bound and for the minimizers. Conjectures are also presented for the corresponding sharp Hausdorff-Young inequality. PACS Typeset using REVTEX Email address: [email protected] Let ψ be in a square integrable function in R, to represent the wave function of a quantum-mechanical particle, and let ρ be its normalized probability density, to wit, ρ(x) = |ψ(x)|/||ψ||2, where ||ψ||p denotes the p-norm ( ∫ |ψ(x)|pdx). The information entropy of ψ (or ρ) is defined as S(ψ) = − ∫ log(ρ(x))ρ(x)dx . (1) It measures the localization of the state in configuration space. A high entropy implies a low spatial localization and vice versa. Likewise, one can consider the wave function in momentum space, defined by the Fourier transform of ψ, that is (Fψ)(x) = ∫ eψ(y)dy (2) for ψ integrable. (The normalization of F corresponds to using units 2πh̄ = 1.) We will often use the notation ψ̃ for the Fourier transform of ψ. Again, its information entropy S(ψ̃) is a measure of its momentum space localization. As shown by Hirschman [1] in one dimension and by Bia lynicki-Birula and Mycielski in the n-dimensional case [2], the basic uncertainty relations of position and momentum in quantum mechanics can be derived from the following sharp bound in L2(R): S(ψ) + S(ψ̃) ≥ n(1− log 2) . (3) Indeed, this inequality puts a bound on the maximum localization in phase space and, in particular, it can be shown to imply the uncertainty relations of Heisenberg (Weyl-Heinsenberg inequality) [1,2]. As stressed by Deutsch [3], entropic uncertainty relations among observables are a more faithful expression of the quantum-mechanical uncertainty principle than the customary generalized Heisenberg relations. (See also [4–6] for further details and applications.) The equality in (3) is reached by any Gaussian function and moreover these are the unique minimizers [7]. Since the Gaussian can be taken centered at the origin, the same sharp bound holds in the subspace of even functions. Less obvious is the value of the sharp