Relative Fisher information in some central potentials

E. Romera, Pablo Sánchez-Moreno, J. S. Dehesa

E. Romera, Pablo Sánchez-Moreno, J. S. Dehesa

An uncertainty Fisher information relation in quantum mechanics is derived for multidimensional single-particle systems with central potentials. It is based on the concept of Fisher information in the two complementary position and momentum spaces, which is a gradient functional of the corresponding probability distributions. The lower bound of the product of position and momentum Fisher informations is shown to depend on the orbital and magnetic quantum numbers of the physical state and the space dimensionality. Applications to various elementary systems is discussed.

Pablo Sánchez-Moreno, Rosario González‐Férez, J. S. Dehesa

The Heisenberg and Fisher-information-based uncertainty relations are improved for stationary states of single-particle systems in a D -dimensional central potential. The improvement increases with the squared orbital hyperangular quantum number. The new uncertainty relations saturate for the isotropic harmonic oscillator wavefunction.

R. C. Venkatesan, Angel Ricardo Plastino

Takuya Yamano

J. S. Dehesa, Rosario González‐Férez, Pablo Sánchez-Moreno

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.