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International Journal of Quantum Chemistry

Heisenberg uncertainty relations can be replaced by stronger ones

Lubomír Skála, V. Kapsa

Published Dec 23, 2008• 8 citations
Publisher

Abstract

Abstract Two uncertainty relations, one related to the probability density current and the other one related to the probability density, are derived and discussed. Both relations are stronger than the Heisenberg uncertainty relations. Their generalization to the multidimensional case and to the mixed states is also discussed. © 2008 Wiley Periodicals, Inc. Int J Quantum Chem, 2009

DOI
10.1002/qua.21948

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Abstract Two uncertainty relations, one related to the probability density current and the other one related to the probability density, are derived and discussed. Both relations are stronger than the Heisenberg uncertainty relations. Their generalization to the multidimensional case and to the mixed states is also discussed. © 2008 Wiley Periodicals, Inc. Int J Quantum Chem, 2009
It is known that the Heisenberg and Robertson-Schr\"odinger uncertainty relations can be replaced by sharper relations in which the ``classical'' (depending on the gradient of the phase of the wave function) and ``quantum'' (depending on the gradient of the envelope of the wave function) parts of the variances $\ensuremath{\langle}{(\ensuremath{\Delta}x)}^{2}\ensuremath{\rangle}$ and $\ensuremath{\langle}{(\ensuremath{\Delta}p)}^{2}\ensuremath{\rangle}$ are separated. In this paper, multidimensional generalization of these relations is discussed.
Recently, Maccone and Pati [Phys. Rev. Lett. {\bf 113}, 260401 (2014)] derived few inequalities among variances of incompatible operators which they called stronger uncertainty relations, stronger than Heisenberg-Robertson or Schrodinger uncertainty relations. Here we generalize their study to get infinite number of such inequalities and propose that only one of them may be the correct uncertainty relation between incompatible operators. We get back well known uncertainty relations of Heisenberg-Robertson and Schrodinger under certain limits. We also reexamine the conclusions of Maccone and Pati and comment on their work.
The Heisenberg uncertainty relation sets a fundamental limit for quantum measurement of incompatible observables. Its standard form derived by Weyl and Robertson is of purely quantum nature when the state is pure. However, for mixed states, because the variance involving a mixed state is a hybrid of classical and quantum uncertainty, the conventional uncertainty relation is of a ``mixed'' flavor. It is desirable to seek some decomposition of variance into classical and quantum parts and to cast the Heisenberg uncertainty relation for mixed states in a more quantum form. By use of the skew information introduced by Wigner and Yanase in 1963, we make such an attempt and establish a different uncertainty relation which is stronger than the conventional one.
The analytic information theory of quantum systems includes the exact determination of their spatial extension or multidimensional spreading in both position and momentum spaces by means of the familiar variance and its generalization, the power and logarithmic moments, and, more appropriately, the Shannon entropy and the Fisher information. These complementary uncertainty measures have a global or local character, respectively, because they are power-like (variance, moments), logarithmic (Shannon) and gradient (Fisher) functionals of the corresponding probability distribution. Here we explicitly discuss all these spreading measures (and their associated uncertainty relations) in both position and momentum for the main prototype in D-dimensional physics, the hydrogenic system, directly in terms of the dimensionality and the hyperquantum numbers which characterize the involved states. Then, we analyze in detail such measures for s-states, circular states (i.e., single-electron states of highest angular momenta allowed within an electronic manifold characterized by a given principal hyperquantum number), and Rydberg states (i.e., states with large radial hyperquantum numbers n). © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2010
Abstract Exact time‐dependent density functionals depend on both the entire history of the density and the initial wave function. Recently we have shown that the two effects are entangled by an exact condition that should be useful as a test in building accurate memory‐dependent functionals. Here we consider this condition for small perturbations and derive an exact relation connecting the exchange‐correlation kernel with initial‐state derivatives. We discuss other exact conditions on the initial‐state derivatives which shed some light on the elusive memory‐dependence in time‐dependent density functional theory. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2005