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Physics Letters A

Uncertainty relations for modified isotropic harmonic oscillator and Coulomb potentials

S. H. Patil, K. D. Sen

Published Nov 29, 2006• 50 citations
Publisher
DOI
10.1016/j.physleta.2006.11.045

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Abstract The Heisenberg uncertainty relations are cornerstones of quantum theory. Yet care is required in applying them. While they determine the lowest energy level in some systems, such as the isotropic three-dimensional harmonic oscillator, they fail to do so in other systems, notably, the hydrogen atom. This failure has been overlooked in many textbooks and pedagogic articles. The failure arises due to the singularity of the Coulomb potential. To elucidate the point, we present four subfamilies of position probability distributions. By delineating the limitations in applying position-momentum uncertainty to the hydrogen atom, we hope to improve the teaching and learning of modern physics.
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The D-dimensional harmonic system (i.e., a particle moving under the action of a quadratic potential) is, together with the hydrogenic system, the main prototype of the physics of multidimensional quantum systems. In this work, we rigorously determine the leading term of the Heisenberg-like and entropy-like uncertainty measures of this system as given by the radial expectation values and the Rényi entropies, respectively, at the limit of large D. The associated multidimensional position-momentum uncertainty relations are discussed, showing that they saturate the corresponding general ones. A conjecture about the Shannon-like uncertainty relation is given, and an interesting phenomenon is observed: the Heisenberg-like and Rényi-entropy-based equality-type uncertainty relations for all of the D-dimensional harmonic oscillator states in the pseudoclassical ( D → ∞ ) limit are the same as the corresponding ones for the hydrogenic systems, despite the so different character of the oscillator and Coulomb potentials.