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Calculation of moments and uncertainty in position space

Anil Kumar

Published Jan 1, 2014• 0 citations
OpenAlex

Abstract

The moments are calculated for some levels of Poschl-Teller potentials. Using isospectral Hamiltonians, the calculations are exte nded to deformed wave functions and their properties are graphically demonstrated. Two Hamiltonians are said to be strictly isospectral, if they have exactly same eigenvalue s pectrum and S-matrix. The wave functions and their dependent quantities are differ ent but related. This property is utilized to obtain the uncertainty in position space. Due to the presence of a deformation parameter, the uncertainty in position space is rea rranged.

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The moments are calculated for some levels of Poschl-Teller potentials. Using isospectral Hamiltonians, the calculations are exte nded to deformed wave functions and their properties are graphically demonstrated. Two Hamiltonians are said to be strictly isospectral, if they have exactly same eigenvalue s pectrum and S-matrix. The wave functions and their dependent quantities are differ ent but related. This property is utilized to obtain the uncertainty in position space. Due to the presence of a deformation parameter, the uncertainty in position space is rea rranged.
We present a construction of semi-classical states for Pöschl-Teller potentials based on a supersymmetric quantum mechanics approach. The parameters of these "coherent" states are points in the classical phase space of these systems. They minimize a special uncertainty relation. Like standard coherent states they resolve the identity with a uniform measure. They permit to establish the correspondence (quantization) between classical and quantum quantities. Finally, their time evolution is localized on the classical phase space trajectory.
Quantum Information Entropies for the ℓ\ellℓ-state Pöschl-Teller-type potential
In this study, we obtained the position-momentum uncertainties and some uncertainty relations for the Pöschl-Teller-type potential for any ℓ\ellℓ. The radial expectation values of r−2r^{-2}r−2, r2r^{2}r2 and p2p^{2}p2 are obtained from which the Heisenberg Uncertainty principle holds for the potential model under consideration. The Fisher information is then obtained and it is observed that the Fisher-information-based uncertainty relation and the Cramer-Rao inequality hold for this even power potential. Some numerical and graphical results are displayed.
The aim of Part II of this paper is to try to describe wave functions on q-deformed versions of position and momentum space. This task is done within the framework developed in Part I of the paper. In order to make Part II self-contained the most important results of Part I are reviewed. Then it is shown that q-deformed exponentials and q-deformed delta functions play the role of momentum and position eigenfunctions, respectively. Their completeness and orthonormality relations are derived. For both bases of eigenfunctions matrix elements of position and momentum operators are calculated. A q-deformed version of the spectral decomposition of multiplication operators is discussed and q-analogs of Heaviside functions are proposed. Interpreting the results from the point of view provided by the concept of quasipoints gives the formalism a physical meaning. The definition of expectation values and the calculation of probability densities are explained in detail. Finally, it is outlined how the considerations so far carry over to antisymmetrized spaces.