Information-theoretic measures for Morse and Pöschl–Teller potentials

Abstract

The spreading of the quantum-mechanical probability cloud for the ground state of the Morse and modified Pöschl–Teller potentials, which controls the chemical and physical properties of some molecular systems, is studied in position and momentum space by means of global (Shannon's information entropy, variance) and local (Fisher's information) information-theoretic measures. We establish a general relation between variance and Fisher's information, proving that, in the case of a real-valued and symmetric wavefunction, the well-known Cramer–Rao and Heisenberg uncertainty inequalities are equivalent. Finally, we discuss the asymptotics of all three information measures, showing that the ground state of these potentials saturates all the uncertainty relations in an appropriate limit of the parameter.

Keywords:

• Morse potential
• Pöschl–Teller potential
• Shannon's entropy
• Fisher's information
• Uncertainty relations

Acknowledgement

This work was partially supported by the European Research Network ‘Network on Constructive Complex Approximation (NeCCA)’, INTAS 03-51-6637, the Ministry of Science and Technology (MCYT) of Spain, through grants FIS2005-00973 (J.S.D.) and MTM2005-08648-C02-01 (A.M.F.), and NATO Collaborative Linkage Grant ‘Orthogonal Polynomials: Theory, Applications and Generalizations’, ref. PST.CLG.979738 (J.S.D., A.M.F.). The authors belong to the research groups FQM207 (J.S.D.) and FQM229 (A.M.F.) of Junta de Andaluc′a, Spain.