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Annalen der Physik

Maximum entropy approach to Schrödinger's radial equation

F. Garcías, M. Casas, Angel Ricardo Plastino

Published Jan 1, 1995• 3 citations
Publisher

Abstract

Abstract From the sole knowledge (at a finite number of points) of the numerical values of the potential V(r) corresponding to Schrödinger's radial equation, it is found that recourse to Information Theory (IT) concepts allows one to infer the pertinent wave functions (and eigenvalues) without attempting to solve the concomitant differential equation. Moreover, the underlying IT ideas allow for an analytical treatment that yields exact wave functions of the maximum (quantal) entropy form in a number of cases of interest.

DOI
10.1002/andp.19955070406

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Abstract From the sole knowledge (at a finite number of points) of the numerical values of the potential V(r) corresponding to Schrödinger's radial equation, it is found that recourse to Information Theory (IT) concepts allows one to infer the pertinent wave functions (and eigenvalues) without attempting to solve the concomitant differential equation. Moreover, the underlying IT ideas allow for an analytical treatment that yields exact wave functions of the maximum (quantal) entropy form in a number of cases of interest.
The radial Schrödinger equation was solved with the combination of three important potentials with [Formula: see text] as deformed parameter via the parametric Nikiforov–Uvarov method and the energy equation as well as the corresponding normalized radial wave function were obtained in close and compact form. The energy equation obtained was used to study eight molecules. The effect of the deformed parameter on energy eigenvalues was also studied numerically. The subset of the combined potential was also studied numerically and the results were found to be in agreement with the previous results. To extend the application of our work, the wave function obtained was used to calculate some theoretic quantities such as the Tsallis entropy, Rényi entropy and information energy. By putting the Tsallis index to 2, we deduced the information energy from Tsallis entropy. Finally, the effect of the deformed parameter and screening parameter, respectively, on the theoretic quantities were also studied.
We solved the Schrödinger equation with a certain approximation to the centrifugal term for an arbitrary angular momentum state with the Eckart Manning Rosen potential. The bound-state energy eigenvalues and the corresponding wave functions have been approximately obtained using the parametric Nikiforov Uvarov method. The solutions of the Schrödinger equation for the Eckart potential, Manning Rosen potential, and Hulthén potential have been obtained using a certain transformation. The concepts of the Shannon entropy and the Fisher information of a system under the Eckart Manning Rosen potential are investigated in detail. The behavior of the screening parameter and the quantum number n for Fisher information and the Shannon entropy are also investigated.
Recently, there has been some interest in the literature in calculating temperature dependent quantum mechanical wavefunctions using the principle of maximum entropy. The idea seems to be to obtain a differential equation, similar to the Schrodinger equation, but with temperature dependent terms. It should be noted that this does not follow von Neumann statistical quantum mechanics. It should also be noted that other authors in the literature maximize or minimize other quantities related to information theory to calculate a wavefunction. The objective of this note is not to examine the maximization of entropy as applied to the calculation of a wavefunction, but to suggest a different form for entropy. Traditionally, authors seem to use the Shannon form -k W*(x)W(x) ln[ W*(x)W*(x)] as a spatial entropy and exchange W(x) for fp for the momentum one, with W(x) being the wavefunction. It is the spatial form which they seem to maximize in calculations. In this note, we consider a probability P(x intersection x) = W(x) fp sin(px) and use it in Shannon’’s entropy. Maximization of entropy then leads to an integral equation which, it seems, needs to be solved numerically, We can however, multiply such an equation by fp and sum over p to obtain results which can be compared with those of authors who use -k W*(x)W(x) ln[ W*(x)W*(x)] . NOTE: In later preprints, we have used W(x) fp exp(ipx) as probability and calculated Shannon's entropy. This has given an overall result for entropy including a term for Shannon's entropy for W(x), another term with Shannon's entropy for fp and a final term W(x) x d/dx W(x). This last term, linked to exp(ipx), is also equal to the covariance of x and d/dx ln(W(x)) which is used in the derivation of the Cramer-Rao inequality for Fisher information.