Abstract
The entropic perspective on the molecular electronic structure is investigated. Information-theoretic description of electron probabilities is extended to cover the complex amplitudes (wave functions) of quantum mechanics. This analysis emphasizes the entropic concepts due to the phase part of electronic states, which generates the probability current density, thus allowing one to distinguish the information content of states generating the same electron density and differing in their current densities. The classical information measures of Fisher and Shannon, due to the probability/density distributions themselves, are supplemented by the nonclassical terms generated by the wave-function phase or the associated probability current. A complementary character of the Fisher and Shannon information measures is explored and the relationship between these classical information densities is derived. It is postulated to characterize also their nonclassical (phase/current-dependent) contributions. The continuity equations of the generalized information densities are examined and the associated nonclassical information sources are identified. The variational rules involving the quantum-generalized Shannon entropy, which generate the stationary and time-dependent Schrodinger equations from the relevant maximum entropy principles, are discussed and their implications for the system “thermodynamic” equilibrium states are examined. It is demonstrated that the lowest, stationary “thermodynamic” state differs from the true ground state of the system, by exhibiting the space-dependent phase, linked to the modulus part of the wave function, and hence also a nonvanishing probability current.
Highlights
In the quantum mechanical description of, say, a single particle the probability density, which defines the classical (Fisher [1] or Shannon [2]) information content of the particle spatial distribution, is determined solely by the moduluspart of the complex wave function
We recall that the current concept, a crucial component of the probability continuity equation in quantum mechanics, emerges in the context of the time dependence of the system wave function, determined by the Schrödinger equation or its complex conjugate: H(r)ψ(r) = i h ∂ψ(r)/∂t and H(r)ψ∗(r) = −i h ∂ψ∗(r)/∂t
The quantum generalizations of the classical Fisher and Shannon information measures, functionals of the particle probability distribution, have been introduced and discussed. They are applicable to the complex probability amplitudes of the quantum mechanical description
Summary
In the quantum mechanical description of, say, a single (spinless) particle the probability density, which defines the classical (Fisher [1] or Shannon [2]) information content of the particle spatial distribution, is determined solely by the moduluspart of the complex wave function. The intrinsic accuracy and Shannon entropy describe the complementary facets of the probability distribution These classical measures probe the real probability amplitude A(r), while the appropriate quantum extensions, e.g., that defined in Eq (3), introduce an additional dependence upon the probability current [3,6] related to the gradient of the phase factor of molecular states. We recall that the current concept, a crucial component of the probability continuity equation in quantum mechanics, emerges in the context of the time dependence of the system wave function, determined by the Schrödinger equation or its complex conjugate: H(r)ψ(r) = i h ∂ψ(r)/∂t and H(r)ψ∗(r) = −i h ∂ψ∗(r)/∂t.
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