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Universal trend of the information entropy of a fermion in a mean field.pdf

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Universal trend of the information entropy of a fermion in a mean field

a r X i v : n u c l - t h / 0 0 0 7 0 6 4 v 1 2 7 J u l 2 0 0 0 Universal trend of the information entropy of a fermion in a mean field C.P. Panos, S.E. Massen and C.G. Koutroulos Department of Theoretical Physics, Aristotle University of Thessaloniki, 54006 Thessaloniki, Greece We calculate the information entropy of single-particle states in position-space Sr and momentum- space Sk for a nucleon in a nucleus, a Λ particle in a hypernucleus and an electron in an atomic cluster. It is seen that Sr and Sk obey the same approximate functional form as functions of the number of particles, Sr (or Sk) = a + bN 1/3 in all of the above many-body systems in position- and momentum- space separately. The net information content Sr +Sk is a slowly varying function of N of the same form as above. The entropy sum Sr + Sk is invariant to uniform scaling of coordinates and a characteristic of the single-particle states of a specific system. The order of single-particle states according to Sr + Sk is the same as their classification according to energy keeping the quantum number n constant. The spin-orbit splitting is reproduced correctly. It is also seen that Sr + Sk enhances with excitation of a fermion in a quantum-mechanical system. Finally, we establish a relationship of Sr + Sk with the energy of the corresponding single-particle state i.e. Sr + Sk = k ln(μE + ν). This relation holds for all the systems under consideration. PACS: 89.70.+c; 36.40.+d; 31.10.+z; 21.60.-n I. INTRODUCTION The information entropy for a continuous probability distribution p(x) in one dimension is defined by the expression: S = ? ∫ p(x) ln(p(x))dx (1) where ∫ p(x)dx = 1. S is measured in bits if the base of the logarithm is 2 and nats (natural units of information) if the logarithm is natural. It represents the information content of a probability distribution as well as a measure of uncertainty of the corresponding state. We note that the information and thermodynamic entropy are different conc