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arXiv:cond-mat/0701745 · Modern Physics Letters B

QUANTUM MECHANICAL TREATMENT OF THE PROBLEM OF CONSTRAINTS IN NON-EXTENSIVE FORMALISM REVISITED

G. Baris Bagci, Altuğ Arda, R. Sever

Published Jul 10, 2007• 3 citations
arXivHTMLPDF

Abstract

The purity of Werner state in non-extensive formalism associated with two different constraints has been calculated in a previous paper by Bağci et al.17 Two different results have been obtained corresponding to ordinary probability and escort probability. The former has been shown to result in negative values thereby leading authors to deduce the advantage of escort probabilities over ordinary probabilities. However, these results have only been for a limited interval of q values which lie between 0 and 1. In this paper, we solve the same problem for all values of non-extensive index q by using a perturbative approach and show that the simultaneous use of both types of constraint is necessary in order to obtain the solution for a whole spectrum of non-extensive index. In this sense, the existence of these different constraints in non-extensive formalism must not be seen as a deficiency in the formalism but rather must be welcomed as a means of providing solution for all values of parameter q.

DOI
10.1142/s0217984907013390

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The purity of Werner state in non-extensive formalism associated with two different constraints has been calculated in a previous paper by Bağci et al.17 Two different results have been obtained corresponding to ordinary probability and escort probability. The former has been shown to result in negative values thereby leading authors to deduce the advantage of escort probabilities over ordinary probabilities. However, these results have only been for a limited interval of q values which lie between 0 and 1. In this paper, we solve the same problem for all values of non-extensive index q by using a perturbative approach and show that the simultaneous use of both types of constraint is necessary in order to obtain the solution for a whole spectrum of non-extensive index. In this sense, the existence of these different constraints in non-extensive formalism must not be seen as a deficiency in the formalism but rather must be welcomed as a means of providing solution for all values of parameter q.
Relative entropy (divergence) of Bregman type recently proposed by T. D. Frank and Jan Naudts is considered and its quantum counterpart is used to calculate purity of the Werner state in nonextensive formalism. It has been observed that two different expressions arise due to two different forms of quantum divergences. It is then argued that the difference is due to the fact that the relative entropy of Bregman type is related to the first choice thermostatistics whereas one of Csiszàr type is related to the third-choice thermostatistics. The superiority of the third-choice thermostatistics to the first-choice thermostatistics has been deduced by noticing that the expression obtained by using the Bregman type leads to negative values for q ∈ (0, 1) and fidelity F smaller than 1 whereas the one obtained by using Csiszàr type is free from such anomalies. Moreover, it has been noted that these two measures show different qualitative behavior with respect to F. Contrary to the classical case, the violation of the positive definiteness of the relative entropy immediately results in a choice between the two constraints without any need of more abstract Shore–Johnson axioms. The possibility of writing a relative entropy of Bregman type compatible with the third choice has been investigated further. The answer turns out to be negative as far as the usual transformation from ordinary probabilities to the escort probabilities are considered.
This paper presents a nonperturbative treatment of strong-coupling induced effects in atom-field systems which cannot be seen in traditional perturbative treatments invoking compromising assumptions such as the Born-Markov, rotating wave, or Fermi Golden rule. We consider an atom whose internal degrees of freedom are modeled by a harmonic oscillator which is bilinearly coupled to a scalar quantum field, representing one of the two polarizations of an electromagnetic field. Because the whole system is Gaussian we can solve this problem exactly. Using the open quantum system conceptual framework and the influence functional formalism we derive the dynamics of the reduced density matrix for the atom which enables the calculation of atomic transition probability and other relevant physical quantities. Finding an exact solution to this problem has the distinct advantage of enabling one to capture fully the strong coupling regime and discover interesting effects such as spontaneous ground state excitation [R. Passante, T. Petrosky, and I. Prigogine, Long-time behaviour of self-dressing and indirect spectroscopy, Physica (Amsterdam) 218A, 437 (1995).] which is unfathomable in perturbative treatments. The conventional description of atomic-optical activities is predicated on the assumption that the state of the total atom-field system is a product state of the atomic excitations and the photon number states, an assumption which is valid only for vanishingly weak coupling. The correct energy eigenfunctions to use should be that of the Hamiltonian of the combined atom-field system. Other features associated with finite to strong coupling effects such as resonance peak broadening and transition from a gapped to a gapless spectrum can all be understood from this perspective. Finally, to put the issues in a proper perspective we take the perturbative limit of the exact results and compare them with those from conventional time-dependent perturbation theory (TDPT). This enables one to pin-point where the deficiencies of TDPT lie as one removes the ultraweak coupling assumption.
In this work, we present a rigorous and complete non-relativistic treatment of the problem of a diatomic molecule which moves under the Hua potential in a D-dimensional Euclidean space. The radial Green’s function associated with this potential is constructed within the framework of the Feynman path integral. Different cases of this potential which are distinguished by the values of the deformation parameter q are considered separately. For [Formula: see text] q < 1 and r ∈ ] r0, +∞[, we adopt an appropriate Pekeris-type approximation for the centrifugal potential term to obtain the approximate spectrum of bound states lD ≠ 0. Some numerical results of the energy levels of certain diatomic molecules are calculated by using the correct minimum value of the deformation parameter q given by [Formula: see text]. For q < 0 or [Formula: see text], the problem is solvable only in a three-dimensional space. In both cases, the energy quantization condition is a transcendental equation that can be solved numerically. In the limit [Formula: see text], the energy spectrum and the corresponding wave functions of the radial Morse potential are regained.