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Journal of Physics Conference Series

Cooper pair of superconductivity in the coordinate representation andq-deformed harmonic oscillator

Man Van Ngu, Ngo Gia Vinh, Nguyen Tri Lan, Luu Thi Kim Thanh, Nguyễn Ái Việt

Published Jun 1, 2016• 1 citation
Publisher

Abstract

In this work we study the similarity between the wave functions of q - deformed harmonic oscillator and wave functions of Cooper pair. The wave functions of Cooper pairs in coordinate-space have an "onion-like" layered structure with exponent decay (Boltzmann) envelope modulation. The ground state wave function of q - deform harmonic oscillator has the form of oscillate functions with Gaussian decay envelope modulation. The corresponding between Boltzmann and Gaussian forms of envelope functions and their quantum similarity are discussed.

DOI
10.1088/1742-6596/726/1/012017

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In this work we study the similarity between the wave functions of q - deformed harmonic oscillator and wave functions of Cooper pair. The wave functions of Cooper pairs in coordinate-space have an "onion-like" layered structure with exponent decay (Boltzmann) envelope modulation. The ground state wave function of q - deform harmonic oscillator has the form of oscillate functions with Gaussian decay envelope modulation. The corresponding between Boltzmann and Gaussian forms of envelope functions and their quantum similarity are discussed.
Relations between the first, the second quantized representations and deform algebra are investigated. In the case of harmonic oscillator, the axiom of first quantization (the commutation relation between coordinate and momentum operators) and the axiom of second quantization (the commutation relation between creation and annihilation operators) are equivalent. We shown that in the case of q -deformed harmonic oscillator, a violence of the axiom of second quantization leads to a violence of the axiom of first quantization, and inverse. Using the coordinate representation, we study fine structures of the vacuum state wave function depend in the deformation parameter q . A comparison with fine structures of Cooper pair of superconductivity in the coordinate representation is also performed.
It is shown, using the quasi-spin operator, that the BCS superconducting ground-state wave function is the direct product of single Cooper-pair SU(2) coherent state wave functions, and in certain conditions, is SU(2) coherent state wave function of the Cooper-pair system. If two superconductors, both in BCS ground state, are coupled together, the coupled system is also in SU(2) coherent state. Under the contraction of group SU(2) to harmonic-oscillator group, the SU(2) coherent states contract to Glauber coherent states. The quantum fluctuations, distributions, and second correlations of Cooper pairs and Josephson superfluidity in the two cases are discussed.
The phase space representation for a q -deformed model of the quantum harmonic oscillator is constructed. We have found explicit expressions for both the Wigner and Husimi distribution functions for the stationary states of the q -oscillator model under consideration. The Wigner function is expressed as a basic hypergeometric series, related to the Al-Salam-Chihara polynomials. It is shown that, in the limit case h → 0 ( q → 1), both the Wigner and Husimi distribution functions reduce correctly to their well-known non-relativistic analogues. Surprisingly, examination of both distribution functions in the q -deformed model shows that, when q ≪ 1, their behaviour in the phase space is similar to the ground state of the ordinary quantum oscillator, but with a displacement towards negative values of the momentum. We have also computed the mean values of the position and momentum using the Wigner function. Unlike the ordinary case, the mean value of the momentum is not zero and it depends on q and n . The ground-state-like behaviour of the distribution functions for excited states in the q -deformed model opens quite new perspectives for further experimental measurements of quantum systems in the phase space.
A wide class of q-deformed harmonic oscillators including those of Macfarlane type and of Dubna type is shown to be describable in a unified way. The Hamiltonian of the oscillator is assumed to be given by a q-deformed anti-commutator of the q-deformed ladder operators. By solving qqq-difference equations, explicit coordinate representations of ladder operators and wave functions are derived, and unified parametric representations are found for qqq-Hermite functions and related formulas for the oscillators of Macfarlane and Dubna types. In addition to the well-known solutions with globally periodic structure, there exist an infinite number of solutions with globally aperiodic structure.
A q-deformed version of standard quantum mechanics in the coordinate Schrodinger picture is obtained by replacing the ordinary coordinate derivative by the so-called q-discrete derivative as the representative of the momentum operator. The chosen q-discrete derivative is symmetric with respect to the exchange of q and q -1 . Under the usually adopted assumptions a q-deformed Schrodinger equation is derived for a harmonic oscillator. The complete set of eigenfunctions can be explicitly constructed as special q-functions and the corresponding energy eigenvalues are identical to those obtained by Biedenharn in his pioneering work (1989). This q-deformed oscillator exhibits a rich novel structure including dynamical symmetry and in the limit q to 1 it reveals some hitherto unknown features of the harmonic oscillator eigenfunctions.