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Few-Body Systems

Non-symmetrized Basis Function for Identical Particles

M. Gattobigio, Alejandro Kievsky, M. Viviani, Paolo Barletta

Published Feb 23, 2009• 14 citations
Publisher
DOI
10.1007/s00601-009-0045-4

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Physica Scripta•Apr 29, 2014•3 citations
We consider entanglement in a system with a fixed number of identical particles. Since any operation should be symmetrized over all the identical particles and there is the precondition that the spatial wave functions overlap, the meaning of identical-particle entanglement is fundamentally different from that of distinguishable particles. The identical-particle counterpart of the Schmidt basis is shown to be the single-particle basis in which the one-particle reduced density matrix is diagonal. But it does not play a special role in the issue of entanglement, which depends on the single-particle basis chosen. The nonfactorization due to (anti)symmetrization is naturally excluded by using the (anti)symmetrized basis or, equivalently, the particle number representation. The natural degrees of freedom in quantifying the identical-particle entanglement in a chosen single-particle basis are occupation numbers of different single-particle basis states. The entanglement between effectively distinguishable spins is shown to be a special case of the occupation-number entanglement.
Investigating few-body systems with identical particles in a hyperspherical basis yields the problem of obtaining symmetrized hyperspherical functions from functions with arbitrary quantum numbers. This article solves the problem of hyperspherical basis symmetrization for four-, five-, and six- body systems using Parentage Scheme of Symmetrization. Parentage coefficients and transformation coefficients corresponding to the [4], [31], [22], [211], representations of S4 groups with specific quantum numbers are obtained, and relationship between parentage coefficients and transformation coefficients between different sets of Jacobi coordinates for six-body systems are derived. Young operators, acting on N = 4, 5, 6 body hyperspherical functions symmetrized with respect to (N-1) particles, are derived. The symmetrized N = 4, 5, 6 body hyperspherical functions are obtained with different values of quantum numbers. The connection between the transformation coefficients for the identical particle systems and the parentage coefficients is demonstrated and the corresponding formulas are introduced.
Quantum tunnelling of a cluster comprised of several identical particles, coupled via an oscillator-type potential, through short-range repulsive barrier potentials is studied in the s-wave approximation of the symmetrized coordinate representation. A procedure is briefly described that allows the construction of states, symmetric or asymmetric with respect to permutations of A identical particles, from the harmonic oscillator basis functions expressed via the newly introduced symmetrized coordinates. In the coupled-channel approximation of the -matrix approach, the effect of quantum transparency is analysed; it manifests itself in non-monotonic resonance dependence of the transmission coefficient upon the energy of the particles, their number A = 3, 4 and the symmetry types of their states. The total transmission coefficient is shown to demonstrate resonance behaviour with probability density growth in the vicinity of the potential energy local minima, which is a manifestation of the barrier quasi-stationary states, embedded in the continuum.