Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1306.2227

Symbolic-numerical algorithm for generating cluster eigenfunctions: identical particles with pair oscillator interactions

Alexander A. Gusev, Sergue Il'ich Vinitsky, Ochbadrakh Chuluunbaatar, V. A. Rostovt︠s︡ev, Lương Lê Hải, Vladimir Leonardovich Derbov, ANDRZEJ GÓŹDŹ, Evgenii Klimov

Published Jun 10, 2013• 3 citations
arXivHTMLPDF

Abstract

The quantum model of a cluster, consisting of A identical particles, coupled by the internal pair interactions and affected by the external field of a target, is considered. A symbolic-numerical algorithm for generating A-1-dimensional oscillator eigenfunctions, symmetric or antisymmetric with respect to permutations of A identical particles in the new symmetrized coordinates, is formulated and implemented using the MAPLE computer algebra system. Examples of generating the symmetrized coordinate representation for A-1 dimensional oscillator functions in one-dimensional Euclidean space are analyzed. The approach is aimed at solving the problem of tunnelling the clusters, consisting of several identical particles, through repulsive potential barriers of a target.

DOI
10.48550/arxiv.1306.2227

Related papers

Resonant tunneling of a few-body cluster through repulsive barriers

Alexander A. Gusev, Sergue Il'ich Vinitsky, Ochbadrakh Chuluunbaatar +4 more

A model for quantum tunnelling of a cluster comprised of A identical particles, interacting via oscillator-type potential, through short-range repulsive barrier potentials is introduced for the first time in symmetrized-coordinate representation and numerically studied in the s -wave approximation. A constructive method for symmetrizing or antisymmetrizing the ( A − 1)-dimensional harmonic oscillator basis functions in the new symmetrized coordinates with respect to permutations of coordinates of A identical particles is described. The effect of quantum transparency, manifesting itself in nonmonotonic resonance-type dependence of the transmission coefficient upon the energy of the particles, their number A = 2, 3, 4 and the type of their symmetry, is analyzed. It is shown that the total transmission coefficient demonstrates the resonance behavior due to the existence of barrier quasi-stationary states, embedded in the continuum.

Physics of Atomic Nuclei•Mar 1, 2014•16 citations

Resonance tunnelling of clusters through repulsive barriers

Alexander A. Gusev, Sergue Il'ich Vinitsky, Ochbadrakh Chuluunbaatar +2 more

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.

Physica Scripta•Apr 29, 2014•3 citations

Quantum Transparency of Barriers and Reflection from Wells for Clusters of Identical Particles

Ochbadrakh Chuluunbaatar, Sergue Il'ich Vinitsky, Alexander A. Gusev +2 more

A method for solving the problem of quantum transmission through potential barriers or potential wells for a compound system consisting of several identical particles coupled via pair oscillator-type potentials in the oscillator representation of the symmetrized coordinates is considered. The efficiency of the proposed approach, algorithms and programs is demonstrated by the examples of calculation of complex energy values and analysis of metastable states of compound systems of two, three, and four identical particles on a straight line, which lead to the effect of quantum transparency of the potential barriers or quantum reflection from the wells.

Bulletin of the Russian Academy of Sciences Physics•Jun 1, 2018•2 citations

Correlations of self-bound identical fermions

Gintautas Kamuntavičius

A method for the calculation of translationally invariant wave functions for systems of identical fermions with arbitrary potential of pair interaction is developed.It is based on the well-known result that the essential dynamic part of Hamiltonian for the system of identical particles is the reduced Hamiltonian operator describing a relative movement of two particles inside the system.The eigenfunctions of this operator take into account all correlations caused by interaction.These eigenfunctions are basic elements for building the components (i.e., the functions with a lower degree of antisymmetry) used to construct the total antisymmetric wave function of the system.The main problem of this approach appears to be the antisymmetrization of the components.The developed universal algorithm for antisymmetrization gives a possibility to carry out this operation in a simple way and to keep numerical approximations under control.

arXiv:nucl-th/0612039•Lithuanian Journal of Physics•Jan 1, 2006•1 citation

Algebraic description of SN-invariant oscillator states

Roberto Floreanini B, Luc Lapointe, Luc Vinet

A group-theoretical description of N identical harmonic oscillators on a line is presented. It provides a scheme for labeling the energy eigenstates that are invariant under the permutation group SN and for obtaining the symmetric operators that transform these degenerate eigenfunctions among themselves. The symmetry algebra that these generators form is in general polynomial. The 2- and 3-particle cases are considered in detail. For the simple 2-body problem the invariance algebra is found to be the cubic SU(2) algebra: [J0,J±]=±J±, [J+,J−]=2J0−αJ03. In the 3-body case, the permutational invariant states are characterized with the help of the subgroup chain U(3)⊃U(2)⊃O(2). The labeling and step operators are obtained from determining an integrity basis for the S3 scalar in U(U(3)). Generating functions techniques are used to that end; an eight-dimensional basis is found whose elements span the symmetry algebra of the three identical oscillator problem. These constants of motion are seen to generate a nonlinear algebra whose representation on the symmetric states is provided.

Journal of Mathematical Physics•Nov 1, 1998•5 citations

Metastable states of a composite system tunneling through repulsive barriers

Alexander A. Gusev, Sergue Il'ich Vinitsky, Ochbadrakh Chuluunbaatar +3 more

We consider a method for solving the problem of quantum tunneling through repulsive potential barriers for a composite system consisting of several identical particles coupled via pair oscillator-type potentials in the oscillator symmetrized-coordinate representation. We confirm the efficiency of the proposed approach by calculating complex energy values and analyzing metastable states of composite systems of three, four, and five identical particles on a line, which leads to the effect of quantum transparency of the repulsive barriers.

Theoretical and Mathematical Physics•Jan 1, 2016•9 citations
The quantum model of a cluster, consisting of A identical particles, coupled by the internal pair interactions and affected by the external field of a target, is considered. A symbolic-numerical algorithm for generating A-1-dimensional oscillator eigenfunctions, symmetric or antisymmetric with respect to permutations of A identical particles in the new symmetrized coordinates, is formulated and implemented using the MAPLE computer algebra system. Examples of generating the symmetrized coordinate representation for A-1 dimensional oscillator functions in one-dimensional Euclidean space are analyzed. The approach is aimed at solving the problem of tunnelling the clusters, consisting of several identical particles, through repulsive potential barriers of a target.
A model for quantum tunnelling of a cluster comprised of A identical particles, interacting via oscillator-type potential, through short-range repulsive barrier potentials is introduced for the first time in symmetrized-coordinate representation and numerically studied in the s -wave approximation. A constructive method for symmetrizing or antisymmetrizing the ( A − 1)-dimensional harmonic oscillator basis functions in the new symmetrized coordinates with respect to permutations of coordinates of A identical particles is described. The effect of quantum transparency, manifesting itself in nonmonotonic resonance-type dependence of the transmission coefficient upon the energy of the particles, their number A = 2, 3, 4 and the type of their symmetry, is analyzed. It is shown that the total transmission coefficient demonstrates the resonance behavior due to the existence of barrier quasi-stationary states, embedded in the continuum.
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.
A method for solving the problem of quantum transmission through potential barriers or potential wells for a compound system consisting of several identical particles coupled via pair oscillator-type potentials in the oscillator representation of the symmetrized coordinates is considered. The efficiency of the proposed approach, algorithms and programs is demonstrated by the examples of calculation of complex energy values and analysis of metastable states of compound systems of two, three, and four identical particles on a straight line, which lead to the effect of quantum transparency of the potential barriers or quantum reflection from the wells.
A method for the calculation of translationally invariant wave functions for systems of identical fermions with arbitrary potential of pair interaction is developed.It is based on the well-known result that the essential dynamic part of Hamiltonian for the system of identical particles is the reduced Hamiltonian operator describing a relative movement of two particles inside the system.The eigenfunctions of this operator take into account all correlations caused by interaction.These eigenfunctions are basic elements for building the components (i.e., the functions with a lower degree of antisymmetry) used to construct the total antisymmetric wave function of the system.The main problem of this approach appears to be the antisymmetrization of the components.The developed universal algorithm for antisymmetrization gives a possibility to carry out this operation in a simple way and to keep numerical approximations under control.
A group-theoretical description of N identical harmonic oscillators on a line is presented. It provides a scheme for labeling the energy eigenstates that are invariant under the permutation group SN and for obtaining the symmetric operators that transform these degenerate eigenfunctions among themselves. The symmetry algebra that these generators form is in general polynomial. The 2- and 3-particle cases are considered in detail. For the simple 2-body problem the invariance algebra is found to be the cubic SU(2) algebra: [J0,J±]=±J±, [J+,J−]=2J0−αJ03. In the 3-body case, the permutational invariant states are characterized with the help of the subgroup chain U(3)⊃U(2)⊃O(2). The labeling and step operators are obtained from determining an integrity basis for the S3 scalar in U(U(3)). Generating functions techniques are used to that end; an eight-dimensional basis is found whose elements span the symmetry algebra of the three identical oscillator problem. These constants of motion are seen to generate a nonlinear algebra whose representation on the symmetric states is provided.
We consider a method for solving the problem of quantum tunneling through repulsive potential barriers for a composite system consisting of several identical particles coupled via pair oscillator-type potentials in the oscillator symmetrized-coordinate representation. We confirm the efficiency of the proposed approach by calculating complex energy values and analyzing metastable states of composite systems of three, four, and five identical particles on a line, which leads to the effect of quantum transparency of the repulsive barriers.