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:quant-ph/0607135

N identical particles under quantum confinement: A many-body dimensional perturbation theory approach II, the lowest-order wave function II

Martin Dunn, D. K. Watson, John G. Loeser

Published Jul 19, 2006• 0 citations
arXivHTMLPDF

Abstract

In this paper, the second in a series of two, we complete the derivation of the lowest-order wave function of a dimensional perturbation theory (DPT) treatment for the N-body quantum-confined system. Taking advantage of the symmetry of the zeroth-order configuration, we use group theoretic techniques and the FG matrix method from quantum chemistry to obtain analytic results for frequencies and normal modes. This method directly accounts for each two-body interaction, rather than an average interaction so that even lowest-order results include beyond-mean-field effects. It is thus appropriate for the study of both weakly and strongly interacting systems and the transition between them. While previous work has focused on energies, lowest-order wave functions yield important information such as the nature of excitations and expectation values of physical observables at low orders including density profiles. Higher orders in DPT also require as input the zeroth-order wave functions. In the earlier paper we presented a program for calculating the analytic normal-mode coordinates of the large-D system and illustrated the procedure by deriving the two simplest normal modes. In this paper we complete this analysis by deriving the remaining, and more complex, normal coordinates of the system.

DOI
10.48550/arxiv.quant-ph/0607135

Related papers

N Identical Particles Under Quantum Confinement: A Many-Body Dimensional Perturbation Theory Approach II, The Lowest-Order Wave Function I

Martin Dunn, D. K. Watson, John G. Loeser

In this paper we continue our development of a dimensional perturbation theory (DPT) treatment of N identical particles under quantum confinement. DPT is a beyond-mean-field method which is applicable to both weakly and strongly-interacting systems and can be used to connect both limits. In a previous paper we developed the formalism for low-order energies and excitation frequencies. This formalism has been applied to atoms, Bose-Einstein condensates and quantum dots. One major advantage of the method is that N appears as a parameter in the analytical expressions for the energy and so results for N up to a few thousand are easy to obtain. Other properties however, are also of interest, for example the density profile in the case of a BEC,and larger N results are desirable as well. The latter case requires us to go to higher orders in DPT. These calculations require as input zeroth-order wave functions and this paper, along with a subsequent paper, addresses this issue.

arXiv:quant-ph/0603158•Mar 17, 2006•0 citations

Analytic, group-theoretic density profiles for confined, correlated N -body systems

W. Blake Laing, Martin Dunn, D. K. Watson

Confined quantum systems involving $N$ identical interacting particles are to be found in many areas of physics, including condensed-matter, atomic, and chemical physics. A beyond-mean-field perturbation method that is applicable, in principle, to weakly, intermediate, and strongly interacting systems has been set forth by the authors in a previous series of papers. Dimensional perturbation theory was used, and in conjunction with group theory, an analytic beyond-mean-field correlated wave function at lowest order for a system under spherical confinement with a general two-body interaction was derived. In the present paper, we use this analytic wave function to derive the corresponding lowest-order, analytic density profile and apply it to the example of a Bose-Einstein condensate.

arXiv:cond-mat/0607600•Physical Review A•Dec 5, 2006•13 citations

Analytic, Group-Theoretic Wave Functions for Confined N-Body Quantum Systems

W. Blake Laing, Martin Dunn, John G. Loeser +1 more

Systems involving N-identical interacting particles under quantum confinement appear in many areas of physics, including chemical, condensed matter, and atomic physics. We discuss a beyondmean- field perturbation method that is applicable to weakly, intermediate and strongly-interacting systems. Group theory is used to derive an analytic beyond-mean-field correlated wave function at zeroth order for a system under spherical confinement. We derive the corresponding zeroth-order analytic density profile and apply it to the example of a Bose-Einstein condensate.

arXiv:physics/0510177•Oct 19, 2005•0 citations

On the use of group theoretical and graphical techniques toward the solution of the general N-body problem

W. Blake Laing, Martin Dunn, D. K. Watson

Group theoretic and graphical techniques are used to derive the N-body wave function for a system of identical bosons with general interactions through first-order in a perturbation approach. This method is based on the maximal symmetry present at lowest order in a perturbation series in inverse spatial dimensions. The symmetric structure at lowest order has a point group isomorphic with the SN group, the symmetric group of N particles, and the resulting perturbation expansion of the Hamiltonian is order-by-order invariant under the permutations of the SN group. This invariance under SN imposes severe symmetry requirements on the tensor blocks needed at each order in the perturbation series. We show here that these blocks can be decomposed into a basis of binary tensors invariant under SN. This basis is small (25 terms at first order in the wave function), independent of N, and is derived using graphical techniques. This checks the N6 scaling of these terms at first order by effectively separating the N scaling problem away from the rest of the physics. The transformation of each binary tensor to the final normal coordinate basis requires the derivation of Clebsch–Gordon coefficients of SN for arbitrary N. This has been accomplished using the group theory of the symmetric group. This achievement results in an analytic solution for the wave function, exact through first order, that scales as N0, effectively circumventing intensive numerical work. This solution can be systematically improved with further analytic work by going to yet higher orders in the perturbation series.

arXiv:0808.2949•Journal of Mathematical Physics•Jun 1, 2009•13 citations

Test of a general symmetry-derived N -body wave function

Martin Dunn, W. Blake Laing, Derrick Toth +1 more

The resources required to solve the general interacting quantum $N$-body problem scale exponentially with $N$, making the solution of this problem very difficult when $N$ is large. In a previous series of papers we develop an approach for a fully interacting wave function for a confined system of identical bosons with a general two-body interaction. This method tames the $N$ scaling by developing a perturbation series that is order-by-order invariant under a point group isomorphic with ${S}_{N}$. Group theory and graphical techniques are then used to solve for the wave function exactly and analytically at each order, yielding a solution for the general $N$-body problem which scales as ${N}^{0}$ at any given order. Recently this formalism has been used to obtain the first-order fully interacting wave function for a system of harmonically confined bosons interacting harmonically. In this paper, we report the application of this $N$-body wave function to a system of $N$ fully interacting bosons in three dimensions. We derive an expression for the density profile for a confined system of harmonically interacting bosons. Choosing this simple interaction is not necessary or even advantageous for our method, however this choice allows a direct comparison of our exact results through first order with exact results obtained in an independent solution. Our density profile for the wave function through first order in three dimensions is indistinguishable from the first-order exact result obtained independently and shows strong convergence to the exact result to all orders.

Physical Review A•Dec 11, 2009•12 citations

Confined quantum systems: Beyond harmonic order in dimensional perturbation theory

W. Blake Laing

We generalize the harmonic-order N-body DPT method for an isotropically confined quantum system of identical interacting bosons to first-anharmonic order and, in principle, higher orders. We introduce a graphical decomposition of the perturbative expansion of the N-body Hamiltonian. We calculate the Clebsch-Gordon coefficient tensors that couple together 3 irreducible representations of SN analytically, and analytically transformed the graphical basis to collective coordinates. We calculate the N-body wavefunction and density profile in general and have demonstrated agreement with an analytic model. We apply this formalism to the exactly-solvable example of a trapped gas of atoms interacting with a (fully-interacting) harmonic-oscillator "Hooke's law" interaction and compare with the dimensional expansion of the exact ground-state wavefunction and density profile. We report progress on the example of a cold gas BEC (with zero angular momentum).

•Jan 1, 2008•0 citations
In this paper, the second in a series of two, we complete the derivation of the lowest-order wave function of a dimensional perturbation theory (DPT) treatment for the N-body quantum-confined system. Taking advantage of the symmetry of the zeroth-order configuration, we use group theoretic techniques and the FG matrix method from quantum chemistry to obtain analytic results for frequencies and normal modes. This method directly accounts for each two-body interaction, rather than an average interaction so that even lowest-order results include beyond-mean-field effects. It is thus appropriate for the study of both weakly and strongly interacting systems and the transition between them. While previous work has focused on energies, lowest-order wave functions yield important information such as the nature of excitations and expectation values of physical observables at low orders including density profiles. Higher orders in DPT also require as input the zeroth-order wave functions. In the earlier paper we presented a program for calculating the analytic normal-mode coordinates of the large-D system and illustrated the procedure by deriving the two simplest normal modes. In this paper we complete this analysis by deriving the remaining, and more complex, normal coordinates of the system.
In this paper we continue our development of a dimensional perturbation theory (DPT) treatment of N identical particles under quantum confinement. DPT is a beyond-mean-field method which is applicable to both weakly and strongly-interacting systems and can be used to connect both limits. In a previous paper we developed the formalism for low-order energies and excitation frequencies. This formalism has been applied to atoms, Bose-Einstein condensates and quantum dots. One major advantage of the method is that N appears as a parameter in the analytical expressions for the energy and so results for N up to a few thousand are easy to obtain. Other properties however, are also of interest, for example the density profile in the case of a BEC,and larger N results are desirable as well. The latter case requires us to go to higher orders in DPT. These calculations require as input zeroth-order wave functions and this paper, along with a subsequent paper, addresses this issue.
Confined quantum systems involving NNN identical interacting particles are to be found in many areas of physics, including condensed-matter, atomic, and chemical physics. A beyond-mean-field perturbation method that is applicable, in principle, to weakly, intermediate, and strongly interacting systems has been set forth by the authors in a previous series of papers. Dimensional perturbation theory was used, and in conjunction with group theory, an analytic beyond-mean-field correlated wave function at lowest order for a system under spherical confinement with a general two-body interaction was derived. In the present paper, we use this analytic wave function to derive the corresponding lowest-order, analytic density profile and apply it to the example of a Bose-Einstein condensate.
Systems involving N-identical interacting particles under quantum confinement appear in many areas of physics, including chemical, condensed matter, and atomic physics. We discuss a beyondmean- field perturbation method that is applicable to weakly, intermediate and strongly-interacting systems. Group theory is used to derive an analytic beyond-mean-field correlated wave function at zeroth order for a system under spherical confinement. We derive the corresponding zeroth-order analytic density profile and apply it to the example of a Bose-Einstein condensate.
Group theoretic and graphical techniques are used to derive the N-body wave function for a system of identical bosons with general interactions through first-order in a perturbation approach. This method is based on the maximal symmetry present at lowest order in a perturbation series in inverse spatial dimensions. The symmetric structure at lowest order has a point group isomorphic with the SN group, the symmetric group of N particles, and the resulting perturbation expansion of the Hamiltonian is order-by-order invariant under the permutations of the SN group. This invariance under SN imposes severe symmetry requirements on the tensor blocks needed at each order in the perturbation series. We show here that these blocks can be decomposed into a basis of binary tensors invariant under SN. This basis is small (25 terms at first order in the wave function), independent of N, and is derived using graphical techniques. This checks the N6 scaling of these terms at first order by effectively separating the N scaling problem away from the rest of the physics. The transformation of each binary tensor to the final normal coordinate basis requires the derivation of Clebsch–Gordon coefficients of SN for arbitrary N. This has been accomplished using the group theory of the symmetric group. This achievement results in an analytic solution for the wave function, exact through first order, that scales as N0, effectively circumventing intensive numerical work. This solution can be systematically improved with further analytic work by going to yet higher orders in the perturbation series.
The resources required to solve the general interacting quantum NNN-body problem scale exponentially with NNN, making the solution of this problem very difficult when NNN is large. In a previous series of papers we develop an approach for a fully interacting wave function for a confined system of identical bosons with a general two-body interaction. This method tames the NNN scaling by developing a perturbation series that is order-by-order invariant under a point group isomorphic with SN{S}_{N}SN​. Group theory and graphical techniques are then used to solve for the wave function exactly and analytically at each order, yielding a solution for the general NNN-body problem which scales as N0{N}^{0}N0 at any given order. Recently this formalism has been used to obtain the first-order fully interacting wave function for a system of harmonically confined bosons interacting harmonically. In this paper, we report the application of this NNN-body wave function to a system of NNN fully interacting bosons in three dimensions. We derive an expression for the density profile for a confined system of harmonically interacting bosons. Choosing this simple interaction is not necessary or even advantageous for our method, however this choice allows a direct comparison of our exact results through first order with exact results obtained in an independent solution. Our density profile for the wave function through first order in three dimensions is indistinguishable from the first-order exact result obtained independently and shows strong convergence to the exact result to all orders.
We generalize the harmonic-order N-body DPT method for an isotropically confined quantum system of identical interacting bosons to first-anharmonic order and, in principle, higher orders. We introduce a graphical decomposition of the perturbative expansion of the N-body Hamiltonian. We calculate the Clebsch-Gordon coefficient tensors that couple together 3 irreducible representations of SN analytically, and analytically transformed the graphical basis to collective coordinates. We calculate the N-body wavefunction and density profile in general and have demonstrated agreement with an analytic model. We apply this formalism to the exactly-solvable example of a trapped gas of atoms interacting with a (fully-interacting) harmonic-oscillator "Hooke's law" interaction and compare with the dimensional expansion of the exact ground-state wavefunction and density profile. We report progress on the example of a cold gas BEC (with zero angular momentum).