A separable model of <i>N</i> interacting Particles

Karwowski, Jacek

doi:10.1002/qua.21751

Cited by 10 publications

(6 citation statements)

References 21 publications

(32 reference statements)

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“…We note that the separability of the Schrödinger equation describing many interacting particles is rather exceptional. In particular, it is possible when interactions between disjoint pairs of particles, interacting by arbitrary two-particle potentials, are harmonic [7,8]. For the HA the Schrödinger equation not only separates exactly but also, for a set of the coupling constants, closed-form analytical solutions exist [9,10].…”

Section: Introductionmentioning

confidence: 99%

Two-Electron Spherical Quantum Dot in a Magnetic Field

Poszwa

2016

Few-Body Syst

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We investigate three-dimensional, two-electron quantum dots in an external magnetic field B. Due to mixed spherical and cylindrical symmetry the Schrödinger equation is not completely separable. Highly accurate numerical solutions, for a wide range of B, have been obtained by the expansion of wavefunctions in double-power series and by imposing on the radial functions appropriate boundary conditions. The asymptotic limit of a very strong magnetic field and the 2D approach have been considered. Ground state properties of the two-electron semiconductor quantum dots are investigated using both the 3D and 2D models. Theoretical calculations have been compared with recent experimental results. IntroductionThe influence of spatial confinement on properties of quantum systems have been widely studied in the literature and remains subject of continuous interest in both theoretical and experimental fields [1][2][3][4][5]. One of the most interesting confined quantum systems is the two-electron Hooke's-law atom (HA) also known as hookium or harmonium [6]. The HA refers to a model system composed of two electrons interacting by the Coulombic potential and confined in an external harmonic potential. Many applications for this system ensues from some unique properties of the HA. For the parabolic confinement the two-electron Hamiltonian separates. This property significantly simplifies the two-particle Schrödinger equation leading, in fact, to the one-dimensional radial problem. We note that the separability of the Schrödinger equation describing many interacting particles is rather exceptional. In particular, it is possible when interactions between disjoint pairs of particles, interacting by arbitrary two-particle potentials, are harmonic [7,8]. For the HA the Schrödinger equation not only separates exactly but also, for a set of the coupling constants, closed-form analytical solutions exist [9,10]. This is of particular importance for the understanding of the role of electron-electron interaction and correlation effects.When the electron mass is replaced by the relevant effective mass and the e-e interaction coupling constant is supplemented by the dielectric constant of semiconductor, the HA atom may be regarded to as a twoelectron quantum dot (QD). During last few decades many theoretical and experimental studies on QDs have been performed within parabolic confinement models as well as beyond this approximation. Both the 2D and 3D QD's have been investigated in a framework of strict quantum-mechanical and semiclassical approaches, including also effects of an external magnetic field [11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30]. Semiclassical solutions for a two-electron QD in a magnetic field have been investigated in terms of action-angle variables using the classical adiabatic approximation [19]. Spin-singlet and spin-triplet transitions have been studied and the magnetic moment and susceptibility have been obtained as functions of the magnetic field [20].A.

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Section: Introductionmentioning

confidence: 99%

Two-Electron Spherical Quantum Dot in a Magnetic Field

Poszwa

2016

Few-Body Syst

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“…Otherwise there are no constraints on masses and arbitrary oscillator intereractions are allowed. A seperable model like the one we discuss here has been studied recently within the context of exactly solvable models [47,48] and applied as a model of two-electron atoms and molecules [49]. In the current work we will be applying this kind of Hamiltonian to other model systems that are of relevance for neutral atoms in cold atomic gases.…”

Section: Introductionmentioning

confidence: 99%

Analytic solutions of topologically disjoint systems

Armstrong¹,

Volosniev²,

Fedorov³

et al. 2015

J. Phys. A: Math. Theor.

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We describe a procedure to solve an up to 2N problem where the particles are separated topologically in N groups with at most two particles in each. Arbitrary interactions are allowed between the (two) particles within one group. All other interactions are approximated by harmonic oscillator potentials. The problem is first reduced to an analytically solvable N -body problem and N independent two-body problems. We calculate analytically spectra, wave functions, and normal modes for both the inverse square and delta-function two-body interactions. In particular, we calculate separation energies between two strings of particles. We find that the string separation energy increases with N and interaction strength.

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“…With the results found in [28,29,31,30], showing that we could treat the individual layers separately, we inserted the exact repulsion in the in-layer energies, coupled with harmonic attraction between the layers. This treatment failed to agree with earlier SVM calculations in Ref.…”

Section: X2mentioning

confidence: 99%

Clusters in Separated Tubes of Tilted Dipoles

et al. 2020

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A few-body cluster is a building block of a many-body system in a gas phase provided the temperature at most is of the order of the binding energy of this cluster. Here we illustrate this statement by considering a system of tubes filled with dipolar distinguishable particles. We calculate the partition function, which determines the probability to find a few-body cluster at a given temperature. The input for our calculations -the energies of few-body clusters -is estimated using the harmonic approximation. We first describe and demonstrate the validity of our numerical procedure. Then we discuss the results featuring melting of the zero-temperature arXiv:2001.07190v1 [cond-mat.quant-gas]

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A separable model of N interacting Particles

Cited by 10 publications

References 21 publications

Two-Electron Spherical Quantum Dot in a Magnetic Field

Two-Electron Spherical Quantum Dot in a Magnetic Field

Analytic solutions of topologically disjoint systems

Clusters in Separated Tubes of Tilted Dipoles

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