The two-dimensional hydrogen atom with a logarithmic potential energy function

Eveker, K.; Grow, D.; Jost, Bradley M.; Monfort, C. E.; Nelson, Ken; Stroh, C.; Witt, R. C.

doi:10.1119/1.16249

Cited by 33 publications

(21 citation statements)

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“…The logarithmic potential in two space dimensions has been widely studied in the past [3][4][5]. In the repulsive case the energy spectrum is continuous, while bound states only show up in the attractive case.…”

Section: -2--22 (Ouau) (Oa ") + Li(r/uouv --½I(ouq)y~v-e(z7a~¢-rn~¢ mentioning

confidence: 99%

The fermion-fermion effective potential in the Maxwell-Chern-Simons theory

Girotti¹,

Gomes²,

Silva³

1992

Physics Letters B

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The effective nonrelativistic potential VT describing the fermion-fermion interaction in the Maxwell-Chern-Simons theory is derived to the lowest order in perturbation theory. As expected, VT is not invariant under parity and time-reversal transformations. The quantum dynamics generated by Vr becomes exactly solvable at the limits where either the Maxwell or the ChernSimons terms disappear; in neither case electron-electron bound states show up. However, numerical calculations indicate that fermion-fermion bound states do exist in the general case.Motivated by recent discussions [1] about the consistency of the nonrelativistic limit of certain relativistically invariant quantum field theories, we consider in this letter the problem of determining the effective electron-electron low energy potential arising from the Maxwell-Chern-Simons (MCS) theory. We shall also investigate whether this potential defines a physically sensible and nontrivial quantum dynamics. In particular, the existence of electronelectron bound states is one of our main concerns in this work.As known [ 2 ], the MCS theory is a (2 + l )-dimensional model describing the coupling of charged fertalons (~, ~) of mass m and electric charge e to the gauge field potential A~ via the lagrangian density -2--22 (OuAU) (O.A ") + li(r/uOuV --½i(Ouq)y~v-e(z7,A~¢-rn~¢ ,where F,, = O~A,-GA,, 0 >/0 is the topological mass, 2 is a gauge parameter and a is a dimensionless real Supported in part by Conselho Nacional de Desenvolvimento Cientifico e Tecnol6gico (CNPq), Brazil.parameter (0 ~< a ~< 1 ) enabling us to modulate the intensity of the Maxwell term. Throughout this paper we use natural units (c--h=l). Our metric is goo=-gl~ = -g22 =l, while for the y matrices we adopt the representation 7° = o3, 71 = ia~, 72 = itr2; ai, i= 1, 2, 3, are the Pauli spin matrices. Neither parity nor time-reversal are, separately, symmetries of the model.The contribution of order e 2 to the elastic scattering amplitude e-+ e-~ e-+ e-(M611er scattering ) is given bywhere _e 2Here, pl, P2 (P~, P~ ) are the on-shell two-momenta of the initial (final) electrons, k-p't -Pl =P2 --P'2 is the transferred momentum and Du~(k) designates the free photon propagator. One can easily check that 0370-2693/92/$ 05.00

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Section: -2--22 (Ouau) (Oa ") + Li(r/uouv --½I(ouq)y~v-e(z7a~¢-rn~¢ mentioning

confidence: 99%

The fermion-fermion effective potential in the Maxwell-Chern-Simons theory

Girotti¹,

Gomes²,

Silva³

1992

Physics Letters B

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“…For the trions, we also employ the diffusion quantum Monte Carlo approach [25,26]. We take into account a specific feature of atomically thin crystals of TMDCs, where, due to the polarizability of atomic orbitals, the interaction between charges qt j is logarithmic, (<M.//r *) ln (r; ;/r *), up to a distance r* much larger than the excitonic Bohr radius [27], as indicated by the com parison of measured [42] and calculated [42][43][44] spectra of ground and excited states of free excitons.…”

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confidence: 99%

Three-Particle Complexes in Two-Dimensional Semiconductors

2015

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We evaluate binding energies of trions X±, excitons bound by a donor or acceptor charge X^{D(A)}, and overcharged acceptors or donors in two-dimensional atomic crystals by mapping the three-body problem in two dimensions onto one particle in a three-dimensional potential treatable by a purposely developed boundary-matching-matrix method. We find that in monolayers of transition metal dichalcogenides the dissociation energy of X^{±} is typically much larger than that of localized exciton complexes, so that trions are more resilient to heating, despite the fact that their recombination line in optics is less redshifted from the exciton line than the line of X^{D(A)}.

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“…The eigenvalue problem of the 2D hydrogen atom (21.48) with an attractive logarithmic potential energy −W (ρ) ∼ ln(ρ/4πα 2D ) can be solved numerically [63,64]. The 2D logarithmic potential has only discrete energy levels since no particle is outside the well, even at infinity ρ → ∞.…”

Section: Excitons In Low-dimensional Systemsmentioning

confidence: 99%

Excitons

Bechstedt¹

2014

Springer Series in Solid-State Sciences

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The general procedure to account for many-body effects in optical and energy-loss spectra consists of three steps, (i) preparation of a starting electronic structure, (ii) its improvement due to quasiparticle effects, and (iii) the inclusion of electron-hole attraction and exchange. Their combination gives rise to novel quasiparticles, the excitons. Singlet and triplet states differ by twice the electron-hole exchange. Instructive exciton models are derived studying only pairs of conduction and valence bands. For large distances of electron and hole in real space the effective mass approximation and a constant screening can be employed. The resulting pairs are hydrogen-like Wannier-Mott excitons, which give rise to Rydberg series below the ionization edge and a Coulomb enhancement, the Sommerfeld factor, of the pair density of states above the edge. Localized excitons such as Frenkel and charge-transfer excitons possess larger binding energies and electron-hole distances of the order of atomic or molecular distances. Exciton binding is increased by spatial confinement as demonstrated for two-dimensional systems. Electron-Hole Pairs: Top-Down Approach TerminologyIn the previous Chaps. 18-20 we have described in detail the response of an electronic system to an external perturbation, which however leaves this system neutral. Especially perturbations which are related to the real or virtual absorption of photons with energy ω or the inelastic scattering of high-energy particles with energy losses of energy ω have been considered as examples. Thereby we have learnt that the many-electron interactions mediated by the Coulomb potential play a central role. We found that the treatment of their influence can be nearly divided into a three-step procedure that is schematically illustrated in Fig. 21.1 for a non-metal. In a first step an electron-hole pair may be actually or virtually excited by a photon of energy ω in the reference electronic structure derived from a Kohn-Sham or generalized Kohn-Sham problem. In the second step the formation of quasiparticles is taken into account. As a consequence an energy shift of the bands is considered resulting in an

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The two-dimensional hydrogen atom with a logarithmic potential energy function

Cited by 33 publications

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The fermion-fermion effective potential in the Maxwell-Chern-Simons theory

The fermion-fermion effective potential in the Maxwell-Chern-Simons theory

Three-Particle Complexes in Two-Dimensional Semiconductors

Excitons

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