First and second-order relativistic corrections to the two and higher-dimensional isotropic harmonic oscillator obeying the spinless Salpeter equation

Edery, Ariel; Laporte, Philippe

doi:10.1088/2399-6528/aaadcd

Cited by 8 publications

(6 citation statements)

References 32 publications

(52 reference statements)

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“…We find first-and second-order corrections to the energy spectrum of the Dirac-Pauli equation according to the definitions [23] ε (1) n =< n|H 1 |n > , (28a)…”

Section: B First and Second-order Corrections To Energy Levelsmentioning

confidence: 99%

Solutions of Pauli-Dirac Equation in terms of Laguerre Polynomials within Perturbative Scheme

Arda

2021

Preprint

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“…We find first-and second-order corrections to the energy spectrum of the Dirac-Pauli equation according to the definitions [23] ε (1) n =< n|H 1 |n > , (28a)…”

Section: B First and Second-order Corrections To Energy Levelsmentioning

confidence: 99%

Solutions of Pauli-Dirac Equation in terms of Laguerre Polynomials within Perturbative Scheme

Arda

2021

Preprint

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“…Considering this approximation, the Schrödinger equation can be solved directly in polar coordinates as [109,110]…”

Section: The Harmonic Potential Approximationmentioning

confidence: 99%

A colloquium on the variational method applied to excitons in 2D materials

Quintela

Peres

2020

Eur. Phys. J. B

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In this colloquium, we review the research on excitons in van der Waals heterostructures from the point of view of variational calculations. We first make a presentation of the current and past literature, followed by a discussion on the connections between experimental and theoretical results. In particular, we focus our review of the literature on the absorption spectrum and polarizability, as well as the Stark shift and the dissociation rate. Afterwards, we begin the discussion of the use of variational methods in the study of excitons. We initially model the electron-hole interaction as a soft-Coulomb potential, which can be used to describe interlayer excitons. Using an ansatz, based on the solution for the two-dimensional quantum harmonic oscillator, we study the Rytova-Keldysh potential, which is appropriate to describe intralayer excitons in two-dimensional (2D) materials. These variational energies are then recalculated with a different ansatz, based on the exact wavefunction of the 2D hydrogen atom, and the obtained energy curves are compared. Afterwards, we discuss the Wannier-Mott exciton model, reviewing it briefly before focusing on an application of this model to obtain both the exciton absorption spectrum and the binding energies for certain values of the physical parameters of the materials. Finally, we briefly discuss an approximation of the electron-hole interaction in interlayer excitons as an harmonic potential and the comparison of the obtained results with the existing values from both first-principles calculations and experimental measurements.

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“…In this regard, the relativistic motion immediately induces anharmonicities even in the case of a quadratic potential with a frequency-amplitude dependence [32,33]. Here we follow the approach of many authors [34][35][36][37][38][39][40][41][42][43], adopting the spinless Salpeter equation [44] with a quadratic external potential as our definition of relativistic harmonic oscillator. The choice is due to its simplicity, where just the Newtonian kinetic energy is replaced by its special relativistic counterpart.…”

Section: Introductionmentioning

confidence: 99%

Relativistic Ermakov–Milne–Pinney Systems and First Integrals

Haas

2021

Physics

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The Ermakov–Milne–Pinney equation is ubiquitous in many areas of physics that have an explicit time-dependence, including quantum systems with time-dependent Hamiltonian, cosmology, time-dependent harmonic oscillators, accelerator dynamics, etc. The Eliezer and Gray physical interpretation of the Ermakov–Lewis invariant is applied as a guiding principle for the derivation of the special relativistic analog of the Ermakov–Milne–Pinney equation and associated first integral. The special relativistic extension of the Ray–Reid system and invariant is obtained. General properties of the relativistic Ermakov–Milne–Pinney are analyzed. The conservative case of the relativistic Ermakov–Milne–Pinney equation is described in terms of a pseudo-potential, reducing the problem to an effective Newtonian form. The non-relativistic limit is considered to be well. A relativistic nonlinear superposition law for relativistic Ermakov systems is identified. The generalized Ermakov–Milne–Pinney equation has additional nonlinearities, due to the relativistic effects.

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First and second-order relativistic corrections to the two and higher-dimensional isotropic harmonic oscillator obeying the spinless Salpeter equation

Cited by 8 publications

References 32 publications

Solutions of Pauli-Dirac Equation in terms of Laguerre Polynomials within Perturbative Scheme

Solutions of Pauli-Dirac Equation in terms of Laguerre Polynomials within Perturbative Scheme

A colloquium on the variational method applied to excitons in 2D materials

Relativistic Ermakov–Milne–Pinney Systems and First Integrals

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