Felipe Silveira | UERJ - Universidade do Estado do Rio de Janeiro / Rio de Janeiro State University (original) (raw)

Thesis Chapters by Felipe Silveira

Research paper thumbnail of Soluções numéricas e analíticas de um modelo de quantum dot com dois elétrons

The main scope of this work is to search numerical an analytical solutions for the Schrödinger e... more The main scope of this work is to search numerical an analytical solutions for the Schrödinger equation that describes a two electron quantum dot. Once confi rmed that both methods are in agreement with the results for the same parameters new forms of the potential energy were considered in order to try to clarify the real form of the Coulombian
potential for the quantum dot in two spatial dimensions. The Coloumbian term of the potential energy in a strictly two-dimensional space is given by ln r (in atomic units) but in the literature the term is usually considered strictly in its three-dimensional form 1=r. This monograph provides numerical results for both potentials which can be further compared with experimental data to prove the actual shape of the potential energy of a quantum dot. The numerical method also allows the determination of new bound states in negative energy regions for the case when the angular momentum quantum number is l = 0.

Papers by Felipe Silveira

Research paper thumbnail of How the inter-electronic potential Ansätze affect the bound state solutions of a planar two-electron quantum dot model

Physica E: Low-dimensional Systems and Nanostructures, 2019

The model of a two-electron quantum dot, confined to move in a two dimensional flat space, in the... more The model of a two-electron quantum dot, confined to move in a two dimensional flat space, in the presence of an external harmonic oscillator potential, is revisited for a specific purpose. Indeed, eigenvalues and eigenstates of the bound state solutions are obtained for any oscillation frequency considering both the 1∕r and ln r Ansätze
for inter-electronic Coulombic-like potentials in 2D. Then, it is pointed out that the significative difference between measurable quantities predicted from these two potentials can shed some light on the problem of space dimensionality as well as on the physical nature of the potential itself.

Research paper thumbnail of Numerical Solutions for a Two-dimensional Quantum Dot Model

Brazilian Journal of Physics, 2019

In this paper, a quantum dot mathematical model based on a two-dimensional Schrödinger equation a... more In this paper, a quantum dot mathematical model based on a two-dimensional Schrödinger equation assuming the 1/r inter-electronic potential is revisited. Generally, it is argued that the solutions of this model obtained by solving a biconfluent Heun equation have some limitations. The known polynomial solutions are confronted with new numerical calculations based on the Numerov method. A good qualitative agreement between them emerges. The numerical method being more general gives rise to new solutions. In particular, we are now able to calculate the quantum dot eigenfunctions for a much larger spectrum of external harmonic frequencies as compared to previous results. Also the existence of bound state for such planar system, in the case l=0, is predicted and its respective eigenvalue is determined.

Research paper thumbnail of Two-electron quantum dot model revisited: bound states and other analytical and numerical solutions

The model of a two-electron quantum dot, confined to move in a two dimensional flat space, is rev... more The model of a two-electron quantum dot, confined to move in a two dimensional flat space, is revisited. Generally, it is argued that the solutions of this model obtained by solving a biconfluent Heun equation have some limitations. In particular, some corrections are also made in previous theoretical calculations. The corrected polynomial solutions are confronted with numerical calculations based on the Numerov method, in a good agreement between both. Then, new solutions considering the 1/r and ln r Coulombian-like potentials in (1+2)D, not yet obtained, are discussed numerically. In particular, we are able to calculate the quantum dot eigenfunctions for a much larger spectrum of external harmonic frequencies as compared to previous results. Also the existence of bound states for such planar system in the case l = 0 is predicted and the respective eigenvalues are determined.

Ph.D. papers by Felipe Silveira

Research paper thumbnail of Corrigendum to ‘‘Solving a two-electron quantum dot model in terms of polynomial solutions of a Biconfluent Heun Equation’’

Research paper thumbnail of Soluções numéricas e analíticas de um modelo de quantum dot com dois elétrons

The main scope of this work is to search numerical an analytical solutions for the Schrödinger e... more The main scope of this work is to search numerical an analytical solutions for the Schrödinger equation that describes a two electron quantum dot. Once confi rmed that both methods are in agreement with the results for the same parameters new forms of the potential energy were considered in order to try to clarify the real form of the Coulombian
potential for the quantum dot in two spatial dimensions. The Coloumbian term of the potential energy in a strictly two-dimensional space is given by ln r (in atomic units) but in the literature the term is usually considered strictly in its three-dimensional form 1=r. This monograph provides numerical results for both potentials which can be further compared with experimental data to prove the actual shape of the potential energy of a quantum dot. The numerical method also allows the determination of new bound states in negative energy regions for the case when the angular momentum quantum number is l = 0.

Research paper thumbnail of How the inter-electronic potential Ansätze affect the bound state solutions of a planar two-electron quantum dot model

Physica E: Low-dimensional Systems and Nanostructures, 2019

The model of a two-electron quantum dot, confined to move in a two dimensional flat space, in the... more The model of a two-electron quantum dot, confined to move in a two dimensional flat space, in the presence of an external harmonic oscillator potential, is revisited for a specific purpose. Indeed, eigenvalues and eigenstates of the bound state solutions are obtained for any oscillation frequency considering both the 1∕r and ln r Ansätze
for inter-electronic Coulombic-like potentials in 2D. Then, it is pointed out that the significative difference between measurable quantities predicted from these two potentials can shed some light on the problem of space dimensionality as well as on the physical nature of the potential itself.

Research paper thumbnail of Numerical Solutions for a Two-dimensional Quantum Dot Model

Brazilian Journal of Physics, 2019

In this paper, a quantum dot mathematical model based on a two-dimensional Schrödinger equation a... more In this paper, a quantum dot mathematical model based on a two-dimensional Schrödinger equation assuming the 1/r inter-electronic potential is revisited. Generally, it is argued that the solutions of this model obtained by solving a biconfluent Heun equation have some limitations. The known polynomial solutions are confronted with new numerical calculations based on the Numerov method. A good qualitative agreement between them emerges. The numerical method being more general gives rise to new solutions. In particular, we are now able to calculate the quantum dot eigenfunctions for a much larger spectrum of external harmonic frequencies as compared to previous results. Also the existence of bound state for such planar system, in the case l=0, is predicted and its respective eigenvalue is determined.

Research paper thumbnail of Two-electron quantum dot model revisited: bound states and other analytical and numerical solutions

The model of a two-electron quantum dot, confined to move in a two dimensional flat space, is rev... more The model of a two-electron quantum dot, confined to move in a two dimensional flat space, is revisited. Generally, it is argued that the solutions of this model obtained by solving a biconfluent Heun equation have some limitations. In particular, some corrections are also made in previous theoretical calculations. The corrected polynomial solutions are confronted with numerical calculations based on the Numerov method, in a good agreement between both. Then, new solutions considering the 1/r and ln r Coulombian-like potentials in (1+2)D, not yet obtained, are discussed numerically. In particular, we are able to calculate the quantum dot eigenfunctions for a much larger spectrum of external harmonic frequencies as compared to previous results. Also the existence of bound states for such planar system in the case l = 0 is predicted and the respective eigenvalues are determined.