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Comparative analysis of electric field influence on the quantum wells with different boundary conditions.: I. Energy spectrum, quantum information entropy and polarization.

Olendski O - Ann Phys (2015)

Bottom Line: Simple two-level approximation elementary explains the negative polarizations as a result of the field-induced destructive interference of the unperturbed modes and shows that in this case the admixture of only the neighboring states plays a dominant role.Different magnitudes of the polarization for different BCs in this regime are explained physically and confirmed numerically.Applications to different branches of physics, such as ocean fluid dynamics and atmospheric and metallic waveguide electrodynamics, are discussed.

View Article: PubMed Central - PubMed

Affiliation: King Abdullah Institute for Nanotechnology, King Saud University P.O. Box 2455, Riyadh, 11451, Saudi Arabia.

ABSTRACT

Analytical solutions of the Schrödinger equation for the one-dimensional quantum well with all possible permutations of the Dirichlet and Neumann boundary conditions (BCs) in perpendicular to the interfaces uniform electric field [Formula: see text] are used for the comparative investigation of their interaction and its influence on the properties of the system. Limiting cases of the weak and strong voltages allow an easy mathematical treatment and its clear physical explanation; in particular, for the small [Formula: see text], the perturbation theory derives for all geometries a linear dependence of the polarization on the field with the BC-dependent proportionality coefficient being positive (negative) for the ground (excited) states. Simple two-level approximation elementary explains the negative polarizations as a result of the field-induced destructive interference of the unperturbed modes and shows that in this case the admixture of only the neighboring states plays a dominant role. Different magnitudes of the polarization for different BCs in this regime are explained physically and confirmed numerically. Hellmann-Feynman theorem reveals a fundamental relation between the polarization and the speed of the energy change with the field. It is proved that zero-voltage position entropies [Formula: see text] are BC independent and for all states but the ground Neumann level (which has [Formula: see text]) are equal to [Formula: see text] while the momentum entropies [Formula: see text] depend on the edge requirements and the level. Varying electric field changes position and momentum entropies in the opposite directions such that the entropic uncertainty relation is satisfied. Other physical quantities such as the BC-dependent zero-energy and zero-polarization fields are also studied both numerically and analytically. Applications to different branches of physics, such as ocean fluid dynamics and atmospheric and metallic waveguide electrodynamics, are discussed.

No MeSH data available.


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Polarizations  for (a) the ground, , (b) first, , and (c) second, , excited states as functions of the normalized electric field . Solid (dotted) lines denote pure Dirichlet (Neumann) configuration while the dashed (dash-dotted) curves are for the DN (ND) geometry. Thin horizontal lines in parts (b) and (c) denote zero polarizations. Note different P scales in each of the panels.
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fig04: Polarizations for (a) the ground, , (b) first, , and (c) second, , excited states as functions of the normalized electric field . Solid (dotted) lines denote pure Dirichlet (Neumann) configuration while the dashed (dash-dotted) curves are for the DN (ND) geometry. Thin horizontal lines in parts (b) and (c) denote zero polarizations. Note different P scales in each of the panels.

Mentions: Energy spectrum as a function of the field is shown in Fig.3 for all four types of the possible combinations of the BCs, and corresponding polarizations for the ground and two lowest excited states are depicted in Fig.4. It is seen that the energies, which at the zero field have the same values for the opposite Zaremba geometries, Eq. (22), at the small voltage move in the opposite directions while at the large they coincide with their counterparts for the uniform BC distribution that is imposed at the right surface of the asymmetric edge requirements. This means that the ND geometry is the only configuration at which the lowest energy at the small fields increases. Ground-state polarizations for all BCs monotonically increase with the field and approach at the strong voltages the asymptotic values that will be discussed below while for the excited levels the polarizations take negative values for the small and moderate fields, and only after reaching minimum they increase with the growing . Classically, this result, which first was predicted for the Dirichlet QW [7], is unexpected since it means that the charged particle moves against the applied force. However, in the quantum mechanical treatment the electron is described as a wave whose behavior is governed by the solution of the corresponding equation inside the domain with the appropriate BCs. For the small electric fields, the energy and polarization dependencies are conveniently described by the standard perturbation theory [1]; namely, considering in the total Hamiltonian from Eq. (4) the second term as the small disturbance of the field-free Hamiltonian , one writes the perturbed energies and wavefunctions as


Comparative analysis of electric field influence on the quantum wells with different boundary conditions.: I. Energy spectrum, quantum information entropy and polarization.

Olendski O - Ann Phys (2015)

Polarizations  for (a) the ground, , (b) first, , and (c) second, , excited states as functions of the normalized electric field . Solid (dotted) lines denote pure Dirichlet (Neumann) configuration while the dashed (dash-dotted) curves are for the DN (ND) geometry. Thin horizontal lines in parts (b) and (c) denote zero polarizations. Note different P scales in each of the panels.
© Copyright Policy - open-access
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4402006&req=5

fig04: Polarizations for (a) the ground, , (b) first, , and (c) second, , excited states as functions of the normalized electric field . Solid (dotted) lines denote pure Dirichlet (Neumann) configuration while the dashed (dash-dotted) curves are for the DN (ND) geometry. Thin horizontal lines in parts (b) and (c) denote zero polarizations. Note different P scales in each of the panels.
Mentions: Energy spectrum as a function of the field is shown in Fig.3 for all four types of the possible combinations of the BCs, and corresponding polarizations for the ground and two lowest excited states are depicted in Fig.4. It is seen that the energies, which at the zero field have the same values for the opposite Zaremba geometries, Eq. (22), at the small voltage move in the opposite directions while at the large they coincide with their counterparts for the uniform BC distribution that is imposed at the right surface of the asymmetric edge requirements. This means that the ND geometry is the only configuration at which the lowest energy at the small fields increases. Ground-state polarizations for all BCs monotonically increase with the field and approach at the strong voltages the asymptotic values that will be discussed below while for the excited levels the polarizations take negative values for the small and moderate fields, and only after reaching minimum they increase with the growing . Classically, this result, which first was predicted for the Dirichlet QW [7], is unexpected since it means that the charged particle moves against the applied force. However, in the quantum mechanical treatment the electron is described as a wave whose behavior is governed by the solution of the corresponding equation inside the domain with the appropriate BCs. For the small electric fields, the energy and polarization dependencies are conveniently described by the standard perturbation theory [1]; namely, considering in the total Hamiltonian from Eq. (4) the second term as the small disturbance of the field-free Hamiltonian , one writes the perturbed energies and wavefunctions as

Bottom Line: Simple two-level approximation elementary explains the negative polarizations as a result of the field-induced destructive interference of the unperturbed modes and shows that in this case the admixture of only the neighboring states plays a dominant role.Different magnitudes of the polarization for different BCs in this regime are explained physically and confirmed numerically.Applications to different branches of physics, such as ocean fluid dynamics and atmospheric and metallic waveguide electrodynamics, are discussed.

View Article: PubMed Central - PubMed

Affiliation: King Abdullah Institute for Nanotechnology, King Saud University P.O. Box 2455, Riyadh, 11451, Saudi Arabia.

ABSTRACT

Analytical solutions of the Schrödinger equation for the one-dimensional quantum well with all possible permutations of the Dirichlet and Neumann boundary conditions (BCs) in perpendicular to the interfaces uniform electric field [Formula: see text] are used for the comparative investigation of their interaction and its influence on the properties of the system. Limiting cases of the weak and strong voltages allow an easy mathematical treatment and its clear physical explanation; in particular, for the small [Formula: see text], the perturbation theory derives for all geometries a linear dependence of the polarization on the field with the BC-dependent proportionality coefficient being positive (negative) for the ground (excited) states. Simple two-level approximation elementary explains the negative polarizations as a result of the field-induced destructive interference of the unperturbed modes and shows that in this case the admixture of only the neighboring states plays a dominant role. Different magnitudes of the polarization for different BCs in this regime are explained physically and confirmed numerically. Hellmann-Feynman theorem reveals a fundamental relation between the polarization and the speed of the energy change with the field. It is proved that zero-voltage position entropies [Formula: see text] are BC independent and for all states but the ground Neumann level (which has [Formula: see text]) are equal to [Formula: see text] while the momentum entropies [Formula: see text] depend on the edge requirements and the level. Varying electric field changes position and momentum entropies in the opposite directions such that the entropic uncertainty relation is satisfied. Other physical quantities such as the BC-dependent zero-energy and zero-polarization fields are also studied both numerically and analytically. Applications to different branches of physics, such as ocean fluid dynamics and atmospheric and metallic waveguide electrodynamics, are discussed.

No MeSH data available.


Related in: MedlinePlus