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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.


Related in: MedlinePlus

Position  (dotted lines), momentum  (dashed lines) and total  (solid curves) entropies as a function of the electric field  for all permutations of the Dirichlet and Neumann BCs. Two characters in the left lower corner of each panel denote a type of the edge requirements while the digits near the curves depict the corresponding quantum number n. Note vertical line breaks from  to .
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fig08: Position (dotted lines), momentum (dashed lines) and total (solid curves) entropies as a function of the electric field for all permutations of the Dirichlet and Neumann BCs. Two characters in the left lower corner of each panel denote a type of the edge requirements while the digits near the curves depict the corresponding quantum number n. Note vertical line breaks from to .

Mentions: Peculiar features of the position and momentum densities exemplified by Figs.5–7 determine the properties of the corresponding entropies and . Fig.8 shows them together with the sum as functions of the electric field for all four possible BCs and several low-lying levels. Characteristic feature is a nonmonotonic dependence of the position and momentum entropies on the small and moderate voltages for the structures with the left Neumann wall (except the ground NN level). For example, the maximum of the lowest ND state position entropy is explained by the sharp decrease of the corresponding function at the left surface for (see the left lower panel of Fig.6) and accompanying decline of its negative contribution to while the corresponding density at the right edge stays almost unchanged. Simultaneously, mentioned above increase of the momentum density results in the contraction of and, as a consequence of these two facts, the lowering of the total entropy that, however, stays above the fundamental limit from the right-hand side of Eq. (32), as expected. For the higher voltages, the accumulating near the right interface function Ψ0 pushes the position space entropy downward forming the extremum seen in the figure. On the contrary, the Dirichlet left interface subdues the entropy swaying, as the corresponding panels of Fig.8 demonstrate. At the large , the entropies are determined by the right-wall BC, as discussed above; accordingly, they are almost equal for the DN and pure Neumann QWs and their growing with the field magnitudes are larger than their (almost identical) counterparts for the ND and pure Dirichlet structures. Similar to the field-free case, the Dirichlet ground state has the lowest total entropy among all other levels and all possible BC combinations. Note its quite small variation with the voltage: from its zero-field value of 2.2120 it saturates to approximately 2.2552 at the large . Growing with the field magnitude of the negative position entropy means smaller uncertainty in determining particle location while the larger values of the positive momentum entropy indicate that the uncertainty in calculating momentum k increases.


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)

Position  (dotted lines), momentum  (dashed lines) and total  (solid curves) entropies as a function of the electric field  for all permutations of the Dirichlet and Neumann BCs. Two characters in the left lower corner of each panel denote a type of the edge requirements while the digits near the curves depict the corresponding quantum number n. Note vertical line breaks from  to .
© Copyright Policy - open-access
Related In: Results  -  Collection

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

fig08: Position (dotted lines), momentum (dashed lines) and total (solid curves) entropies as a function of the electric field for all permutations of the Dirichlet and Neumann BCs. Two characters in the left lower corner of each panel denote a type of the edge requirements while the digits near the curves depict the corresponding quantum number n. Note vertical line breaks from to .
Mentions: Peculiar features of the position and momentum densities exemplified by Figs.5–7 determine the properties of the corresponding entropies and . Fig.8 shows them together with the sum as functions of the electric field for all four possible BCs and several low-lying levels. Characteristic feature is a nonmonotonic dependence of the position and momentum entropies on the small and moderate voltages for the structures with the left Neumann wall (except the ground NN level). For example, the maximum of the lowest ND state position entropy is explained by the sharp decrease of the corresponding function at the left surface for (see the left lower panel of Fig.6) and accompanying decline of its negative contribution to while the corresponding density at the right edge stays almost unchanged. Simultaneously, mentioned above increase of the momentum density results in the contraction of and, as a consequence of these two facts, the lowering of the total entropy that, however, stays above the fundamental limit from the right-hand side of Eq. (32), as expected. For the higher voltages, the accumulating near the right interface function Ψ0 pushes the position space entropy downward forming the extremum seen in the figure. On the contrary, the Dirichlet left interface subdues the entropy swaying, as the corresponding panels of Fig.8 demonstrate. At the large , the entropies are determined by the right-wall BC, as discussed above; accordingly, they are almost equal for the DN and pure Neumann QWs and their growing with the field magnitudes are larger than their (almost identical) counterparts for the ND and pure Dirichlet structures. Similar to the field-free case, the Dirichlet ground state has the lowest total entropy among all other levels and all possible BC combinations. Note its quite small variation with the voltage: from its zero-field value of 2.2120 it saturates to approximately 2.2552 at the large . Growing with the field magnitude of the negative position entropy means smaller uncertainty in determining particle location while the larger values of the positive momentum entropy indicate that the uncertainty in calculating momentum k increases.

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