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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

Momentum space densities  at zero electric field for (a) the Dirichlet, (b) Neumann, and (c) mixed BCs as functions of the momentum k where the solid lines are for the ground states (), dashed curves - for the first excited levels (), dotted lines - for the second excited states (), and dash-dotted curves - for the levels with the quantum number .
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fig02: Momentum space densities at zero electric field for (a) the Dirichlet, (b) Neumann, and (c) mixed BCs as functions of the momentum k where the solid lines are for the ground states (), dashed curves - for the first excited levels (), dotted lines - for the second excited states (), and dash-dotted curves - for the levels with the quantum number .

Mentions: Fig.2 exhibits zero-field momentum densities from Eqs. (41) and (37) for all BCs and several quantum states n. Previous analysis discussed these dependencies for the Dirichlet QW only [41, 44–46]. As is an even function of its argument, , we plot the parts with the non negative k only. Qualitatively, this behavior is the same for any type of the surface requirements; namely, for all excited states, , the momentum density is characterized by the two conspicuous peaks, which are located symmetrically with respect to and are flanked by the series of much smaller extrema. The distance between the sharp maxima decreases for the smaller n, and for the ground state they merge together forming one symmetric peak with its magnitude being the largest as compared to the excited levels. Quantitatively, Neumann maximum is larger than the Dirichlet or Zaremba one, . As the momentum density is smaller than one, , the corresponding entropy from Eq. (27) is always positive. More detailed numerical analysis reveals that it is a growing function of the quantum number n. In contrast, the parts of the position density, which are larger than unity, in their contribution to the integral from Eq. (26) overweigh those with leading in this way to the negative entropies from Eq. (40). Returning to the momentum entropies, we note that despite the fact that , the ground-state entropy is greater than its mentioned above Neumann counterpart what apparently is explained by the far-reaching spread of the corresponding density: the wide sidelobe is clearly seen for the curve in panel (c) of Fig.2. In this way, the Dirichlet ground state with its smallest momentum entropy comes closest to the limit imposed by the entropic uncertainty relation: . Our calculations indicate that the Dirichlet momentum and total entropy for any level n remain the smallest ones among all possible BC combinations: .


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)

Momentum space densities  at zero electric field for (a) the Dirichlet, (b) Neumann, and (c) mixed BCs as functions of the momentum k where the solid lines are for the ground states (), dashed curves - for the first excited levels (), dotted lines - for the second excited states (), and dash-dotted curves - for the levels with the quantum number .
© Copyright Policy - open-access
Related In: Results  -  Collection

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

fig02: Momentum space densities at zero electric field for (a) the Dirichlet, (b) Neumann, and (c) mixed BCs as functions of the momentum k where the solid lines are for the ground states (), dashed curves - for the first excited levels (), dotted lines - for the second excited states (), and dash-dotted curves - for the levels with the quantum number .
Mentions: Fig.2 exhibits zero-field momentum densities from Eqs. (41) and (37) for all BCs and several quantum states n. Previous analysis discussed these dependencies for the Dirichlet QW only [41, 44–46]. As is an even function of its argument, , we plot the parts with the non negative k only. Qualitatively, this behavior is the same for any type of the surface requirements; namely, for all excited states, , the momentum density is characterized by the two conspicuous peaks, which are located symmetrically with respect to and are flanked by the series of much smaller extrema. The distance between the sharp maxima decreases for the smaller n, and for the ground state they merge together forming one symmetric peak with its magnitude being the largest as compared to the excited levels. Quantitatively, Neumann maximum is larger than the Dirichlet or Zaremba one, . As the momentum density is smaller than one, , the corresponding entropy from Eq. (27) is always positive. More detailed numerical analysis reveals that it is a growing function of the quantum number n. In contrast, the parts of the position density, which are larger than unity, in their contribution to the integral from Eq. (26) overweigh those with leading in this way to the negative entropies from Eq. (40). Returning to the momentum entropies, we note that despite the fact that , the ground-state entropy is greater than its mentioned above Neumann counterpart what apparently is explained by the far-reaching spread of the corresponding density: the wide sidelobe is clearly seen for the curve in panel (c) of Fig.2. In this way, the Dirichlet ground state with its smallest momentum entropy comes closest to the limit imposed by the entropic uncertainty relation: . Our calculations indicate that the Dirichlet momentum and total entropy for any level n remain the smallest ones among all possible BC combinations: .

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