30 research outputs found
Comparative analysis of information measures of the Dirichlet and Neumann two-dimensional quantum dots
Analytic representation of both position as well as momentum waveforms of the
two-dimensional (2D) circular quantum dots with the Dirichlet and Neumann
boundary conditions (BCs) allowed an efficient computation in either space of
Shannon , R\'{e}nyi and Tsallis entropies, Onicescu
energies and Fisher informations . It is shown that a transition to the
2D geometry lifts the 1D degeneracy of the position components ,
, . Among many other findings, it is established that
the lower limit of the semi-infinite range of the dimensionless
R\'{e}nyi/Tsallis coefficient where one-parameter momentum entropies exist is
equal to 2/5 for the Dirichlet requirement and 2/3 for the Neumann one. Since
their 1D counterparts are and , respectively, this simultaneously
reveals that this critical value crucially depends not only on the position BC
but the dimensionality of the structure too. As the 2D Neumann threshold
is greater than one half, its R\'{e}nyi uncertainty relation
for the sum of the position and wave vector components
is valid in the
range only with its logarithmic divergence at the right edge whereas
for all other systems it is defined at any coefficient not smaller
than one half. For both configurations, the lowest-energy level at
does saturate R\'{e}nyi and Tsallis entropic inequalities. Other properties are
discussed and analyzed from the mathematical and physical points of view
R\'{e}nyi and Tsallis entropies of the Dirichlet and Neumann one-dimensional quantum wells
A comparative analysis of the Dirichlet and Neumann boundary conditions (BCs)
of the one-dimensional (1D) quantum well extracts similarities and differences
of the R\'{e}nyi as well as Tsallis entropies between
these two geometries. It is shown, in particular, that for either BC the
dependencies of the R\'{e}nyi position components on the parameter are
the same for all orbitals but the lowest Neumann one for which the
corresponding functional is not influenced by the variation of .
Lower limit of the semi infinite range of the dimensionless
R\'{e}nyi/Tsallis coefficient where {\em momentum} entropies exist crucially
depends on the {\em position} BC and is equal to one quarter for the Dirichlet
requirement and one half for the Neumann one. At approaching this
critical value, the corresponding momentum functionals do diverge. The gap
between the thresholds of the two BCs causes different behavior
of the R\'{e}nyi uncertainty relations as functions of . For both
configurations, the lowest-energy level at does saturate either
type of the entropic inequality thus confirming an earlier surmise about it. It
is also conjectured that the threshold of one half is
characteristic of any 1D non-Dirichlet system. Other properties are discussed
and analyzed from the mathematical and physical points of view.Comment: 7 figure
Comment on "On the realisation of quantum Fisher information"
It is shown that calculation of the momentum Fisher information of the
quasione- dimensional hydrogen atom recently presented by Saha et al (2017 Eur.
J. Phys. {\bf 38} 025103) is wrong. A correct derivation is provided and its
didactical advantages and scientific significances are highlighted
Theory of the Robin quantum wall in a linear potential. II. Thermodynamic properties
A theoretical analysis of the thermodynamic properties of the Robin wall
characterized by the extrapolation length in the electric field
that pushes the particle to the surface is presented both in the
canonical and two grand canonical representations and in the whole range of the
Robin distance with the emphasis on its negative values which for the
voltage-free configuration support negative-energy bound state. For the
canonical ensemble, the heat capacity at exhibits a nonmonotonic
behavior as a function of the temperature with its pronounced maximum
unrestrictedly increasing for the decreasing fields as and
its location being proportional to . For the
Fermi-Dirac distribution, the specific heat per particle is a
nonmonotonic function of the temperature too with the conspicuous extremum
being preceded on the axis by the plateau whose magnitude at the vanishing
is defined as , with being a number of the
particles. The maximum of is the largest for and, similar to the
canonical ensemble, grows to infinity as the field goes to zero. For the
Bose-Einstein ensemble, a formation of the sharp asymmetric feature on the
- dependence with the increase of is shown to be more prominent at
the lower voltages. This cusp-like dependence of the heat capacity on the
temperature, which for the infinite number of bosons transforms into the
discontinuity of , is an indication of the phase transition to the
condensate state. Qualitative and quantitative explanation of these physical
phenomena is based on the variation of the energy spectrum by the electric
field
Influence of the interplay between de Gennes boundary conditions and cubicity of Ginzburg-Landau equation on the properties of superconducting films
Exact solutions of the Ginzburg-Landau (GL) equation for the straight film
subjected at its edges to the Robin-type boundary conditions characterized by
the extrapolation length are analyzed with the primary emphasis on
the interaction between the coefficient of the cubic GL term and the de
Gennes distance and its influence on the temperature of the
strip. Very substantial role is played also by the carrier density that
naturally emerges as an integration constant of the GL equation. Physical
interpretation of the obtained results is based on the -dependent
effective potential created by the nonlinear term and its
influence on the lowest eigenvalue of the corresponding Schr\"{o}dinger
equation. In particular, for the large cubicities, the temperature becomes
independent linearly decreasing function of the growing since
in this limit the boundary conditions can not alter very strong . It
is shown that the temperature increase, which is produced in the linear GL
regime by the negative de Gennes distance, is wiped out by the growing
cubicity. In this case, the decreasing passes through its bulk value
at the unique density only, and the corresponding extrapolation
length is an analytical function of whose properties
are discussed in detail. For the densities smaller than , the
temperature stays above saturating for the large cubicities to the value
determined by and negative while for the
superconductivity is destroyed by the growing GL nonlinearity at some
temperature , which depends on , and . It is
proved that the concentration transforms for the large cubicities
into the density of the bulk sample.Comment: 26 pages, 8 figure
Comparative analysis of electric field influence on the quantum wells with different boundary conditions. II. Thermodynamic properties
Thermodynamic properties of the one-dimensional (1D) quantum well (QW) with
miscellaneous permutations of the Dirichlet (D) and Neumann (N) boundary
conditions (BCs) at its edges in the perpendicular to the surfaces electric
field are calculated. For the canonical ensemble, analytical
expressions involving theta functions are found for the mean energy and heat
capacity for the box with no applied voltage. Pronounced maximum
accompanied by the adjacent minimum of the specific heat dependence on the
temperature for the pure Neumann QW and their absence for other BCs are
predicted and explained by the structure of the corresponding energy spectrum.
Applied field leads to the increase of the heat capacity and formation of the
new or modification of the existing extrema what is qualitatively described by
the influence of the associated electric potential. A remarkable feature of the
Fermi grand canonical ensemble is, at any BC combination in zero fields, a
salient maximum of observed on the axis for one particle and its
absence for any other number of corpuscles. Qualitative and quantitative
explanation of this phenomenon employs the analysis of the chemical potential
and its temperature dependence for different . It is proved that critical
temperature of the Bose-Einstein (BE) condensation increases with the
applied voltage for any number of particles and for any BC permutation except
the ND case at small intensities what is explained again by the
modification by the field of the interrelated energies. It is shown that even
for the temperatures smaller than the total dipole moment may become negative for the quite moderate . For either
Fermi or BE system, the influence of the electric field on the heat capacity is
shown to be suppressed with growing
Theory of the Robin quantum wall in a linear potential. I. Energy spectrum, polarization and quantum-information measures
Information-theoretical concepts are employed for the analysis of the
interplay between a transverse electric field applied to a
one-dimensional surface and Robin boundary condition (BC), which with the help
of the extrapolation length zeroes at the interface a linear
combination of the quantum mechanical wave function and its spatial derivative,
and its influence on the properties of the structure. For doing this, exact
analytical solutions of the corresponding Schr\"{o}dinger equation are derived
and used for calculating energies, dipole moments, position and momentum
quantum information entropies and their Fisher information and
and Onicescu information energies and counterparts. It is
shown that the weak (strong) electric field changes the Robin wall into the
Dirichlet, (Neumann, ), surface. This
transformation of the energy spectrum and associated waveforms in the growing
field defines an evolution of the quantum-information measures; for example, it
is proved that for the Dirichlet and Neumann BCs the position (momentum)
quantum information entropy varies as a positive (negative) natural logarithm
of the electric intensity what results in their field-independent sum
. Analogously, at and the position and
momentum Fisher informations (Onicescu energies) depend on the applied voltage
as () and its inverse, respectively,
leading to the field-independent product (). Peculiarities of
their transformations at the finite nonzero are discussed and
similarities and differences between the three quantum-information measures in
the electric field are highlighted with the special attention being paid to the
configuration with the negative extrapolation length.Comment: 10 Figures, 1 Tabl
Electric-Field Control of Bound States and Optical Spectrum in Window-Coupled Quantum Waveguides
Properties of the bound states of two quantum waveguides coupled via the
window of the width in their common boundary are calculated under the
assumption that the transverse electric field is applied to
the structure. It is shown that the increase of the electric intensity brings
closer to each other fundamental propagation thresholds of the opening and the
arms. As a result, the ground state, which in the absence of the field exists
at any nonzero , exhibits the energy decrease for the growing
and in the high-field regime stays practically the same
regardless of the size of the connecting region. It is predicted that the
critical window widths , , at which new excited
localized orbitals emerge, strongly depend on the transverse voltage; in
particular, the field leads to the increase of , and, for quite
strong electric intensities, the critical width unrestrictedly diverges. This
remarkable feature of the electric-field-induced switching of the bound states
can be checked, for example, by the change of the optical properties of the
structure when the gate voltage is applied; namely, both the oscillator
strength and absorption spectrum exhibit a conspicuous maximum on their
dependence and turn to zero when the electric intensity reaches
its critical value. Comparative analysis of the two-dimensional (2D) and 3D
geometries reveals their qualitative similarity and quantitative differences.Comment: 21 pages, 12 figure
Magnetic field control of the intraband optical absorption in two-dimensional quantum rings
Linear and nonlinear optical absorption coefficients of the two-dimensional
semiconductor ring in the perpendicular magnetic field are calculated
within independent electron approximation. Characteristic feature of the energy
spectrum are crossings of the levels with adjacent nonpositive magnetic quantum
numbers as the intensity changes. It is shown that the absorption
coefficient of the associated optical transition is drastically decreased at
the fields corresponding to the crossing. Proposed model of the Volcano disc
allows to get simple mathematical analytical results, which provide clear
physical interpretation. An interplay between positive linear and
intensity-dependent negative cubic absorption coefficients is discussed; in
particular, critical light intensity at which additional resonances appear in
the total absorption dependence on the light frequency, is calculated as a
function of the magnetic field and levels' broadening.Comment: 17 pages, 5 figure
Quantum information measures of the Dirichlet and Neumann hyperspherical dots
-dimensional hyperspherical quantum dot with either Dirichlet or
Neumann boundary conditions (BCs) allows analytic solution of the
Schr\"{o}dinger equation in position space and the Fourier transform of the
corresponding wave function leads to the analytic form of its momentum
counterpart too. This paves the way to an efficient computation in either space
of Shannon, R\'{e}nyi and Tsallis entropies, Onicescu energies and Fisher
informations; for example, for the latter measure, some particular orbitals
exhibit simple expressions in either space at any BC type. A comparative study
of the influence of the edge requirement on the quantum information measures
proves that the lower threshold of the semi-infinite range of the dimensionless
R\'{e}nyi/Tsallis coefficient where one-parameter momentum entropies exist is
equal to for the Dirichlet hyperball and
for the Neumann one what means that at the
unrestricted growth of the dimensionality both measures have their Shannon
fellow as the lower verge. Simultaneously, this imposes the restriction on the
upper value of the interval inside which the R\'{e}nyi
uncertainty relation for the sum of the position and wave
vector components is defined:
is equal to for the Dirichlet geometry
and to for the Neumann BC. Some other properties
are discussed from mathematical and physical points of view. Parallels are
drawn to the corresponding properties of the hydrogen atom and similarities and
differences are explained based on the analysis of the associated wave
functions.Comment: 5 figures, 2 table