Rényi and Tsallis entropies of the Dirichlet and Neumann one‐dimensional quantum wells

Olendski, O.

doi:10.1002/qua.26220

Cited by 10 publications

(31 citation statements)

References 53 publications

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“…The present research expands the previous 1D endeavor [ 6 ] to the comparative analysis of the influence of the Dirichlet and Neumann BCs on the Rényi and Tsallis entropies of the 2D circular quantum dot. It is revealed that the lowest thresholds of the range, where momentum functionals exist, are not only BC‐dependent, but dimensionality strongly influences them too; namely, the 2D geometry pushes them higher:

α_{TH}^{D} = 2 / 5

and

α_{TH}^{N} = 2 / 3

where, similar to the 1D well, [ 6 ] Dirichlet value is again smaller than its Neumann counterpart. As

α_{TH}^{N}

is greater than one half, the Rényi inequality from Equation ) has its upper boundary at α = 2, at which its left‐hand side diverges, whereas for the Dirichlet configuration, the corresponding sum is defined for the whole interval of the uncertainty relation [1/2, +∞).…”

Section: Introductionmentioning

confidence: 67%

“…Contrary to the 1D well, [ 6 ] the dependence of the Rényi position entropy of the Dirichlet disc

R_{ρ_{italicnm}}^{D} ((), α) = 2 \ln a + \ln π + \frac{1}{1 - α} \ln ((), 2 \int_{0}^{1} z {[\frac{J_{∣ m ∣}^{2} ((), j_{∣ m ∣ n} z)}{J_{∣ m ∣ + 1}^{2} ((), j_{∣ m ∣ n})}]}^{α} italicdz)

is a distinct function of each corresponding level, which is exemplified in Figure 5. As it follows from the most general properties, [ 8 ]

R_{ρ_{italicnm}}^{D} ((), α)

monotonically decreases with its argument increasing.…”

Section: Rényi and Tsallis Entropiesmentioning

confidence: 99%

“…Depending on the BC type, the properties of the structure might differ significantly. [ 2 ] This general statement has found its countless confirmations, [ 3–5 ] with one of the most recent analyses [ 6 ] addressing the influence of the Dirichlet and Neumann surface requirements on the Rényi [ 7,8 ] R and Tsallis [ 9 ] (or, more correctly, Havrda‐Charvát‐Daróczy‐Tsallis [ 10,11 ] ) T entropies of the 1D quantum well. These one‐parameter functionals are defined in the d ‐dimensional position (subscript ρ ) and wave vector ( γ ) spaces:

R_{ρ_{n}} ((), α) = \frac{1}{1 - α} \ln ((), \int_{{scriptD}_{ρ}^{d}} ρ_{n}^{α} ((), r) d boldr)

R_{γ_{n}} ((), α) = \frac{1}{1 - α} \ln ((), \int_{{scriptD}_{γ}^{d}} γ_{n}^{α} ((), k) d boldk)

T_{ρ_{n}} ((), α) = \frac{1}{α - 1} ((), 1 - \int_{{scriptD}_{ρ}^{d}} ρ_{n}^{α} ((), r) d boldr)

T_{γ_{n}} ((), α) = \frac{1}{α - 1} ((), 1 - \int_{{scriptD}_{γ}^{d}} γ_{n}^{α} ((), k) d boldk) .

…”

Section: Introductionmentioning

confidence: 97%

“…In all these equations, integrations are carried out over the whole region

D_{ρ}^{d}

D_{γ}^{d}

, where the functions Ψ n ( r ) or Φ n ( k ) are defined. It was particularly shown [ 6 ] that, for either BC, the dependencies of the Rényi position components on the nonnegative parameter α are the same for all orbitals except the lowest Neumann one, for which the corresponding functional R due to the constant

Ψ_{1}^{N} ((), x)

is not influenced by the variation of α . A crucial difference between the two BCs lies in the fact that the lower‐limit α TH of the semi‐infinite range of the dimensionless Rényi/Tsallis coefficient where momentum entropies exist is equal to one quarter of the Dirichlet requirement and one half for the Neumann one.…”

Section: Introductionmentioning

confidence: 99%

“…[ 16,19 ] Equation ) is a result of logarithmization of the Sobolev inequality of the Fourier transform [ 20 ]

{(\frac{α}{π})}^{d / ((), 4 α)} {[\int_{D_{ρ}^{d}} ρ_{n}^{α} (boldr) d r]}^{1 / ((), 2 α)} \geq {(\frac{β}{π})}^{d / ((), 4 β)} {[\int_{D_{γ}^{d}} γ_{n}^{β} (boldk) d k]}^{1 / ((), 2 β)},

for which, in addition to the requirement from Equation ), an extra constraint

\frac{1}{2} \leq α \leq 1

is imposed, which directly leads to the Tsallis uncertainty relation [ 21 ] :

{(\frac{α}{π})}^{d / ((), 4 α)} {[1 + (1 - α) T_{ρ_{n}} (α)]}^{1 / ((), 2 α)} \geq {(\frac{β}{π})}^{d / ((), 4 β)} {[1 + (1 - β) T_{γ_{n}} (β)]}^{1 / ((), 2 β)},

which is obviously saturated at α = β = 1 when either of its sides is equal to π − d /4 . For the 1D Neumann or Dirichlet well, both Rényi, Equation ) and Tsallis, Equation ), relations are tightened at α = 1/2 by the lowest orbital [ 6 ] when, for example, either side of Equation ) turns to Φ 1 ( 0 ), which appears to be a general property of any magnet...…”

Section: Introductionmentioning

confidence: 99%

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Comparative analysis of information measures of the Dirichlet and Neumann two‐dimensional quantum dots

Olendski

2020

Int J of Quantum Chemistry

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d-dimensional hyperspherical quantum dot with either Dirichlet or Neumann boundary conditions (BCs) allows analytic solution of the Schrödinger 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, Rényi 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 Rényi/Tsallis coefficient where one-parameter momentum entropies exist is equal to d/(d + 3) for the Dirichlet hyperball and d/(d + 1) 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 [1/2, α R ) inside which the Rényi uncertainty relation for the sum of the position R ρ (α) and wave vector R γ α 2α−1 components is defined: α R is equal to d/(d − 3) for the Dirichlet geometry and to d/(d − 1) 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.

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α_{TH}^{D} = 2 / 5

and

α_{TH}^{N} = 2 / 3

where, similar to the 1D well, [ 6 ] Dirichlet value is again smaller than its Neumann counterpart. As

α_{TH}^{N}

Section: Introductionmentioning

confidence: 67%

“…Contrary to the 1D well, [ 6 ] the dependence of the Rényi position entropy of the Dirichlet disc

R_{ρ_{italicnm}}^{D} ((), α) = 2 \ln a + \ln π + \frac{1}{1 - α} \ln ((), 2 \int_{0}^{1} z {[\frac{J_{∣ m ∣}^{2} ((), j_{∣ m ∣ n} z)}{J_{∣ m ∣ + 1}^{2} ((), j_{∣ m ∣ n})}]}^{α} italicdz)

is a distinct function of each corresponding level, which is exemplified in Figure 5. As it follows from the most general properties, [ 8 ]

R_{ρ_{italicnm}}^{D} ((), α)

monotonically decreases with its argument increasing.…”

Section: Rényi and Tsallis Entropiesmentioning

confidence: 99%

R_{ρ_{n}} ((), α) = \frac{1}{1 - α} \ln ((), \int_{{scriptD}_{ρ}^{d}} ρ_{n}^{α} ((), r) d boldr)

R_{γ_{n}} ((), α) = \frac{1}{1 - α} \ln ((), \int_{{scriptD}_{γ}^{d}} γ_{n}^{α} ((), k) d boldk)

T_{ρ_{n}} ((), α) = \frac{1}{α - 1} ((), 1 - \int_{{scriptD}_{ρ}^{d}} ρ_{n}^{α} ((), r) d boldr)

T_{γ_{n}} ((), α) = \frac{1}{α - 1} ((), 1 - \int_{{scriptD}_{γ}^{d}} γ_{n}^{α} ((), k) d boldk) .

…”

Section: Introductionmentioning

confidence: 97%

“…In all these equations, integrations are carried out over the whole region

D_{ρ}^{d}

D_{γ}^{d}

Ψ_{1}^{N} ((), x)

Section: Introductionmentioning

confidence: 99%

“…[ 16,19 ] Equation ) is a result of logarithmization of the Sobolev inequality of the Fourier transform [ 20 ]

{(\frac{α}{π})}^{d / ((), 4 α)} {[\int_{D_{ρ}^{d}} ρ_{n}^{α} (boldr) d r]}^{1 / ((), 2 α)} \geq {(\frac{β}{π})}^{d / ((), 4 β)} {[\int_{D_{γ}^{d}} γ_{n}^{β} (boldk) d k]}^{1 / ((), 2 β)},

for which, in addition to the requirement from Equation ), an extra constraint

\frac{1}{2} \leq α \leq 1

is imposed, which directly leads to the Tsallis uncertainty relation [ 21 ] :

{(\frac{α}{π})}^{d / ((), 4 α)} {[1 + (1 - α) T_{ρ_{n}} (α)]}^{1 / ((), 2 α)} \geq {(\frac{β}{π})}^{d / ((), 4 β)} {[1 + (1 - β) T_{γ_{n}} (β)]}^{1 / ((), 2 β)},

Section: Introductionmentioning

confidence: 99%

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Comparative analysis of information measures of the Dirichlet and Neumann two‐dimensional quantum dots

Olendski

2020

Int J of Quantum Chemistry

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Shannon and von Neumann entropies of multi-qubit Schrödinger's cat states

Jansen

Loucks

Gilbert

et al. 2022

Phys. Chem. Chem. Phys.

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Rydberg multidimensional states: Rényi and Shannon entropies in momentum space

Aptekarev¹,

Belega²,

Dehesa³

2020

J. Phys. A: Math. Theor.

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In this work the momentum spreading of a multidimensional hydrogenic system in highly excited (Rydberg) states is quantified by means of the Rényi and Shannon entropies of its momentum probability density. These quantities, which rest at the core of numerous fields from atomic and molecular physics to quantum technologies, are determined by means of a methodology based on the strong degree-asymptotics of a modified L q -norm of the Gegenbauer polynomials which control the wavefunctions of these states in momentum space. The leading term of these entropic quantities is found from first principles, i.e. by means of the Coulomb potential parameters (space dimensionality, the nuclear charge) and the states’s hyperquantum numbers, in a rigorous simple manner. It is shown that they fulfill a logarithmic growth scaling law with the principal hyperquantum number n which characterizes the Rydberg state.

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Rényi and Tsallis entropies of the Dirichlet and Neumann one‐dimensional quantum wells

Cited by 10 publications

References 53 publications

Comparative analysis of information measures of the Dirichlet and Neumann two‐dimensional quantum dots

Comparative analysis of information measures of the Dirichlet and Neumann two‐dimensional quantum dots

Shannon and von Neumann entropies of multi-qubit Schrödinger's cat states

Rydberg multidimensional states: Rényi and Shannon entropies in momentum space

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