Theory of the Robin quantum wall in a linear potential. I. Energy spectrum, polarization and quantum‐information measures

2016

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A theoretical analysis of the thermodynamic properties of the Robin wall characterized by the extrapolation length Λ in the electric field E 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 Λ < 0 exhibits a nonmonotonic behavior as a function of the temperature T with its pronounced maximum unrestrictedly increasing for the decreasing fields as ln 2 E and its location being proportional to (− ln E ) −1 . For the Fermi-Dirac distribution, the specific heat per particle cN is a nonmonotonic function of the temperature too with the conspicuous extremum being preceded on the T axis by the plateau whose magnitude at the vanishing E is defined as 3(N − 1)/(2N )kB, with N being a number of the particles. The maximum of cN is the largest for N = 1 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 cN -T dependence with the increase of N 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 cN (T ), is an indication of the phase transition to the condensate state. Some other physical characteristics such as the critical temperature Tcr and ground-level population of the Bose-Einstein condensate are calculated and analyzed as a function of the field and extrapolation length. Qualitative and quantitative explanation of these physical phenomena is based on the variation of the energy spectrum by the electric field.

“…for the finite nonzero extrapolation length Λ is determined from the transcendental equation [2] 2meE 2…”

Section: Introductionmentioning

confidence: 99%

Theory of the Robin quantum wall in a linear potential. II. Thermodynamic properties

2016

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“…Another method to arrive at it is to zero the denominator in Equation (56) what results in infinite specific heat at the transition point. This highly asymmetric cusp-like shape of the specific heat was intensively analyzed theoretically [4,5,65,67,69,70,[91][92][93][94] and demonstrated experimentally. At the infinite number of bosons, N = ∞, the cusp-like shape at T = T cr becomes a discontinuity disclosing in this way a phase transition; namely, in this particular situation, it is a transition from the BE condensate to the normal phase of the noninteracting corpuscles in the asymmetric Robin QW.…”

Section: Bosonsmentioning

confidence: 93%

“…and for the canonical ensemble it is expressed with the help of the fluctuationdissipation theorem: [1] c can (β) = β 2 E 2 can − E 2 can (5) For N noninteracting particles in the system, the right-hand sides of Equations (2)-(5) have to be multiplied by N. Recent comparative analysis of the 1D quantum well (QW) with miscellaneous permutations of the Dirichlet and Neumann boundary conditions (BCs) [2] confirmed that the energy spacing between the orbitals δ E n = E n+1 − E n (6) plays a crucial role in the thermodynamic properties dependence on temperature; in particular, since this quantity is, at the fixed n (specifically, at n = 0), the smallest for the pure Neumann structure as compared to other two geometries, [3] its heat capacity exhibits a salient maximum as a function of T accompanied by the broad minimum at higher temperatures whereas for any other BC configuration the c V −T characteristics is a smooth line. [2] www.advancedsciencenews.com www.ann-phys.org…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…(7) Just this parameter was used for explaining a giant enhancement of the specific heat of the attractive Robin wall in vanishingly small electric fields [4] when it is practically the same for many quantum orbitals with n ≥ 1. [5] In the present research, we apply the methodology developed before [2,4] for analyzing thermodynamics of the Robin QW. Robin BC [6] for the wave function…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Thermodynamic Properties of the 1D Robin Quantum Well

2018

The thermodynamic properties of the Robin quantum well with extrapolation length are analyzed theoretically both for the canonical and two grand canonical ensembles, with special attention being paid to the situation when the energies of one or two lowest-lying states are split off from the rest of the spectrum by the large gap that is controlled by the varying . For the single split-off level, which exists for the geometry with the equal magnitudes but opposite signs of the Robin distances on the confining interfaces, the heat capacity c V of the canonical averaging is a nonmonotonic function of the temperature T with its salient maximum growing to infinity as ln 2 for the decreasing to zero extrapolation length and its position being proportional to 1/( 2 ln ). The specific heat per particle c N of the Fermi-Dirac ensemble depends nonmonotonically on the temperature too, with its pronounced extremum being foregone on the T axis by the plateau, whose value at the dying is (N − 1)/(2N)k B , with N being the number of fermions. The maximum of c N , similar to the canonical averaging, unrestrictedly increases as goes to zero and is largest for one particle. The most essential property of the Bose-Einstein ensemble is the formation, for a growing number of bosons, of the sharp asymmetric shape on the c N −T characteristics, which is more protrusive at the smaller Robin distances. This cusp-like structure is a manifestation of the phase transition to the condensate state. For two split-off orbitals, one additional maximum emerges whose position is shifted to colder temperatures with the increase of the energy gap between these two states and their higher-lying counterparts and whose magnitude approaches a -independent value. All these physical phenomena are qualitatively and quantitatively explained by the variation of the energy spectrum by the Robin distance. Parallels with other structures are drawn and similarities and differences between them are highlighted. Generalization to higher dimensions is also provided.

Theory of the Robin quantum wall in a linear potential. I. Energy spectrum, polarization and quantum‐information measures

2016

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Information-theoretical concepts are employed for the analysis of the interplay between a transverse electric field E applied to a one-dimensional surface and Robin boundary condition (BC), which with the help of the extrapolation length Λ 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 Schrödinger equation are derived and used for calculating energies, dipole moments, position Sx and momentum S k quantum information entropies and their Fisher information Ix and I k and Onicescu information energies Ox and O k counterparts. It is shown that the weak (strong) electric field changes the Robin wall into the Dirichlet, Λ = 0 (Neumann, Λ = ∞), 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 Sx + S k . Analogously, at Λ = 0 and Λ = ∞ the position and momentum Fisher informations (Onicescu energies) depend on the applied voltage as E 2/3 (E 1/3 ) and its inverse, respectively, leading to the field-independent product IxI k (OxO k ). Peculiarities of their transformations at the finite nonzero Λ 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.