Effect of the velocity-dependent potentials on the energy eigenvalues of the Morse potential

We investigate the effect of the isotropic velocity-dependent potentials on the bound state energy eigenvalues of the Morse potential for any quantum states. When the velocity-dependent term is used as a constant parameter, ρ(r) = ρ ₀, the energy eigenvalues can be obtained analytically by using the Pekeris approximation. When the velocity-dependent term is considered as an harmonic oscillator type, ρ(r) = ρ ₀ r ², we show how to obtain the energy eigenvalues of the Morse potential without any approximation for any n and ℓ quantum states by using numerical calculations. The calculations have been performed for different energy eigenvalues and different numerical values of ρ ₀, in order to show the contribution of the velocity-dependent potential on the energy eigenvalues of the Morse potential.     

Bistability of quantum magnetotransport in a multilayer Ge/p-Ge1−x Six heterostructure with wide potential wells

Two metastable states of a multilayer Ge/p-Ge_1−x Si_x heterosystem with wide (∼ 35 nm) potential wells (Ge) are observed in strong magnetic fields B at low temperatures. In the first state, the Hall resistivity exhibits an inflection near the value ρ_xy=h/e ² scaled to one Ge layer. The longitudinal magnetoresistivity ρ_xx(B) possesses a minimum in the range of fields where this inflection occurs. The temperature evolution of the inflection in ρ_xy(B), the minimum of ρ _xx(B), and the value of ρ_xy at the inflection indicates a weakly expressed state of the quantum Hall effect with a uniform current distribution over the layers. In the second metastable state, an unusually wide plateau near h/2e ² with a very weak field dependence is observed in ρ_xy(B). Estimates show that in these samples the Fermi level lies below but close to the top of the inflection in the bottom of the well. For this reason, the second state can be explained by separation of a hole gas in the Ge layers into two sublayers, and the saturation of ρ_xy(B) near h/2e ² can be explained by the formation of a quantum Hall insulator state. Pis’ma Zh. éksp. Teor. Fiz. 70, No. 4, 290–297 (25 August 1999)     

Quantum oscillations of the resistivity and hall coefficient and the quantum limit in Bi<Subscript>0.93</Subscript>Sb<Subscript>0.07</Subscript> alloys in a magnetic field along the trigonal axis

The dependences of the electrical resistivity ρ and the Hall coefficient R on the magnetic field have been measured for single-crystal samples of the n-Bi_0.93Sb_0.07 semiconductor alloys with electron concentrations in the range 1 × 10¹⁶ cm⁻³ < n < 2 × 10¹⁸ cm⁻³. It has been found that the measured dependences exhibit Shubnikov-de Haas quantum oscillations. The magnetic fields corresponding to the maxima of the quantum oscillations of the electrical resistivity are in good agreement with the calculated values of the magnetic fields in which the Landau quantum level with the number N intersects the Fermi level. The quantum oscillations of the Hall coefficient with small numbers are characterized by a significant spin splitting. In a magnetic field directed along the trigonal axis, the quantum oscillations of the resistivity ρ and the Hall coefficient R are associated with electrons of the three-valley semiconductor and are in phase with the magnetic field. In the case of a magnetic field directed parallel to the binary axis, the quantum oscillations associated both with electrons of the secondary ellipsoids in weaker magnetic fields and with electrons of the main ellipsoid in strong magnetic fields (after the overflow of electrons from the secondary ellipsoids to the main ellipsoid) are also in phase. In magnetic fields of the quantum limit ħω _c /2 ≥ E _F, the electrical conductivity increases with an increase in the magnetic field: σ₂₂(H) ∼ H ^k . A theoretical evaluation of the exponent in this expression for a nonparabolic semiconductor leads to values of k close to the experimental values in the range 4 ≤ k ≤ 4.6, which were obtained for samples of the semiconductor alloys with different electron concentrations. A further increase in the magnetic field results in a decrease of the exponent k and in the transition to the inequality σ₂₂(H) ≤ σ₂₁(H).     

Local Asymptotic Normality for Finite Dimensional Quantum Systems

Previous results on local asymptotic normality (LAN) for qubits [16, 19] are extended to quantum systems of arbitrary finite dimension d. LAN means that the quantum statistical model consisting of n identically prepared d-dimensional systems with joint state converges as n → ∞ to a statistical model consisting of classical and quantum Gaussian variables with fixed and known covariance matrix, and unknown means related to the parameters of the density matrix ρ. Remarkably, the limit model splits into a product of a classical Gaussian with mean equal to the diagonal parameters, and independent harmonic oscillators prepared in thermal equilibrium states displaced by an amount proportional to the off-diagonal elements. As in the qubits case [16], LAN is the main ingredient in devising a general two step adaptive procedure for the optimal estimation of completely unknown d-dimensional quantum states. This measurement strategy shall be described in a forthcoming paper [18].     

Evaluated cross sections of the σ(γ, nX) and σ(γ, 2nX) reactions on 112, 114, 116, 117, 118, 119, 120, 122, 124Sn isotopes

A combined analysis of experimental data on total and partial photoneutron reaction cross sections, obtained using bremsstrahlung γ-radiation and quasi-monoenergetic annihilation photon beams, was performed for nine Sn isotopes. The partial reactions σ^eval(γ, nX) and σ^eval(γ, 2nX) cross sections were evaluated using an approach free of the shortcomings of experimental neutron multiplicity sorting methods. This approach involves calculations within the photonuclear reaction model, based on Fermi gas densities and considering the effects of nucleus deformation, the isospin splitting of its giant dipole resonance (GDR), and experimental data on the total photoneutron cross sections σ^exp(γ, xn) = σ^exp(γ, nX) + 2σ^exp(γ, 2nX) = σ^exp(γ, n) + σ^exp(γ, np) + … + 2σ^exp(γ, 2n) + 2σ^exp(γ, 2np) + …. The evaluated σ^eval(γ, nX) and σ^eval(γ, 2nX) reactions cross sections were obtained using the introduced transition multiplicity functions F ^theor = σ^theor(γ, 2nX)/σ^theor(γ, xn) = σ^theor(γ, 2nX)/[σ^theor(γ, nX) + 2σ^theor(γ, 2nX) + …]; and σ^eval.(γ, 2nX) = F ^theor.σ^exp.(γ, xn) = σ^eval(γ, nX) = (1 − 2F ^theor)σ^exp(γ, xn). The evaluated partial reaction cross sections were used to assess the total photoneutron reaction cross sections σ^eval(γ, sn) = σ^eval(γ, nX) + σ^eval(γ, 2nX) + … as functions of the mass number A. The GDR features of ^{112, 114, 116, 117, 118, 119, 120, 122, 124}Sn isotopes were studied and are discussed here.     

New data for the 197Au(γ, nX) and 197Au(γ, 2nX) reaction cross sections

The results from experimental and theoretical studies of the total and partial cross sections of photoneutron reactions on the ¹⁹⁷Au isotope were analyzed. The cross sections for reactions σ(γ, nX) = σ(γ, n) + σ(γ, np) + … + σ(γ, 2nX) = σ(γ, 2n) + σ(γ, 2np) + … were evaluated in the energy range 7 ≤ E _γ ≤ 30 MeV using an approach free of the shortcomings of experimental photoneutron multiplicity sorting methods. The total photoneutron reaction cross sections σ^exp(γ, xn) = σ^exp(γ, nX) + 2σ^exp(γ, 2nX) + … = σ^exp(γ, n) + σ^exp(γ, np) + 2σ^exp(γ, 2n) + 2σ^exp(γ, 2np) + … were used as the initial experimental data. The contributions from the cross sections σ(γ, nX) and σ(γ, 2nX) to the cross sections σ^exp(γ, xn) were separated using the multiplicity transition functions F ₁ ^theor = σ^theor(γ, 1nX)/σ^theor(γ, xn) and F ₂ ^theor = σ^theor(γ, 2nX)/σ^theor(γ, xn), calculated within an updated version of the pre-equilibrium model of photonuclear reactions. New evaluated data for both partial reaction cross sections, i.e., σ^eval (γ, 1nX) = F ₁ ^theorσ^exp(γ, xn) and σ^eval(γ, 2nX) = F ₂ ^theorσ^exp(γ, xn), were obtained. The cross sections σeval(γ, nX) and σ^eval.(γ, 2nX) evaluated using the theoretically calculated functions F _1,2^theor are consistent with the Livermore data, but substantially contradict the Saclay data.     

Observational constraints on decaying vacuum dark energy model

The decaying vacuum model (DV), treating dark energy as a varying vacuum, has been studied well recently. The vacuum energy decays linearly with the Hubble parameter in the late-times, ρ _Λ (t)∝H(t), and produces the additional matter component. We constrain the parameters of the DV model using the recent data-sets from supernovae, gamma-ray bursts, baryon acoustic oscillations, CMB, the Hubble rate and X-rays in galaxy clusters. It is found that the best fit of the matter density contrast Ω _m in the DV model is much lager than that in ΛCDM model. We give the confidence contours in the Ω _m –h plane up to 3σ confidence level. Besides, the normalized likelihoods of Ω _m and h are presented, respectively.     

Quadratic spherical bessel functions and the inverse scattering problem

In the course of inverting the partial-wave Born approximation, a new expression for the inverse function ofj _l ² (ρ) was obtained. Using this result, one can also derive two expressions involving the binomial coefficients. Finally, a particular differential operator whose effect onj _l ² (ρ) was previously investigated by Mavromatis and Jalal is shown to have similar effects onn _l ² (ρ) andn _l (ρ)j _l (ρ).     

Generators of KMS Symmetric Markov Semigroups on {\mathcal{B}({\rm h})} Symmetry and Quantum Detailed Balance

We find the structure of generators of norm-continuous quantum Markov semigroups on B(h){\mathcal{B}({\rm h})} that are symmetric with respect to the scalar product tr (ρ ^1/2 x*ρ ^1/2 y) induced by a faithful normal invariant state ρ and satisfy two quantum generalisations of the classical detailed balance condition related with this non-commutative notion of symmetry: the so-called standard detailed balance condition and the standard detailed balance condition with an antiunitary time reversal.     

Dynamics of atom scattering from 4He nanoclusters

We present a microscopic theory and results of atom scattering calculations to determine the dispersion of surface modes (ripplons) of superfluid helium-4 nanodroplets, expanding previous work [J. Chem. Phys. 115, 10161 (2001)]. A quantum transport formalism is adapted to the many-body scattering problem, yielding both elastic and inelastic fluxes. We demonstrate that, in analogy to the dynamic structure function S(k,ω) obtained from neutron scattering, a dynamic structure function σ(k,ω) can be obtained from ³He scattering. The ³He dynamic structure function σ(k,ω) is sensitive to surface dynamics, whereas the neutron dynamic structure function S(k,ω) is dominated by bulk-like excitations, in particular by rotons. Unlike for neutron-scattering, the total inelastic cross section for atom-scattering on ⁴He nanodroplets is large which we believe makes experimental detection feasible. We also show that scattering identical particles, i.e. ⁴He atoms, does not provide information about the dispersion of surface modes. Instead, inelastically scattered ⁴He atoms preferably lose roughly half their energy.     

Zero bias anomaly in the density of states of low-dimensional metals

We consider the effect of Coulomb interactions on the average density of states (DOS) of disordered low-dimensional metals for temperatures T and frequencies ω smaller than the inverse elastic life-time 1/τ. Using the fact that long-range Coulomb interactions in two dimensions (2d) generate ln²-singularities in the DOS ν(ω) but only ln-singularities in the conductivity σ(ω), we can re-sum the most singular contributions to the average DOS via a simple gauge-transformation. If σ(ω) > 0, then a metallic Coulomb gapν(ω) ∝ |ω|/e ⁴ appears in the DOS at T = 0 for frequencies below a certain crossover frequency Ω ₂ which depends on the value of the DC conductivity σ(0). Here, - e is the charge of the electron. Naively adopting the same procedure to calculate the DOS in quasi 1d metals, we find ν(ω) ∝ (|ω|/Ω ₁)^1/2exp(- Ω ₁/|ω|) at T = 0, where Ω ₁ is some interaction-dependent frequency scale. However, we argue that in quasi 1d the above gauge-transformation method is on less firm grounds than in 2d. We also discuss the behavior of the DOS at finite temperatures and give numerical results for the expected tunneling conductance that can be compared with experiments. Received 28 August 2001 / Received in final form 28 January 2002 Published online 9 July 2002     

Tensor product composition of algebras in Bose and Fermi canonical formalism

We consider a graded algebra with two products (σ, α) over anε-factor commutation. One of the products (σ) isε-commutative, but, in general non-associative; and the other (α) is a graded Lieε-product and a gradedε-derivative with respect to the first (σ). Using the obvious mathematical condition, namely—the tensor product of two graded algebras with the sameε-factors is another with the sameε-factor, we determine the complete structure of a two-product (σ, α) graded algebra. When theε-factors are taken to be unity and the gradation structure is ignored, we recover the algebras of the physical variables of classical and quantum systems, considered by Grgin and Petersen. With the retention of the gradation structure and the possible choice of two ε-factors we recover the algebras of the canonical formalism of boson and fermion systems for the above classical and quantum theories. We also recover in this case the algebra of anticommutative classical systems considered by Martin along with its quantum analogue.     

Probabilistic Averages of Jacobi Operators

I study the Lyapunov exponent and the integrated density of states for general Jacobi operators. The main result is that questions about these can be reduced to questions about ergodic Jacobi operators. I use this to show that for finite gap Jacobi operators, regularity implies that they are in the Cesàro–Nevai class, proving a conjecture of Barry Simon. Furthermore, I use this to study Jacobi operators with coefficients a(n) = 1 and b(n) = f(n ^ρ (mod 1)) for ρ > 0 not an integer.     

Two-site quantum random walk

We study the measure theory of a two-site quantum random walk. The truncated decoherence functional defines a quantum measure μ _n on the space of n-paths, and the μ _n in turn induce a quantum measure μ on the cylinder sets within the space Ω of untruncated paths. Although μ cannot be extended to a continuous quantum measure on the full σ-algebra generated by the cylinder sets, an important question is whether it can be extended to sufficiently many physically relevant subsets of Ω in a systematic way. We begin an investigation of this problem by showing that μ can be extended to a quantum measure on a “quadratic algebra” of subsets of Ω that properly contains the cylinder sets. We also present a new characterization of the quantum integral on the n-path space.     

Typicality of Pure States Randomly Sampled According to the Gaussian Adjusted Projected Measure

Consider a mixed quantum mechanical state, describing a statistical ensemble in terms of an arbitrary density operator ρ of low purity, tr ρ ² ≪1, and yielding the ensemble averaged expectation value tr (ρ A) for any observable A. Assuming that the given statistical ensemble ρ is generated by randomly sampling pure states |ψ〉 according to the corresponding so-called Gaussian adjusted projected measure (Goldstein et al. in J. Stat. Phys. 125:1197, 2006), the expectation value 〈ψ|A|ψ〉 is shown to be extremely close to the ensemble average tr (ρ A) for the overwhelming majority of pure states |ψ〉 and any experimentally realistic observable A. In particular, such a ‘typicality’ property holds  whenever the Hilbert space ℋ of the system contains a high dimensional subspace ℋ₊⊂ℋ with the property that all |ψ〉∈ℋ₊ are realized with equal probability and all other |ψ〉∈ℋ are excluded.     

Equation of state for the Universe from similarity symmetries

In this paper we proposed to use the group of analysis of symmetries of the dynamical system to describe the evolution of the Universe. This method is used in searching for the unknown equation of state. It is shown that group of symmetries enforce the form of the equation of state for noninteracting scaling multifluids. We showed that symmetries give rise to the equation of state in the form p =-Λ + w ₁ρ(a) + w ₂ a ^β + 0 and energy density ρ = Λ+ρ₀₁ a ^-3(1+w) +ρ₀₂ a ^α +ρ₀₃ a ^-3, which is commonly used in cosmology. The FRW model filled with scaling fluid (called homological) is confronted with the observations of distant type Ia supernovae. We found the class of model parameters admissible by the statistical analysis of SNIa data.We showed that the model with scaling fluid fits well to supernovae data. We found that Ω_m,0 ≃ 0.4 and n ≃ -1 (β = -3n), which can correspond to (hyper) phantom fluid, and to a high density universe. However if we assume prior that Ω_m,0 = 0.3 then the favoured model is close to concordance ΛCDM model. Our results predict that in the considered model with scaling fluids distant type Ia supernovae should be brighter than in the ΛCDM model, while intermediate distant SNIa should be fainter than in the ΛCDM model. We also investigate whether the model with scaling fluid is actually preferred by data over ΛCDM model. As a result we find from the Akaike model selection criterion: it prefers the model with noninteracting scaling fluid.     

(GL(<Emphasis Type="Italic">n</Emphasis> + 1, ℝ), GL(<Emphasis Type="Italic">n</Emphasis>, ℝ)) is a generalized Gelfand pair

Denote by G = GL(n + 1, ℝ) the group of invertible (n + 1) × (n + 1) matrices with real entries, acting on ℝ ⁿ⁺¹ in the usual way, and let H ₁ = GL(n, ℝ) be the stabilizer of the first unit vector e ₀. Let H ₀ = GL(1, ℝ) and set H = H ₀ × H ₁. It is known that the pair (G,H) is a generalized Gelfand pair. Define a character χ of H by χ(h) = χ(h ₀ h ₁) = χ0(h ₀) where χ0 is a unitary character of H ₀ (h ₀ ∈ H ₀, h ₁ ∈ H ₁). Let σ be the anti-involution on G given by σ(g) = ^t g. In this note, we show that any distribution T on G satisfying T(h ₁ gh ₂) = χ(h ₁ h ₂) T(g) (g ∈ G; h ₁, h ₂ ∈ H) is invariant under the anti-involution σ. This result implies that (G,H ₁) is a generalized Gelfand pair.     

Optimal Estimation of Qubit States with Continuous Time Measurements

We propose an adaptive, two step strategy, for the estimation of mixed qubit states. We show that the strategy is optimal in a local minimax sense for the trace norm distance as well as other locally quadratic figures of merit. Local minimax optimality means that given n identical qubits, there exists no estimator which can perform better than the proposed estimator on a neighborhood of size n ^−1/2 of an arbitrary state. In particular, it is asymptotically Bayesian optimal for a large class of prior distributions. We present a physical implementation of the optimal estimation strategy based on continuous time measurements in a field that couples with the qubits. The crucial ingredient of the result is the concept of local asymptotic normality (or LAN) for qubits. This means that, for large n, the statistical model described by n identically prepared qubits is locally equivalent to a model with only a classical Gaussian distribution and a Gaussian state of a quantum harmonic oscillator. The term ‘local’ refers to a shrinking neighborhood around a fixed state ρ ₀. An essential result is that the neighborhood radius can be chosen arbitrarily close to n ^−1/4. This allows us to use a two step procedure by which we first localize the state within a smaller neighborhood of radius n ^−1/2+ϵ, and then use LAN to perform optimal estimation.     

Biexcitonic absorption in a disk-shaped GaN quantum dot

The present work investigates the nonlinear optical properties of a GaN quantum dot in the disk limit via the exciton and biexciton states using the compact density matrix formalism. Based on this model, we calculate the ground state energy of the exciton and biexciton states by the variation method, within envelope function and effective mass approximations. Linear and nonlinear optical absorption (α ⁽¹⁾, α ⁽³⁾) and oscillator strengths attributed to the optical transitions are obtained. The details of the behaviour of α ⁽¹⁾ and α ⁽³⁾ around the resonance frequencies and for different quantum dot geometries are presented. It is found that the size of quantum dot and the optical intensity have a remarkable effect on the optical absorption, and the biexcitonic two-photon absorption coefficient(K ₂) has also been calculated in this system. The results show that this parameter is strongly affected by the size of the quantum dot.     

Correction to Entropy of Schwarzschild Black Hole by Modified Dispersion Relation

Taking WKB approximation to solve the scalar field equation in the Schwarzschild black hole spacetime, we can get the classical momenta. Substituting the classical momenta into state density equation corrected by the modified dispersion relation, we will obtain the number of quantum states with energy less than ω. Then, it is used to calculate the statistical-mechanical entropy of the scalar field in the Schwarzschild black hole spacetime. By taking exact method, we obtained the leader term of entropy which is proportional to the event horizon area and correction terms take the forms of ln A, A ⁻¹ln A, A ⁻¹ and so on.