Shannon information entropies for rectangular multiple quantum well systems with constant total lengths

We first study the Shannon information entropies of constant total length multiple quantum well systems and then explore the effects of the number of wells and confining potential depth on position and momentum information entropy density as well as the corresponding Shannon entropy.We find that for small full width at half maximum(FWHM) of the position entropy density,the FWHM of the momentum entropy density is large and vice versa.By increasing the confined potential depth,the FWHM of the position entropy density decreases while the FWHM of the momentum entropy density increases.By increasing the potential depth,the frequency of the position entropy density oscillation within the quantum barrier decreases while that of the position entropy density oscillation within the quantum well increases.By increasing the number of wells,the frequency of the position entropy density oscillation decreases inside the barriers while it increases inside the quantum well.As an example,we might localize the ground state as well as the position entropy densities of the1 st,2 nd,and 6 th excited states for a four-well quantum system.Also,we verify the Bialynicki–Birula–Mycieslki(BBM)inequality.     

Tripartite Entanglement Measures of Generalized GHZ State in Uniform Acceleration

Using the single-mode approximation,we study entanglement measures including two independent quantities;i.e.,negativity and von Neumann entropy for a tripartite generalized Greenberger-Horne-Zeilinger(GHZ)state in noninertial frames.Based on the calculated negativity,we study the whole entanglement measures named as the algebraic average π3-tangle and geometric average Π3-tangle.We find that the difference between them is very small or disappears with the increase of the number of accelerated qubits.The entanglement properties are discussed from one accelerated observer and others remaining stationary to all three accelerated observers.The results show that there will always exist entanglement,even if acceleration r arrives to infinity.The degree of entanglement for all 1-1 tangles are alwa.ys equal to zero,but 1-2 tangles always decrease with the acceleration parameter r.We notice that the von Neumann entropy increases with the number of the accelerated observers and Sκ_Ιζ_Ι(κ,ζ∈(A,B,C)) first increases and then decreases with the acceleration parameter r.This implies that the subsystem ρκΙζΙ is first more disorder and then the disorder will be reduced as the acceleration parameter r increases.Moreover,it is found that the von Neumann entropies SABCI,SABICI and SAIBICI always decrease with the controllable angle θ,while the entropies of the bipartite subsystems S2-2_(non)(two accelerated qubits),S2-1_(non)(one accelerated qubit) and S2-0_(non)(without accelerated qubit) first increase with the angle θ and then decrease with it.     

Tetrapartite entanglement measures of generalized GHZ state in the noninertial frames

Using a single-mode approximation, we carry out the entanglement measures, e.g., the negativity and von Neumann entropy when a tetrapartite generalized GHZ state is treated in a noninertial frame, but only uniform acceleration is considered for simplicity. In terms of explicit negativity calculated, we notice that the difference between the algebraic average $\pi_{4}$ and geometric average $\varPi_{4}$ is very small with the increasing accelerated observers and they are totally equal when all four qubits are accelerated simultaneously. The entanglement properties are discussed from one accelerated observer to all four accelerated observers. It is shown that the entanglement still exists even if the acceleration parameter $r$ goes to infinity. It is interesting to discover that all 1-1 tangles are equal to zero, but 1-3 and 2-2 tangles always decrease when the acceleration parameter $r$ increases. We also study the von Neumann entropy and find that it increases with the number of the accelerated observers. In addition, we find that the von Neumann entropy $S_{\text{ABCDI}}$, $S_{\text{ABCIDI}}$, $S_{\text{ABICIDI}}$ and $S_{\text{AIBICIDI}}$ always decrease with the controllable angle $\theta$, while the entropies $S_{3-3~\rm non}$, $S_{3-2~\rm non}$, $S_{3-1~\rm non}$ and $S_{3-0~\rm non}$ first increase with the angle $\theta$ and then decrease with it.     

The Origin and Mathematical Characteristics of the Super-Universal Associated-Legendre Polynomials

We study the mathematical characteristics of the super-universal associated-Legendre polynomials arising from a kind of double ring-shaped potentials and obtain their polar angular wave functions with certain parity. We find that there exists the even or odd parity for the polar angular wave functions when the parameter η is taken to be positive integer, while there exist both even and odd parities when η is taken as positive non-integer real values. The relations among the super-universal associated-Legendre polynomials, the hypergeometric polynomials, and the Jacobi polynomials are also established.     

研究分子振动光谱的Fermi共振代数模型

提出了一种描述多原子分子振动能谱的Fermi共振U( 2 )代数模型 .在这一模型中 ,用U( 2 )代数描述键的振动 ,并考虑了伸缩振动和弯曲振动间的Fermi共振相互作用 .成功地把这一模型应用到H₂ O和AsH₃ 的最新观测的振动谱 ,并与其他模型的结果进行了比较 .计算结果表明代数模型能以较小的标准偏差描述分子的振动谱     

Exact solutions of the Schrödinger equation for a class of hyperbolic potential well

We propose a new scheme to study the exact solutions of a class of hyperbolic potential well. We first apply different forms of function transformation and variable substitution to transform the Schrödinger equation into a confluent Heun differential equation and then construct a Wronskian determinant by finding two linearly dependent solutions for the same eigenstate. And then in terms of the energy spectrum equation which is obtained from the Wronskian determinant, we are able to graphically decide the quantum number with respect to each eigenstate and the total number of bound states for a given potential well. Such a procedure allows us to calculate the eigenvalues for different quantum states via Maple and then substitute them into the wave function to obtain the expected analytical eigenfunction expressed by the confluent Heun function. The linearly dependent relation between two eigenfunctions is also studied.     

Analytic solutions of the double ring-shaped Coulomb potential in quantum mechanics

The exact solutions of the Schr6dinger equation with the double ring-shaped Coulomb potential are presented, including the bound states, continuous states on the ＂k/2π scale＂, and the calculation formula of the phase shifts. The polar angular wave functions are expressed by constructing the so-called super-universal associated Legendre polynomials. Some special cases are discussed in detail.     

Semi-exact Solutions of Konwent Potential

In this work we study the quantum system with the symmetric Konwent potential and show how to find its exact solutions. We find that the solutions are given by the confluent Heun function. The eigenvalues have to be calculated numerically because series expansion method does not work due to the variable z ≥ 1. The properties of the wave functions depending on the potential parameter A are illustrated for given potential parameters V_0 and a. The wave functions are shrunk towards the origin with the increasing |A|. In particular, the amplitude of wave function of the second excited state moves towards the origin when the positive parameter A decreases. We notice that the energy levels ε_i increase with the increasing potential parameter |A| ≥ 1, but the variation of the energy levels becomes complicated for |A| ∈(0, 1), which possesses a double well. It is seen that the energy levels ε_i increase with |A| for the parameter interval A ∈(-1, 0), while they decrease with |A| for the parameter interval A ∈(0, 1).     

Morse Potential in the Momentum Representation

The momentum representation of the Morse potential is presented analytically by hypergeometric function. The properties with respect to the momentum p and potential parameter β are studied. Note that | Ψ(p) | is a nodeless function and the mutual orthogonality of functions is ensured by the phase functions arg[Ψ(p)]. It is interesting to see that the | Ψ(p) | is symmetric with respect to the axis p = 0 and the number of wave crest of | Ψ(p) | is equal to n + 1. We also study the variation of | Ψ(p) | with respect to β. The amplitude of | Ψ(p) | first increases with the quantum number n and then deceases. Finally, we notice that the discontinuity in phase occurs at some points of the momentum p and the position and momentum probability densities are symmetric with respect to their arguments.     

Quantum information entropy for one-dimensional system undergoing quantum phase transition

Calculations of the quantum information entropy have been extended to a non-analytically solvable situation. Specifically, we have investigated the information entropy for a one-dimensional system with a schematic "Landau" potential in a numerical way. Particularly, it is found that the phase transitional behavior of the system can be well expressed by the evolution of quantum information entropy. The calculated results also indicate that the position entropy S_x and the momentum entropy S_p at the critical point of phase transition may vary with the mass parameter M but their sum remains as a constant independent of M for a given excited state. In addition, the entropy uncertainty relation is proven to be robust during the whole process of the phase transition.