Deriving new operator identities by alternately using normally,antinormally,and Weyl ordered integration technique

Dirac's ket-bra formalism is the language of quantum mechanics.We have reviewed how to apply Newton-Leibniz integration rules to Dirac's ket-bra projectors in previous work.In this work,by alternately using the technique of integration within normal,antinormal,and Weyl ordering of operators we not only derive some new operator ordering identities,but also deduce some new integration formulas regarding Laguerre and Hermite polynomials.This may open a new route of directly deriving some complicated mathematical integration formulas by virtue of the quantum mechanical operator ordering technique,without really performing the integrations in the ordinary way.     

Simple approach to deriving operator identities and mathematical formulas regarding to two-variable Hermite polynomials

By virtue of operator ordering technique and the generating function of polynomials, we provide a simple and neat approach to studying operator identities and mathematical formulas regarding to two-variable Hermite polynomials, which differs from the existing mathematical ways. We not only derive some new integration formulas and summation relations about two-variable Hermite polynomial, but also draw a conclusion that two-variable Hermite polynomial excitation of two-mode squeezed vacuum state is a squeezed two-mode number state. This may open a new route of developing mathematics by virtue of the quantum mechanical representations and operator ordering technique.     

Multi-variable special polynomials using an operator ordering method

Using an operator ordering method for some commutative superposition operators, we introduce two new multi-variable special polynomials and their generating functions, and present some new operator identities and integral formulas involving the two special polynomials. Instead of calculating complicated partial differential, we use the special polynomials and their generating functions to concisely address the normalization, photocount distributions and Wigner distributions of several quantum states that can be realized physically, the results of which provide real convenience for further investigating the properties and applications of these states.     

SOME NEW QUANTUM OPERATOR FORMULAS --NEW APPLICATIONS OF THE COHERENT STATE

We present a novel method for deriving some new quantum operator formulas by carrying out,the integrations within a normal product symbol. The idea underlying this method is com- bining the properties of normal product and of coherent state together. For example, we can deduce operator ordering theorems, the Fujiwara formula, and some disentangling theorems simply and directly. Bymeans of this method we also get the explicit operator form and coherent state form for the Wigner function with these new forms, the wigner function's eigen-state is found and its zero point value property is discussed. Furthermore some famous quantum unitary transformations and the transformed matrix element for functional obtained conveniently. integral may be also     

光场位相算符和逆算符的Weyl编序展开

用有序算符内的积分技术, 推导了光场位相算符和逆算符的Weyl编序展开形式, 并利用该结果获得了相算符的经典对应以及某些新的特殊函数的生成函数和新的积分公式, 尤其是导出了带负次幂的复高斯积分的积分公式. 关键词： Weyl编序 位相算符 有序算符内的积分     

Sufficient Conditions of the Same State Order Induced by Coherence

In this paper, we study coherence-induced state ordering with Tsallis relative entropy of coherence, relative entropy of coherence and l1 norm of coherence, and give the sufficient conditions of the same state order induced by above coherence measures. First, we show that the above measures give the same ordering for single-qubit states in some special cases. Second, we consider some special states in a d-dimensional quantum system. We show that the above measures generate the same ordering for these special states. Finally, we discuss dynamics of coherence-induced state ordering under Markovian channels. We find amplitude damping channel changes the coherence-induced ordering even though for single-qubit states with fixed mixedness.     

Normal Ordering Expansion of Hermite Polynomials of Radial Coordinate Operator in Three-Dimensional Coordinate Space

For Hermite polynomials of radial coordinate operator in three-dimensional coordinate space we derive its normal ordering expansion, which are new operator identities. This is done by virtue of the technique of integration within an ordered product of operators. Application of the new formulas is briefly discussed.     

New Operator Identities and Integration Formulas Regarding to Hermite Polynomials Obtained via the Operator Ordering Method

Based on the technique of integration within an ordered product (IWOP) of operators we show that the operator ordering method can lead us to derive new operator identities and new integration formulas regarding to Hermite polynomials. Work supported by specialized research fund for the doctoral progress of higher education of China.     

δ-Function Operator Formulas of Photon Creation and Annihilation Operators

Using the technique of integration within an ordered product of operators we derive some &function operator formulas of photon creation and annihilation operators. Applications of these new formulas in simplifying some Louisell's operator theorems and in discussing the eigenket of creation operator are presented.     

Sufficient Conditions of the Same State Order Induced by Coherence

In this paper, we study coherence-induced state ordering with Tsallis relative entropy of coherence, relative entropy of coherence and l₁ norm of coherence, and give the sufficient conditions of the same state order induced by above coherence measures. First, we show that the above measures give the same ordering for single-qubit states in some special cases. Second, we consider some special states in a d-dimensional quantum system. We show that the above measures generate the same ordering for these special states. Finally, we discuss dynamics of coherence-induced state ordering under Markovian channels. We find amplitude damping channel changes the coherence-induced ordering even though for single-qubit states with fixed mixedness.     

Quantum affine algebras and deformations of the Virasoro and 237-1237-1237-1

Using the Wakimoto realization of quantum affine algebras we define new Poisson algebras, which areq-deformations of the classicalW. We also define their free field realizations, i.e. homomorphisms into some Heisenberg-Poisson algebras. The formulas for these homomorphisms coincide with formulas for spectra of transfer-matrices in the corresponding quantum integrable models derived by the Bethe-Ansatz method.Partially supported by NSF grants DMS-9205303 and DMS-9296120.     

Rich-club and page-club coefficients for directed graphs

Rich-club and page-club coefficients and their null models are introduced for directed graphs. Null models allow for a quantitative discussion of the rich-club and page-club phenomena. These coefficients are computed for four directed real-world networks: Arxiv High Energy Physics paper citation network, Web network (released from Google), Citation network among US Patents, and email network from a EU research institution. The results show a high correlation between rich-club and page-club ordering. For journal paper citation network, we identify both rich-club and page-club ordering, showing that “elite” papers are cited by other “elite” papers. Google web network shows partial rich-club and page-club ordering up to some point and then a narrow declining of the corresponding normalized coefficients, indicating the lack of rich-club ordering and the lack of page-club ordering, i.e. high in-degree (PageRank) pages purposely avoid sharing links with other high in-degree (PageRank) pages. For UC patents citation network, we identify page-club and rich-club ordering providing a conclusion that “elite” patents are cited by other “elite” patents. Finally, for email communication network we show lack of both rich-club and page-club ordering. We construct an example of synthetic network showing page-club ordering and the lack of rich-club ordering.     

Effects of pumped styles on power conversion efficiency and gain characteristics of phosphate glass Er3+–Yb3+-co-doped waveguide amplifiers

In order to analyze the power conversion efficiency (PCE) and the gain characteristics of the phosphate glass Er³⁺–Yb³⁺-co-doped waveguide amplifier (EYCDWA), formulas are derived from the rate equations and the light propagation equations under the uniform dopant and the steady-state conditions. In the derivation of these formulas, we have neglected the amplified spontaneous emission (ASE) and have introduced the initial energy transfer efficiency. By using these formulas, the effects of the pump power, signal power and waveguide length on the PCE and the gain characteristics of the EYCDWA are analyzed, and some useful results are obtained in three kinds of pumped styles, i.e. the forward-, backward- and bidirectional pumped styles.     

A microscopic mechanism for rejuvenation and memory effects in spin glasses

Aging in spin glasses (and in some other systems) reveals astonishing effects of `rejuvenation and memory' upon temperature changes. In this paper, we propose microscopic mechanisms (at the scale of spin-spin interactions) which can be at the origin of such phenomena. Firstly, we recall that, in a frustrated system, the effective average interaction between two spins may take different values (possibly with opposite signs) at different temperatures. We give simple examples of such situations, which we compute exactly. Such mechanisms can explain why new ordering processes (rejuvenation) seem to take place in spin glasses when the temperature is lowered. Secondly, we emphasize the fact that inhomogeneous interactions do naturally lead to a wide distribution of relaxation times for thermally activated flips. `Memory spots' spontaneously appear, in the sense that the flipping time of some spin clusters becomes extremely long when the temperature is decreased. Such memory spots are capable of keeping the memory of previous ordering at a higher temperature while new ordering processes occur at a lower temperature. After a qualitative discussion of these mechanisms, we show in the numerical simulation of a simplified example that this may indeed work. Our conclusion is that certain chaos-like phenomena may show up spontaneously in any frustrated and inhomogeneous magnetic system, without impeding the occurrence of memory effects. Received 5 February 2001 and Received in final form 27 April 2001     

On the “Matrix Approach” to Interacting Particle Systems

Derrida et al. and Schütz and Stinchcombe gave algebraic formulas for the correlation functions of the partially asymmetric simple exclusion process. Here we give a fairly general recipe of how to get these formulas and extend them to the whole time evolution (starting from the generator of the process), for a certain class of interacting systems. We then analyze the algebraic relations obtained to show that the matrix approach does not work with some models such as the voter and the contact processes.     

Magnetic Properties of CaMnO<Subscript>3</Subscript> Layers on the (100) Surface of BaTiO<Subscript>3</Subscript>

“Ab-initio” calculations of the electronic structure of the heterojunction (001) between cubic CaMnO₃ and BaTiO₃ perovskites are performed on the basis of density functional theory for different variants of magnetic ordering in calcium manganite. The paper considers some cases of ferromagnetic and antiferromagnetic A-type ordering. Comparison of the total energies of these structures shows that antiferromagnetic ordering in CaMnO3 is the most favorable. All the studied structures in a ferromagnetic state are half-metallic ferromagnets.     

Normal ordering and antinormal ordering of the operator （fQ＋gP）n and some of their applications

In this paper by virtue of the technique of integration within an ordered product (IWOP) of operators and the intermediate coordinate-momentum representation in quantum optics, we derive the normal ordering and antinormal ordering products of the operator (f Q + gP )n when n is an arbitrary integer. These products are very useful in calculating their matrix elements and expectation values and obtaining some useful mathematical formulae. Finally, the applications of some new identities are given.     

The Klein-Gordon Operator on Möbius Strip Domains and the Klein Bottle in ℝ n

In this paper we present explicit formulas for the fundamental solution to the Klein-Gordon operator on some higher dimensional generalizations of the Möbius strip and the Klein bottle with values in distinct pinor bundles. The fundamental solution is described in terms of generalizations of the Weierstraß $\wp $ -function that are adapted to the context of these geometries. The explicit formulas for the kernel then allow us to express all solutions to the homogeneous and inhomogeneous Klein-Gordon problem with given boundary data in the context of these manifolds. In the case of the Klein bottle we are able to describe all null solutions of the Klein-Gordon equation in terms of finite linear combinations of the fundamental solution and its partial derivatives.     

Planck's Spectral Distribution Law in N Dimensions

Planck's spectral distribution law is derived in N-dimensional space. Some relevant formulas are thus obtained and analyzed. The behavior of these formulas in higher dimensions is examined and some numerical values are calculated.     

基于坐标表象的脊波变换研究

在小波变换量子力学机制的启发下, 通过采用Fock空间里双模坐标本征态改写经典Ridgelet变换, 定义了量子光学态的Ridgelet变换. 然后利用IWOP技术给出不对称积分算符的显式, 并推导出了两个有用的双模算符正规乘积公式. 在此基础上, 通过选取双模“墨西哥帽”母小波函数, 分析了相干态、特殊压缩相干态、中介纠缠态表象的Ridgelet变换. 关键词： IWOP技术 Ridgelet变换 相干态