New decomposition of the Fresnel operator corresponding to the optical transformation in AB CD-systems

By virtue of the coherent state representation of the newly introduced Fresnel operator and its group product property we obtain new decomposition of the Fresnel operator as the product of the quadratic phase operator, the squeezing operator, and the fractional Fourier transformation operator, which in turn sheds light on the matrix optics design of ABCD-systems The new decomposition for the two-mode Fresnel operator is also obtained by the use of entangled state representation.     

Operator Fredholm Integration Equation in Intermediate Coordinate-Momentum Representation and Its Analogue to Thermo Conduction Equation

Using the technique of integration within an ordered product （IWOP） of operators we construct intermediate coordinate-momentum representation, with which we build a type of operator Fredholm integration equation that is an operator generalization of the solution of thermo conduction equation. Then we seach for the solution of operator Fredholm integration equations, which provides us with a new approach for deriving some operator identities.     

Squeezed displaced Wigner operator and its applications

In this paper, we introduce the squeezed displaced Wigner operator. We proved that the squeezed displaced Wigner operator can bring more convenience to calculate the Wigner functions of squeezed states. Finally, we give some new applications of such squeezed displaced Wigner operator.     

New fractional derivative with sigmoid function as the kernel and its models

Based on the idea of the fractional derivative with respect to another function, a new fractional derivative operator with sigmoid function as the kernel in this article, is proposed for the first time. Then, we make use of this new fractional operator to model various nonlinear phenomena from different fields of applications in science, such as the population growth, the shallow water wave phenomena and reaction-diffusion processes, and so on. As a result, we hope that the new fractional operator can be used to discover more evolutionary mechanisms of these phenomena.     

Quantum mechanics in discrete space and angular momentum

Recently we have studied quantum mechanics of bounded operators with a discrete spectrum. In particular, we derived an expression for the commutator[Q, P] of two bounded operators whose spectrum is discrete, and we showed that in the limit of a continuous spectrum the commutator becomes the standard one of Heisenberg. In this paper we show that the angular momentum operator and the phase operator satisfy the new commutation relation. We also briefly discuss the problem of the canonical phase operator conjugate to the number operator.     

Detaching two single-mode squeezing operators from the two-mode squeezing operator

It has been common knowledge that the single-mode squeezing operator and the two-mode squeezing operator are independent of each other. However, in this work we find that after using the technique of integration within Ω-ordering and β-ordering, we can detach two single-mode squeezing operators from the two-mode squeezing operator. In other words, we show that the two-mode squeezing operator can be split into a β-ordered two-mode squeezing operator （with a new squeezing parameter） and two single-mode squeezing operators （with another squeezing parameter）. This tells us that the two-mode squeezing mechanism also involves some single-mode squeezing.     

The Nondynamical r-Matrix Structure of the Elliptic Calogero–Moser Model

In this Letter, we construct a new Lax operator for the elliptic Calogero–Moser model with N=2. The nondynamical r-matrix structure of this Lax operator is also studied. The relation between our Lax operator and the Lax operator given by Krichever is also obtained.     

Normally Ordered Operator Fredholm Equation and Its Applications in Quantum Mechanics

By virtue of the technique of integration within an ordered product of operators we construct the normally ordered operator fiedholm equation. We use it to derive some new operator formulas. For Weyl correspondence, operator fiedholm equation can also be constructed. Some applications of the operator Fkedholm equation are given.     

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.     

New optical field operator expansion in number state representation

By virtue of the Weyl ordering method,we find a new formalism of optical field operator expansion in number state representation.Miscellaneous optical fields’(coherent state,squeezed field,Wigner operator,etc.) new expansions are therefore exhibited.Some new generating functions of special polynomials are derived herewith.     

Relation between Characteristic Function of Density Operator and Tomogram

In terms of the intermediate coordinate-momentum representation （Chin. Phys. Lett. 18 （2001） 850） and using the technique of integration within an ordered product of operators, we put the tomography theory into operator version. We reveal the new relation between the tomogram and the characteristic function of the density operator. The new expansion of the density operator in terms of the intermediate coordinate-momentum representation is also obtained.     

Symmetries and Similarity Reductions of Nonlinear Diffusion Equation

The inverse recursion operator, three new sets of symmetries, and infinite-dimensional Lie algebras for the nonlinear diffusion equation are given. Some nonlocal symmetries related to eigenvectors of the recursion operator with the eigenvalue λi are also obtained with the help of the recursion operator φi=φ-λi. Using a part of these symmetries we get twelve types of nontrivial new similarity reduction.     

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.     

Fractional corresponding operator in quantum mechanics and applications: A uniform fractional Schrödinger equation in form and fractional quantization methods

In this paper we use Dirac function to construct a fractional operator called fractional corresponding operator, which is the general form of momentum corresponding operator. Then we give a judging theorem for this operator and with this judging theorem we prove that R–L, G–L, Caputo, Riesz fractional derivative operator and fractional derivative operator based on generalized functions, which are the most popular ones, coincide with the fractional corresponding operator. As a typical application, we use the fractional corresponding operator to construct a new fractional quantization scheme and then derive a uniform fractional Schrödinger equation in form. Additionally, we find that the five forms of fractional Schrödinger equation belong to the particular cases. As another main result of this paper, we use fractional corresponding operator to generalize fractional quantization scheme by using Lévy path integral and use it to derive the corresponding general form of fractional Schrödinger equation, which consequently proves that these two quantization schemes are equivalent. Meanwhile, relations between the theory in fractional quantum mechanics and that in classic quantum mechanics are also discussed. As a physical example, we consider a particle in an infinite potential well. We give its wave functions and energy spectrums in two ways and find that both results are the same.     

New Coherent States for the BDS-Hamiltonian

In the paper we construct a new set of coherent states for a deformed Hamiltonian of the harmonic oscillator, previously introduced by Beckers, Debergh, and Szafraniec, which we have called the BDS-Hamiltonian. This Hamiltonian depends on the new creation operator a ⁺, i.e. the usual creation operator displaced with the real quantity . In order to construct the coherent states, we use a new measure in the Hilbert space of the Hamiltonian eigenstates, in fact we change the inner product. This ansatz assures that the set of eigenstates be orthonormalized and complete. In the new inner product space the BDS-Hamiltonian is self-adjoint. Using these coherent states, we construct the corresponding density operator and we find the P-distribution function of the unnormalized density operator of the BDS-Hamiltonian. Also, we calculate some thermal averages related to the BDS-oscillators system which obey the quantum canonical distribution conditions.     

Optical complex integration-transform for deriving complex fractional squeezing operator

We find a new complex integration-transform which can establish a new relationship between a two-mode operator's matrix element in the entangled state representation and its Wigner function. This integration keeps modulus invariant and therefore invertible. Based on this and the Weyl–Wigner correspondence theory, we find a two-mode operator which is responsible for complex fractional squeezing transformation. The entangled state representation and the Weyl ordering form of the two-mode Wigner operator are fully used in our derivation which brings convenience.     

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     

The length operator in Loop Quantum Gravity

The dual picture of quantum geometry provided by a spin network state is discussed. From this perspective, we introduce a new operator in Loop Quantum Gravity—the length operator. We describe its quantum geometrical meaning and derive some of its properties. In particular we show that the operator has a discrete spectrum and is diagonalized by appropriate superpositions of spin network states. A series of eigenstates and eigenvalues is presented and an explicit check of its semiclassical properties is discussed.     

Thermal Wigner Operator in Coherent Thermal State Representation and Its Application

In the coherent thermal state representation we introduce thermal Wigner operator and find that it is “squeezed” under the thermal transformation.The thermal Wigner operator provides us with a new direct and neat approach for deriving Wigner functions of thermal states.