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.     

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.     

Uniformization of nonlocal elliptic operators and KK-theory

By a pseudodifferential uniformization of a nonlocal elliptic operator we mean the procedure of reducing the operator to a pseudodifferential operator with a controlled modification of the index. In the paper, we suggest an approach to solving the uniformization problem; this approach uses the reduction of the symbol of a nonlocal operator to the symbol of a pseudodifferential operator. The technical apparatus here is Kasparov’s KK-theory.     

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.     

Strings in the operator formalism

We consider string theories in the operator formalism. In particular we develop Polyakov strings completely in the operator language. The operator approach turns out to be economical, self-contained and manifestly modular invariant.     

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.     

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.     

Introducing the general condition for an operator in curved space to be unitary

We investigate the general condition for an operator to be unitary. This condition is introduced according to the definition of the position operator in curved space. In a particular case, we discuss the concept of translation operator in curved space followed by its relation with an anti-Hermitian generator. Also we introduce a universal formula for adjoint of an arbitrary linear operator. Our procedure in this paper is totally different from others, as we explore a general approach based only on the algebra of the operators. Our approach is only discussed for the translation operators in one-dimensional space and not for general operators.     

Nijenhuis Operators on n-Lie Algebras

In this paper, we study (n-1)-order deformations of an n-Lie algebra and introduce the notion of a Nijenhuis operator on an n-Lie algebra, which could give rise to trivial deformations. We prove that a polynomial of a Nijenhuis operator is still a Nijenhuis operator. Finally, we give various constructions of Nijenhuis operators and some examples.     

Algebraic aspects of the Hirzebruch signature operator and applications to transitive Lie algebroids

The index of the classical Hirzebruch signature operator on a manifold M is equal to the signature of the manifold. The examples of Lusztig ([10], 1972) and Gromov ([4], 1985) present the Hirzebruch signature operator for the cohomology (of a manifold) with coefficients in a flat symmetric or symplectic vector bundle. In [6], we gave a signature operator for the cohomology of transitive Lie algebroids. In this paper, firstly, we present a general approach to the signature operator, and the above four examples become special cases of a single general theorem.     

On the algebra of symmetries of Laplace and Dirac operators

We consider a generalization of the classical Laplace operator, which includes the Laplace–Dunkl operator defined in terms of the differential-difference operators associated with finite reflection groups called Dunkl operators. For this Laplace-like operator, we determine a set of symmetries commuting with it, in the form of generalized angular momentum operators, and we present the algebraic relations for the symmetry algebra. In this context, the generalized Dirac operator is then defined as a square root of our Laplace-like operator. We explicitly determine a family of graded operators which commute or anticommute with our Dirac-like operator depending on their degree. The algebra generated by these symmetry operators is shown to be a generalization of the standard angular momentum algebra and the recently defined higher-rank Bannai–Ito algebra.     

Representation of the coherent state for a beam splitter operator and its applications

A beam splitter operator is a very important linear device in the field of quantum optics and quantum information. It can not only be used to prepare complete representations of quantum mechanics, entangled state representation, but it can also be used to simulate the dissipative environment of quantum systems. In this paper, by combining the transform relation of the beam splitter operator and the technique of integration within the product of the operator, we present the coherent state representation of the operator and the corresponding normal ordering form. Based on this, we consider the applications of the coherent state representation of the beam splitter operator, such as deriving some operator identities and entangled state representation preparation with continuous-discrete variables. Furthermore, we extend our investigation to two single and two-mode cascaded beam splitter operators, giving the corresponding coherent state representation and its normal ordering form. In addition, the application of a beam splitter to prepare entangled states in quantum teleportation is further investigated, and the fidelity is discussed. The above results provide good theoretical value in the fields of quantum optics and quantum information.     

On Bosonic Magnetic Flux Operator and Bosonic Faraday Operator Formula

In the literature about mesoscopic Josephson devices the magnetic flux is considered as an operator, the fundamental commutative relation between the magnetic flux operator and the Cooper-pair charge operator is usually preengaged. In this paper we show that such a relation can be deduced from the basic Bose operators＇ commutative relation through the entangled state representation. The Faraday formula in bosonic form is then equivalent to the second Josephson equation. The current operator equation for LC mesoscopic circuit is also derived.     

Shifts on a Deformed Hilbert Space

In this paper we observe that the creation operator on a deformed Hilbert spaceis the product of an ordinary shift and a diagonal operator.     

A Construction of Critical GJMS Operators Using Wodzicki's Residue

For an even dimensional, compact, conformal manifold without boundary we construct a conformally invariant differential operator of order the dimension of the manifold. In the conformally flat case, this operator coincides with the critical GJMS operator of Graham-Jenne-Mason-Sparling. We use the Wodzicki residue of a pseudo-differential operator of order −2, originally defined by A. Connes, acting on middle dimension forms.     

Hermitian Operators Conjugate to Two-Mode Number-Difference Operator Studied in Entangled State Representation

In similar to the derivation of phase angle operator conjugate to the number operator by Arroyo Carrasco-Moya Cessay we deduce the Hermitian phase operators that are conjugate to the two-mode number-difference operator and the three-mode number combination operator. It is shown that these operators are on the same footing in the entangled state representation as the one of Turski in the coherent state representation.     

探析算符运算问题

在量子力学中,利用算符可以较好地反映整体特征,利用它可以从状态(波函数)里提取力学性质;而解决或推导算符的关系时,除了要掌握各有关算符的意义和性质外,还应了解一些基本的算符运算方法.本文通过举例介绍了量子力学中算符常用的几种运算方法:直接法、作用法、参数微分法、待定算符法、积分变换法、数学归纳法等.     

Quantum Phase and Phase Difference for Fermion

In this paper, from the index theorem, we present the Hermitian phase operator for fermion, give its eigenvalues and orthogonal, normalized and complete eigenstates. Furthermore, we also get the phase difference operator for two fermions and discuss its quantum properties.     

Fractional Number Operator and Associated Fractional Diffusion Equations

In this paper, we study the fractional number operator as an analog of the finite-dimensional fractional Laplacian. An important relation with the Ornstein-Uhlenbeck process is given. Using a semigroup approach, the solution of the Cauchy problem associated to the fractional number operator is presented. By means of the Mittag-Leffler function and the Laplace transform, we give the solution of the Caputo time fractional diffusion equation and Riemann-Liouville time fractional diffusion equation in infinite dimensions associated to the fractional number operator.     

Radiation field eigenstates and its unitary equivalence to coordinate eigenstates

In this paper, we reconstruct the explicit representation of the radiation field eigenstates in Fock space by decomposing the normally ordered Gaussian operator. Then with the help of the technique of integration within an ordered product of operators, the phase-shifting operator in quantum optics has been expressed through the Dirac's representation theory. In addition, the unitarily equivalent relation between the radiation field eigenstates and the coordinate eigenstates has been naturally established by the phase-shifting operator in quantum optics. These results deepen people's understanding to the radiation field eigenstates and phase-shifting operator in quantum optics.