Number-Difference Orthonormal Representation

We study the properties of two-mode phase operator √(a⁺+b)/(a+b⁺) and the phase state by constructing the number-difference orthonormal state |q, г). We prove that |q, г ) constitutes a complete and orthonormal representation (number-difference representation). We show that two-mode phase operator √(a⁺+b)/(a+b⁺) exhibits its phase behavior explicitly in the number-difference representation, which resembles the single-mode phase operator e^iψ ≡ (1/√a⁺a+1)a. The corresponding phase state is also studied.     

Recapitulating Quantum Phase Space Representation by Virtue of Normally Ordered Gaussian Form of Wigner Operator

Using the normally ordered Gaussian form of the Wigner operator we recapitulate the quantum phase space representation, we derive a new formula for searching for the classical correspondence of quantum mechanical operators; we also show that if there exists the eigenvector ｜q〉λ,v of linear combination of the coordinate and momentum operator, （λQ ＋ vP）, where λ,v are real numbers, and ｜q〉λv is complete, then the projector ｜q〉λ,vλ,v〈q｜ must be the Radon transform of Wigner operator. This approach seems concise and physical appealing.     

Optical scaled Fresnel--Fourier transform obtained via intermediate coordinate-- momentum representation

Using the intermediate coordinate--momentum representation |x>_s,r, we introduce a new Hadamard transform. It is found that the operator U corresponding to this transform can be considered as a combination of the Fresnel operator F(r,s) and the Fourier transform operator F by decomposing U. We also find that the matrix element _s,r< x| U|f> just corresponds to an optical scaled Fresnel--Fourier transform.     

Wigner function for the generalized excited pair coherent state

This paper introduces the generalized excited pair coherent state （GEPCS）. Using the entangled state 〈η〉 representation of Wigner operator, it obtains the Wigner function for the GEPCS. In the ρ-γ phase space, the variations of the Wigner function distributions with the parameters q, α, k and l are discussed. The tomogram of the GEPCS is calculated with the help of the Radon transform between the Wigner operator and the projection operator of the entangled state ｜η1, η2, τ1, τ2｜. The entangled states ｜η〉 and ｜η1, η2, τ1, τ2〉 provide two good representative space for studying the Wigner functions and tomograms of various two-mode correlated quantum states.     

On the Bosonic Phase Operator Realization for Josephson Hamiltonian Model

On the assumption that a Cooper pair acts as a Bose particle and based on the newly estabished (η|representation,which is the common eigenvector of two particles‘ relative position and total momentum,we introduce a mesoscopic Josephson junction Hamiltonian constituted by two-mode Bose phase operator and number-difference operator,The number-difference-phase uncertainty relation can then be set up,which implies the existence of Josephson current.     

A new finite-dimensional pair coherent state studied by virtue of the entangled state representation and its statistical behavior

In this paper we construct a new type of finite-dimensional pair coherent states |ξ, q〉 as realizations of SU(2) Lie algebra. Using the technique of integration within an ordered product of operator, the nonorthogonality and completeness properties of the state |ξ, q〉 are investigated. Based on the Wigner operator in the entangled state |τ〉 representation, the Wigner function of |ξ, q〉 is obtained. The properties of |ξ, q〉 are discussed in terms of the negativity of its Wigner function. The tomogram of |ξ, q〉 is calculated with the aid of the Radon transform between the Wigner operator and the projection operator of the entangled state |η, κ₁, κ₂〉. In addition, using the entangled state |τ〉 representation of |ξ, q〉 to show that the states |ξ, q〉 are just a set of energy eigenstates of time-independent two coupled oscillators.     

Parseval Theory of Complex Wavelet Transform for Wavelet Family Including Rotational Parameters

A rotational parameter R_θ has been introduced to complex wavelet transform (CWT). The rotational CWT (RCWT) corresponds to a matrix element 〈ψ|U₂ (θ;μ;k)|F〉in the context of quantum mechanics, where U₂(θ;μ;k) is a two-mode rotational displacing-squeezing operator in the 〈η| representation. Based on this, the Parseval theorem and the inversion formula of RCWT have been proved. The concise proof not only manifestly shows the merit of Dirac's representation theory but also leads to a new orthogonal property of complex mother wavelets in parameter space.     

Operator Realization of Radial- and Azimuthal- Differentiations Obtained by Using the Entangled State Representation

We find Bose operator realization of radial- and azimuthal-differential operations in polar coordinate system by virtue of the entangled state |η〉 representation, which indicates that |η〉 representation just fits to describe the polar coordinate operators in quantum mechanics. The Bose operator corresponding to the Laplacian operation $\frac{\partial^{2}}{\partial r^{2} }+\frac{1}{r}\frac{\partial}{\partial r}+\frac{1}{r^{2}}\frac{\partial^{2} }{\partial\varphi^{2}}$ for 2-dimensional system and its eigenvector are also obtained. Their new applications are partly presented.     

Two-Mode Wigner Operator in 〈ξ| Representation and 〈τ| Representation

We derive the two-mode Wigner operator in the 〈ξ| representation and 〈τ| representation, where |ξ〉 is common eigenvector of the mass-weighted relative coordinate and the mass-combinatorial momentum.and |τ〉 common eigenvector of the mass-weighted relative coordinate and the mass-combinatorial momentum. As an application,we calculate the Wigner function of some two-mode state.     

Coordinate-Momentum Intermediate Representation and Marginal Distributions of Quantum Mechanical Bivariate Normal Distribution

We introduce bivariate normal distribution operator for state vector [ψ） and find that its marginal distribution leads to one-dimensional normal distribution corresponding to the measurement probability ｜λ,v〈x｜.ψ〉｜^2, where ｜x〉λ,v is the coordinate-momentum intermediate representation. As a by-product, the one-dimensional normal distribution in statistics can be explained as a Radon transform of two-dimensional Gaussian function.     

Two-variable Hermite Polynomial State and Its Wigner Function

In this paper we obtain the Wigner functions of two-variable Hermite polynomial states (THPS) and their marginal distribution using the entangled state |ξ〉 representation. Also we obtain tomogram of THPS by virtue of the Radon transformation between the Wigner operator and the projection operator of another entangled state |η,τ ₁,τ ₂〉.     

Options for angular momentum projection II: A finite sum representation of the Hill-Wheeler projector

An exact and finite sum representation of the Hill-Wheeler projection operator is obtained under the provision that the state on which the operator acts can be expanded as $$\left| {\psi _\alpha } \right\rangle = \sum\limits_{J = J_{\min } }^{J_{\max } } {c_J^\alpha \left| {J,\alpha } \right\rangle .} $$ The result provides a definite advantage over numerical integration methods, especially if high spin states are considered.     

Research on the Dynamic Problems of 3D Cross Coupling Quantum Harmonic Oscillator by Virtue of Intermediate Representation |<Emphasis Type="Italic">x</Emphasis>〉<Subscript><Emphasis Type="Italic">λ</Emphasis>,<Emphasis Type="Italic">ν</Emphasis></Subscript>

The intermediate representation (namely intermediate coordinate-momentum representation) |x〉 _λ,ν are introduced and employed to research the expression of the operator in intermediate representation |x〉 _λ,ν . The systematic Hamilton operator of 3D cross coupling quantum harmonic oscillator was diagonalized by virtue of quadratic form theory. The quantity of λ,ν,τand σ were figured out. The dynamic problems of 3D cross coupling quantum harmonic oscillator are researched by virtue of intermediate representation. The energy eigen-value and eigenwave function of 3D cross coupling quantum harmonic oscillator were obtained in intermediate representation. The importance of intermediate representation was discussed. The results show that the Radon transformation of Wigner operator is just the projectional operator |x〉 _{λ,ν λ,ν} 〈x|, and the Radon transformation of Wigner function is just a margin distribution.     

Obtaining Multimode Entangled State Representation by Generalized Radon Transformation of the Wigner Operator

In a preceding paper (Fan and Lv in J. Math. Phys. 50:102108, 2009), the phase-space integration corresponding to the straight line characteristic of two different real parameters λ,τ over the Wigner operator (i.e. the Radon transformation) leads to pure-state density operator |u〉 _{λ,τ  λ,τ} 〈u|, where |u〉 _λ,τ is just the coordinate-momentum intermediate representation. In this work we show that generalized Radon transformation of the Wigner operator yields multimode density operator of continuum variables. This provides us with a new approach for obtaining multimode entangled state representation. The Weyl ordering of the Wigner operator is used in our discussions.     

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.     

New Approach for Solving Master Equations in Quantum Optics and Quantum Statistics by Virtue of Thermo-Entangled State Representation

By introducing a fictitious mode to be a counterpart mode of the system mode under review we introduce the entangled state representation <η |, which can arrange master equations of density operators ρ(t) in quantum statisticsas state-vector evolution equations due to the elegant properties of <η |. In this way many master equations (respectively describing damping oscillator, laser, phase sensitive, and phase diffusion processes with different initial density operators) can be concisely solved. Specially, for a damping process characteristic of thedecay constant κ we find that the matrix element of ρ (t) at time t in <η| representation is proportional to that of the initial ρ₀ in the decayed entangled state <ηe^-κt| representation, accompanying with a Gaussian damping factor. Thus we have a new insight about the nature of the dissipative process. We also set up the so-called thermo-entangled state representation of density operators, ρ=∫(d²η /π)< η | ρ> D(η), which is different from all the previous known epresentations.     

Landau Wavefunction Immediately Derived by Virtue of the <λ| Representation

We show that by virtue of the <λ| representation (Hong-Yi Fan, Phys. Lett. A 126 (1987) 150) the Landau wavefunctions of an electron in a uniform magnetic field can be immediately derived without solving the corresponding Schrödinger equation. It turns out that the differential operation form of the electron's dynamic Hamiltonian, adopted in standard quantum mechanics textbooks, is actually expressed in <λ| representation.     

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.     

Phase Properties of Two-Mode Squeezing-Rotating Entangled Representation

By virtue of the squeezing-rotating entangled representation, we mainly establish thc new two-mode phase operator and phase angle operator, which is a general form including the foregoing formalist in two-mode Fock space.In addition, the corresponding phase distribution function is given in the entangled representation. In terms of this definition, we also analyze the phase behavior of some simple two-mode states such as squeezing-rotating coherent state,squeezing-rotating vacuum state, and so on. It is found that the results exactly agree with the foregoing phase theory.     

Normally-Ordered Time Evolution Operator for Mass-Varying Harmonic Oscillator and Wigner Function of Squeezed Number State

For investigating dynamic evolution of a mass-varying harmonic oscillator we constitute a ket-bra integration operator in coherent state representation and then perform this integral by virtue of the technique ofintegration within an ordered product of operators. The normally orderedtime evolution operator is thus obtained. We then derive the Wigner functionof$u(t)| n>, where | n> is a Fock state, which exhibits a generalized squeezing, the squeezing effect is related to the varying mass with time.