涉及Hermite多项式的二项式定理和Laguerre多项式的负二项式定理

提出量子力学算符Hermite多项式方法,即将若干常用的特殊函数的宗量由普通数变为算符,并用它来发现涉及Hermite多项式（单变数和双变数）的二项式定理和涉及Laguerre多项式的负二项式定理,它们在计算若干量子光场的物理性质时有实质性的应用. 该方法不但具有简捷的优点,而且能导出很多新的算符恒等式,成为发展数学物理理论的一个重要分支. 关键词： 量子力学 Hermite多项式 二项式定理 Laguerre多项式     

New operator-ordering identities and associative integration formulas of two-variable Hermite polynomials for constructing non-Gaussian states

For directly normalizing the photon non-Gaussian states(e.g., photon added and subtracted squeezed states), we use the method of integration within an ordered product(IWOP) of operators to derive some new bosonic operator-ordering identities. We also derive some new integration transformation formulas about one- and two-variable Hermite polynomials in complex function space. These operator identities and associative integration formulas provide much convenience for constructing non-Gaussian states in quantum engineering.     

厄米多项式的定义及其他

分析了关于厄米多项式的争论.介绍了厄米多项式的两种定义.     

Some new generating function formulae of the two-variable Hermite polynomials and their application in quantum optics

We explore the theoretical possibility of extending the usual squeezed state to those produced by nonlinear singlemode squeezing operators. We derive the wave functions of exp[-（ig/2）（（1-X2）1/2P ＋ P（1-X2）1/2）]｜0 in the coordinate representation. A new operator＇s disentangling formula is derived as a by-product.     

量子光学理论中双模厄米多项式的新应用

强调双模厄米多项式在量子光学理论中的地位,认为它是研究连续变量纠缠态和压缩态的必要函数,具有明确的物理意义。利用双模厄米多项式,结合有序算符内的积分技术,给出了若干新的算符恒等式和互逆的积分变换公式,证明了压缩双模粒子数态恰好是双变量厄米多项式激发压缩真空态。     

On entropy and information of quantum states

Shannon entropy and information are applied to study the properties of quantum states of a system in the probability representation of quantum mechanics. Examples of spin states and mixed Gaussian states of the two-mode system are considered. The relationship between the new entropy and the von Neumann entropy is reviewed. Two tomographic maps are considered within the framework of the star-product quantization. The explicit expression of tomographic entropy associated with photon-number tomogram of the two-mode state of photons is obtained in terms of Hermite polynomials of four variables. Based on a contribution to the International Conference “New Trends in Quantum Mechanics. Fundamental Aspects and Applications” (Palermo, Italy, November 2005).     

New Bosonic Operator Ordering Identities Gained by the Entangled State Representation and Two-Variable Hermite Polynomials

Based on the technique of integration within an ordered product of operators, we derive new bosonicoperators‘ ordering identities by using entangled state representation and the properties of two-variable Hermite poly-nomials H and vice versa. In doing so, some concise normally (antinormally) ordering operator identities, such asa man =:Hm,n(a ,a):, ana m = (-i)m n:Hm,n(ia ,ia): are obtained.     

Contextuality and the probability representation of quantum states

We consider the notions of contextuality and noncontextuality within the framework of the probability representation of quantum states. We present an example of qutrit states and violation of the noncontextuality inequalities using the spin tomogram and tomographic symbols of the observables.     

Classical probability density distributions with uncertainty relations for ground states of simple non-relativistic quantum-mechanical systems

The probability density distributions for the ground states of certain model systems in quantum mechanics and for their classical counterparts are considered. It is shown that classical distributions are remarkably improved by incorporating into them the Heisenberg uncertainty relation between position and momentum. Even the crude form of this incorporation makes the agreement between classical and quantum distributions unexpectedly good, except for the small area, where classical momenta are large. It is demonstrated that the slight improvement of this form makes the classical distribution very similar to the quantum one in the whole space. The obtained results are much better than those from the WKB method. The paper is devoted to ground states, but the method applies to excited states too.     

New operator identities with regard to the two-variable Hermite polynomial by virtue of entangled state representation

By virtue of the entangled state representation we concisely derive some new operator identities with regard to the two-variable Hermite polynomial （TVHP）. By them and the technique of integration within an ordered product （IWOP） of operators we further derive new generating function formulas of the TVHP. They are useful in quantum optical theoretical calculations. It is seen from this work that by combining the IWOP technique and quantum mechanical representations one can derive some new integration formulas even without really performing the integration.     

Probability Representation of Classical States

Probability representation of classical states described by symplectic tomograms is discussed. Tomographic symbols of classical observables which are functions on phase-space are studied. Explicit form of kernel of commutative star-product of the tomographic symbols is obtained.     

Roy-型奇偶非线性相干态的位相概率分布

借助于数值计算方法，研究了Roy-型奇偶非线性相干态的位相概率分布特性.结果表明，它们的位相概率分布不同；与通常的奇偶相干态不同，在这种新的奇偶非线性相干态中，Pegg-Barnett位相概率分布能明显地反映出不同的量子干涉特性. 关键词： Roy-型奇偶非线性相干态 Pegg-Barnett位相算符公式 位相概率分布     

Spin and quark states in the probability representation of quantum mechanics

A short review of the general principles of constructing tomograms of spin and quark states is presented.     

New q‐Hermite polynomials: characterization,operators algebra and associated coherent states

This paper addresses a construction of new q‐Hermite polynomials with a full characterization of their main properties and corresponding raising and lowering operator algebra. The three‐term recursive relation as well as the second‐order differential equation obeyed by these new polynomials are explicitly derived. Relevant operator actions, including the eigenvalue problem of the deformed oscillator and the self‐adjointness of the related position and momentum operators, are investigated and analyzed. The associated coherent states are constructed and discussed with an explicit resolution of the induced moment problem. The phase collapse in a q‐deformed boson system is studied.     

Chebyshev polynomials and Fourier transform of <Emphasis Type="Italic">SU</Emphasis>(2) irreducible representation character as spin tomographic star-product kernel

Spin tomographic symbols of qudit states and spin observables are studied. Spin observables are associated with the functions on a manifold whose points are labeled by the spin projections and sphere S ² coordinates. The star-product kernel for such functions is obtained in an explicit form and connected with the Fourier transform of characters of the SU(2) irreducible representation. The kernels are shown to be in close relation to the Chebyshev polynomials. Using specific properties of these polynomials, we establish the recurrence relation between the kernels for different spins. Employing the explicit form of the star-product kernel, a sum rule for Clebsch–Gordan and Racah coefficients is derived. Explicit formulas are obtained for the dual tomographic star-product kernel as well as for intertwining kernels which relate spin tomographic symbols and dual tomographic symbols.     

Generating function of product of bivariate Hermite polynomials and their applications in studying quantum optical states

By virtue of the operator-Hermite-polynomial method, we derive some new generating function formulas of the product of two bivariate Hermite polynomials. Their applications in studying quantum optical states are presented.     

Path integral representation for arbitrary matrix elements and wavefunctions of excited states in quantum mechanics

Arbitrary matrix elements and wavefunctions of the excited states have been expressed in terms of euclidean functional integrals. Thus, the range of applicability of the path integral method in quantum mechanics is enlarged. An example, demonstrating the convenience of the new formulas for semiclassical calculations is considered.