Classical and quantum q-deformed physical systems

On the basis of non-commutative q-calculus, we investigate a q-deformation of the classical Poisson bracket in order to formulate a generalized q-deformed dynamics in the classical regime. The obtained q-deformed Poisson bracket appears invariant under the action of the q-symplectic group of transformations. Within this framework we introduce the q-deformed Hamilton equations and we derive the evolution equation for some simple q-deformed mechanical systems governed by a scalar potential dependent only on the coordinate variable. It appears that the q-deformed Hamiltonian, which is the generator of the equation of motion, is generally not conserved in time but, in correspondence, a new constant of motion is generated. Finally, by following the standard canonical quantization rule, we compare the well-known q-deformed Heisenberg algebra with the algebra generated by the q-deformed Poisson bracket. PACS 02.45.Gh, 45.20.-d, 03.65.-w, 02.20.Uw     

The q-Deformed Energy Gap Equation in SLq(2) Superconductive Thermodynamic Model

The q-deformed energy gap equations in SL_q(2) superconductive thermodynamic model are derived by using quantum group method.     

Central extensions of classical and quantum q-Virasoro algebras

We investigate the central extensions of the q-deformed (classical and quantum) Virasoro algebras constructed from the elliptic quantum algebra A_q,p(sl(N))_c. After establishing the expressions of the cocycle conditions, we solve them, both in the classical and in the quantum case (for sl(2)). We find that the consistent central extensions are much more general that those found previously in the literature.     

q-Deformed Differential Operator Superalgebra on Quantum Superplane

The q-deformed differential operator superalgebra is proposed, and these differential operators are analyzed in the framework of quantum superplane. At the same time, a solution of graded Yang-Baxter equation is obtained.     

Spinor and Oscillator Representations of Quantum Enveloping Algebras of Type Bn,Cn and Dn

With the help of the q-deformed bosonic and fermionic oscillation operators, which can be conetructed from the ordinary ones, the quantum enveloping algebras of the classical Lie algebras B, C and D are written down explicitly. Under these representations the highest roots are given.     

q形变光场与级联型三能级原子相互作用中场的压缩效应

建立了q形变光场与级联型三能级原子相互作用的非线性理论,求得相互作用绘景中薛定谔方程的形式解,利用数值计算讨论了q形变对相互作用中场压缩效应的影响并与q-偶相干光场的情形做了对比分析。研究发现,q偏离1的程度越大,q形变对场压缩效应的调控能力越强,反映出q形变的非线性行为对量子相干性的干扰以及对量子特性的影响。当q→1时,恢复为普通线性理论。和q-偶相干光场的情形做的对比分析,反映出q-奇相干光场不仅不存在压缩效应,还会弱化压缩效应。这与q-偶相干光场和q-奇相干光场存在不同的量子效应的结论完全一致。     

The q-Deformation of the Superoscillators and Their q-Supercoherent States

We discuss the superoscillators in thc supersymmetric quantum systems, and give the solution to the system. By making use of the annihilation and creation operators and their q-deformed operators, we also give the q-deformed quantum harmonic oscillator model in the supersymmetric quantum systems. The q-supercoherent states associated with the q-deformed supersymmetric quantum harmonic oscillators are constructed explicitly, and their properties are investigated. The uncertainty relations for q-supercoherent states are discussed as well.     

q-奇相干场与级联型三能级原子相互作用中场的反聚束效应

提出了q-奇相干场与级联型三能级原子相互作用的非线性理论,求得了在相互作用绘景中薛定谔方程的形式解及其在其态下的期望值。利用数值计算讨论了q形变对相互作用中场的反聚束效应的影响。经研究发现,q形变对反聚束效应的调制能力很强,反映出了q形变的非线性行为对量子相干性的干扰以及对量子特性的影响。当q→1时,恢复为普通的线性理论。     

Extending Newton's law from nonlocal-in-time kinetic energy

We study a new equation of motion derived from a context of classical Newtonian mechanics by replacing the kinetic energy with a form of nonlocal-in-time kinetic energy. It leads to a hypothetical extension of Newton's second law of motion. In a first stage the obtainable solution form is studied by considering an unknown value for the nonlocality time extent. This is done in relation to higher-order Euler-Lagrange equations and a Hamiltonian framework. In a second stage the free particle case and harmonic oscillator case are studied and compared with quantum mechanical results. For a free particle it is shown that the solution form is a superposition of the classical straight line motion and a Fourier series. We discuss the link with quanta interpretations made in Pais-Uhlenbeck oscillators. The discrete nature emerges from the continuous time setting through application of the least action principle. The harmonic oscillator case leads to energy levels that approximately correspond to the quantum harmonic oscillator levels. The solution to the extended Newton equation also admits a quantization of the nonlocality time extent, which is determined by the classical oscillator frequency. The extended equation suggests a new possible way for understanding the relationship between classical and quantum mechanics.     

Z_n格点模型的量子仿射变形代数

应用两种不同的星三角关系及其对应的Ｂｏｌｔｚｍａｎｎ面权，通过反对称聚合，构造出了在椭圆情形下的ｑ变形仿射代数．     

A q-Deformed osp(1,2) Superalgebra and Its Coherent States

In this paper we present a q-deformed osp(1, 2) superalgebra, construct a type of coherent state representations and obtain a q-differential realization for the q-deformed superalgebra.We also study some quantum optical properties of the q-deformed osp(1, 2) coherent states.     

Quantum Group Structure of the q-Deformed Virasoro Algebra

In this Letter, the quantum group structure of the q-deformed Virasoro algebra Vir_q will be given.     

Non-Bosonic Excitations in q-Oscillator System for q a Root of Unity

The systems of q-deformed oscillators of type two are studied in the case of q being the roots of unity. Non-bosonic excitations are found in (quantum mechanical) q-oscillators, and the classical mechanical counterparts and the saturation property are given. Eepecially when rank k is near 4, the excitations are found violating the Pauli exclusion principle, and possible physical interpretation is investigated.     

Exact solution of Schrdinger equation with q-deformed quantum potentials using Nikiforov–Uvarov method

In this paper,we present the exact solution of the one-dimensional Schrdinger equation for the q-deformed quantum potentials via the Nikiforov–Uvarov method.The eigenvalues and eigenfunctions of these potentials are obtained via this method.The energy equations and the corresponding wave functions for some special cases of these potentials are briefly discussed.The PT-symmetry and Hermiticity for these potentials are also discussed.     

Quantum matrices inN dimensions

We obtain the commutation relation between rows and columns ofN ×N quantum matrices. We derive the quantum determinant and discuss its property in terms of the q-deformed Levi-Civita symbol. We find the inverse and trace ofN×N quantum matrices. Finally we discuss the q-deformed complexification of the quantum matrices.     

q变形量子振子的Glauber相干态

利用具有量子群对称性的q变形量子谐振子Fock态│n>_q生成q变形量子谐振子的Glauber相干态(q-Glauber相干态│a>_q)。还讨论了q-Glauber相干态│a>_q的完备性、粒子数分布、振子强度分布和最小测不准关系,指出q-Glauber相干态│a>_q的相干程度可以用q参数来描述。     

q变形叠加态及其特性

利用ｑ－振子代数提出了一种由ｑ变形振子的本征态（ｑ变形光子数态）叠回由成的态。给出了这种ｑ有叠加态的定义,研究了它们的量子统计性质。也讨论了这种态的压缩和ｑ压缩,并进一步研究了它们的寻常高阶压缩和ｑ高阶压缩。ｑ变形振子是一种非线性振子,通过这些讨论,推广通常的叠加态。     

q变形氢原子的能谱

用ＳＯ_q（４）量子代数得到ｑ变形三维氢原子的能谱及简并度，还讨论了二维ｑ变形氢原子的能谱和简并度。 关键词：     

Quantum corrected cell model for an anharmonic generalized Lennard-Jones solid

A simple method is proposed to dispose the quantum effect and anharmonic effect at the same time. Considering the quantum effect is remarkable only at low temperature, and tends to zero at high temperature, the potential energy of an atom is expanded harmonically to consider the quantum effect of solids within the harmonic oscillator framework. The anharmonic effect is remarkable only at high temperature, and tends to zero at low temperature, it was disposed by using a classical approximation. The universal formalism is applied to the generalized Lennard-Jones solid. The comparison shows that the results with and without anharmonic effect are in agreement with each other at some low temperature, to which the Einstein model is applicable. The results without anharmonic effect become divergent at slightly higher temperatures; however, the results including anharmonic effect are in good agreement with the experimental data of solid xenon. The method proposed in this paper can be extended to other potentials to develop practical molecular thermodynamic equations of state for solids.