An Alternative Approach to Obtain the Exact Wave Functions of Time-Dependent Hamiltonian Systems Involving Quadratic, Inverse Quadratic, and (1/x)p+p(1/x) Terms

By using the Lewis-Riesenfeld theory and algebraic method, we present an alternative approach to obtain the exact solution of time-dependent Hamiltonian systems involving quadratic, inverse quadratic and (1/x)p+p(1/x) terms. This solution is discussed and compared with that obtained by Choi, J. R. (2003). International Journal of Theoretical Physics 42, 853]. PACS: 03.65Ge; 03.65Fd; 03.65Bz     

Excitonic polaron in the molecular crystal model: Nonlocal dynamic coherent-potential approximation

A nonlocal dynamic coherent-potential approximation is formulated as a further development of the dynamic coherent-potential method. The nonlocal dynamic coherent-potential approximation is an efficient method of determining the one-exciton Green’s function in a model with the Hamiltonian in the strong-coupling approximation, where a spectrum of optical phonons is assumed, and the exciton-phonon interaction operator is linear or quadratic in the phonon operators. A system of recursion equations is derived, from which the coherent potential is found as a function of the energy E and the wave vector k. An analytical expression is derived for the one-exciton Green’s function in the case of narrow (in comparison with the phonon energy) exciton bands and exciton-phonon interaction linear in the phonon operators. For broader exciton bands and more complex exciton-phonon interaction the system of equations determining the coherent potential represents a recursion algorithm, which can be effectively implemented by numerical means. Fiz. Tverd. Tela (St. Petersburg) 39, 1560–1563 (September 1997)     

Velocity Quantization Approach of the One-Dimensional Dissipative Harmonic Oscillator

Given a constant of motion for the one-dimensional harmonic oscillator with linear dissipation in the velocity, the problem to get the Hamiltonian for this system is pointed out, and the quantization up to second order in the perturbation approach is used to determine the modification on the eigenvalues when dissipation is taken into consideration. This quantization is realized using the constant of motion instead of the Hamiltonian. PACS: 03.20.+i, 03.30.+p, 03.65.−w,03.65.Ca     

Generalized two-mode harmonic oscillator model: squeezed number state solutions and nonadiabatic Berry’s phase

A generalized two-mode harmonic oscillator model is investigated within the framework of its general dynamical algebra so(3,2). Two types of eigenstates, formulated as extended su(1,1), su(2) squeezed number states are found respectively. The nonadiabatic Berrys phase for this system with the cranked time-dependent Hamiltonian is also given.Received: 16 January 2004, Published online: 10 August 2004PACS: 42.50.Dv Nonclassical states of the electromagnetic field, including entangled photon states; quantum state engineering and measurements - 03.65.Fd Algebraic methods - 03.65.Vf Phases: geometric; dynamic or topological     

Separation of Variables for Symplectic Characters

We perform separation of variables for the symplectic Weyl character using Sklyanin’s scheme. Viewing the characters as eigenfunctions of a quantum integrable system, we explicitly construct the separating operator using the Q-operator method. We also construct the inverse of the separating operator, as well as the factorised Hamiltonian.     

Construction of New Variable Separation Excitations via Extended Projective Ricatti Equation Expansion Method in (2+ 1)- Dimensional Dispersive Long Wave Systems

Utilizing the extended projective Ricatti equation expansion method, abundant variable separation solutions of the (2+1)-dimensional dispersive long wave systems are obtained. From the special variable separation solution (38) and by selecting appropriate functions, new types of interaction between the multi-valued and the single-valued solitons, such as semi-foldon and dromion, semi-foldon and peakon, semi-foldon and compacton are found. Meanwhile, we conclude that the solution v is essentially equivalent to the ’universal” formula (1). PACS numbers 05.45.Yv, 02.30.Jr, 03.65.Ge     

Lattice Approach to Wigner-Type Theorems

The Wigner's Theorem states that a bijective transformation of the set of all one-dimensional linear subspaces of a complex Hilbert space which preserves orthogonality is induced by either a unitary or an anti-unitary operator. There exist many Wigner-type theorems, in particular in indefinite metric spaces, von Neumanns algebras and Banach spaces and we try to find a common origin of all these results by using properties of the lattice subspaces of certain topological vector spaces. We prove a Wigner-type theorem for a pair of dual spaces which allows us to obtain, as particular cases, the usual Wigner's Theorem and some of its generalizations. PACS: 02.40.Dr, 03.65.Fd,03.65.Ta AMS Subject Classification (1991): 06C15, 46A20, 81P10.     

Computation of bound states for 2D potentials using discrete basis

Using a suitable Laguerre basis set that ensures a tridiagonal matrix representation of the reference Hamiltonian, we were able to evaluate in closed form the matrix representation of the associated Hamiltonian for two exactly solvable 2D potentials. This enabled us to treat analytically the full Hamiltonian and compute the associated bound states spectrum as the eigenvalues of the associated analytical matrix representing their Hamiltonians. Finally we compared our results satisfactorily with those obtained using the Gauss quadrature numerical integration approach.

PACS numbers: 03.65.Ge, 34.20.Cf, 03.65.Nk, 34.20.Gj     

Marginal Distributions of Wigner Function in a Mesoscopic L-C Circuit at Finite Temperature and Thermal Wigner Operator

Wigner function in phase space has its physical meaning as marginal probability distribution in coordinate space and momentum space respectively, here we endow the Wigner function with a new physical meaning, i.e., its marginal distributions’ statistical average for q ²/(2C) and p ²/(2L) are the energy stored in capacity and in inductance of a mesoscopic L-C circuit at finite temperature, respectively. PACS numbers: 03.65.-w, 73.21.-b     

Bell and Zeno

Bell's inequalities and related inequalities of Wigner, Clauser–Horne–Shimony–Holt, Accardi–Fedullo, Gudder–Zanghi, Herbert–Peres, Khrennikov, others, are shown to be contained within a general operator trigonometry developed by this author starting in 1967. These inequalities are improved here to useful quantum spin correlation identities. Secondly, the Zeno problems from quantum measurement theory are traced from early work by this author starting in 1974, to the present. A Zeno Alternative that stresses domain-theoretic properties as essential to distinguishing reversible from irreversible quantum evolutions is presented. PACS: 03.65.Ud, 03.65.Xp,02.30.Tb.     

SU(1,1) Lie Algebra Applied to the General Time-dependent Quadratic Hamiltonian System

Exact quantum states of the time-dependent quadratic Hamiltonian system are investigated using SU(1,1) Lie algebra. We realized SU(1,1) Lie algebra by defining appropriate SU(1,1) generators and derived exact wave functions using this algebra for the system. Raising and lowering operators of SU(1,1) Lie algebra expressed by multiplying a time-constant magnitude and a time-dependent phase factor. Two kinds of the SU(1,1) coherent states, i.e., even and odd coherent states and Perelomov coherent states are studied. We applied our result to the Caldirola–Kanai oscillator. The probability density of these coherent states for the Caldirola–Kanai oscillator converged to the center as time goes by, due to the damping constant γ. All the coherent state probability densities for the driven system are somewhat deformed. PACS Numbers: 02.20.Sv, 03.65.-w, 03.65.Fd     

Symmetry and Topology in Quantum Logic

A test space is a collection of non-empty sets, usually construed as the catalogue of (discrete) outcome sets associated with a family of experiments. Subject to a simple combinatorial condition called algebraicity, a test space gives rise to a “quantum logic”—that is, an orthoalgebra. Conversely, all orthoalgebras arise naturally from algebraic test spaces. In non-relativistic quantum mechanics, the relevant test space is the set ℱ F(H) of frames (unordered orthonormal bases) of a Hilbert space H. The corresponding logic is the usual one, i.e., the projection lattice L(H) of H. The test space ℱ F(H) has a strong symmetry property with respect to the unitary group of H, namely, that any bijection between two frames lifts to a unitary operator. In this paper, we consider test spaces enjoying the same symmetry property relative to an action by a compact topological group. We show that such a test space, if algebraic, gives rise to a compact, atomistic topological orthoalgebra. We also present a construction that generates such a test space from purely group-theoretic data, and obtain a simple criterion for this test space to be algebraic. PACS: 02.10.Ab; 02.20.Bb; 03.65.Ta.     

Interplay of superconductivity and magnetism in presence of inter sub-lattice effect in cuprates

In the present communication, we report a model Hamiltonian to study the interplay between the two long range orders of anti-ferromagnetism (AFM) and superconductivity (SC) in cuprate superconductors in presence of the intersite pairing effect. The BCS type but non-phonon pairing mechanism is considered among the electrons of two equivalent ‘Cu’ sites. The pairing among the electrons of two nearest neighbour non-equivalent ‘Cu’ sites is included in the Hamiltonian and its effect on the interplay of SC and AFM is investigated. The Hamiltonian is solved by the Green’s function method and the corresponding gap equations are calculated and solved selfconsistently. The influence of model parameters like AFM coupling (λ), SC couling (λ ₁) and the coupling (λ ₂) for intersite superconducting interactions on the gaps (SC and AFM) are studied numerically and the results are reported.     

Beta-transition properties for neutron-rich Sn and Te isotopes by Pyatov method

Based on Pyatov’s method, the low-lying Gamow-Teller (GT) 1⁺ state energies and log(ft) values for ^128,130,132Sb and ^132,134,136I isotopes have been calculated. In this method, the strength parameter of the effective spin-isospin interaction is found by providing the commutativity of the GT operator with the central part of the nuclear Hamiltonian. The problem has been solved within the framework of RPA. The calculation results have been compared with the corresponding experimental data.      

A Non-Commutative Approach to Ordinary Differential Equations

We adapt ideas coming from Quantum Mechanics to develop a non-commutative strategy for the analysis of some systems of ordinary differential equations. We show that the solution of such a system can be described by an unbounded, self-adjoint and densely defined operator H which we call, in analogy with Quantum Mechanics, the Hamiltonian of the system. We discuss the role of H in the analysis of the integrals of motion of the system. Finally, we apply this approach to several examples. Pacs Numbers: 02.30.Hq, 03.65.-w, 03.65.Db     

Born Reciprocity and the 1/r Potential

Many structures in nature are invariant under the transformation pair, (p,r)→(b  r,−p/b), where b is some scale factor. Born’s reciprocity hypothesis affirms that this invariance extends to the entire Hamiltonian and equations of motion. We investigate this idea for atomic physics and galactic motion, where one is basically dealing with a 1/r potential and the observations are very accurate, so as to determine the scale b≡mΩ. We find that an Ω∼1.5×10⁻¹⁵ s⁻¹ has essentially no effect on atomic physics but might possibly offer an explanation for galactic rotation, without invoking dark matter.