Flow equations for band-matrices

Continuous unitary transformations can be used to diagonalize or approximately diagonalize a given Hamiltonian. In the last four years, this method has been applied to a variety of models of condensed matter physics and field theory. With a new generator for the continuous unitary transformation proposed in this paper one can avoid some of the problems of former applications. General properties of the new generator are derived. It turns out that the new generator is especially useful for Hamiltonians with a banded structure. Two examples, the Lipkin model, and the spin-boson model are discussed in detail. Received: 2 February 1998 / Accepted: 17 March 1998     

Landau diamagnetism within Tsallis thermostatistics and quantum groups

In this study, using the base of coherent states, Landau diamagnetism has been generalized within Tsallis Thermostatistics. As far as we know, this is the first attempt to introduce coherent states in this formalism. The magnetization and the susceptibility of the system have been obtained and compared with the standard result to illustrate the effect of nonextensivity. Then, adding a perturbation term to the Hamiltonian of the system, nonextensive effects on diamagnetic susceptibility have been investigated. In addition to this, making use of the -deformed partition function of the -oscillator system, the magnetization for -deformed Landau diamagnetism has been derived, with the aim of comparing the results obtained within both formalisms. Received: 21 July 1997 / Revised: 12 December 1997 / Accepted: 23 January 1998     

Generalization of the Hellmann-Feynman theorem

The well-known Hellmann-Feynman theorem of quantum mechanics connected with the derivative of the eigenvalues with respect to a parameter upon which the Hamiltonian depends, is generalized to include cases in which the domain of definition of the Hamiltonian of the system also depends on that parameter.     

Dynamic behaviour of electron interference in the two-mode SU(1,1) coherent state field

Aharonov-Bohm-type electron interference in the presence of the two-mode SU(1,1) coherent state(CS) field is investigated. The visibility of the time-averaged interference pattern is discussed for this state, and a comparison with classical cases is made. It is shown that the time evolution of the intensity of electron interference exhibits collapse and revival (CR) phenomenon for this state. The fluctuation in electron interference, and its relation to CR phenomenon are also discussed. Received: 22 August 1997 / Accepted: 15 January 1998     

Controllability of multiple qubit systems

This paper explores the problem of manipulating multiple-qubit systems when only single-qubit operations or two-qubit-interactive operations are permitted. It is demonstrated that if there exist 2 directional control Hamiltonian for each individual qubit, and one interactive Hamiltonian for each pair of qubits, then multiple qubit systems are open-loop controllable. An important observation of physical interest is emphasized: when only single-qubit operations or two-qubit-interactive operations are permitted, only n(n+3)/2 control Hamilton may guarantee open-loop controllability of n qubit systems, and n(n+3) is, in the restricted sense, also the lower limit on the number of operators needed for controllability. At last, we demonstrate that an n-quantum-dot system is open-loop controllable even when only single-qubit operations or two-qubit-interactive operations are permitted.     

A unified grand canonical description of the nonextensive thermostatistics of the quantum gases: Fractal and fractional approach

In this paper, the particles of quantum gases, that is, bosons and fermions are regarded as g-ons which obey fractional exclusion statistics. With this point of departure the thermostatistical relations concerning the Bose and Fermi systems are unified under the g-on formulation where a fractal approach is adopted. The fractal inspired entropy, the partition function, distribution function, the thermodynamics potential and the total number of g-ons have been found for a grand canonical g-on system. It is shown that from the g-on formulation; by a suitable choice of the parameters of the nonextensivity q, the parameter of the fractional exclusion statistics g, nonextensive Tsallis as well as extensive (q=1) standard thermostatistical relations of the Bose and Fermi systems are recovered. Received 17 September 1999     

q-deformed Fermions

This is a study of q-Fermions resulting from q-deformed algebra of harmonic oscillators arising from two distinct algebras. Employing the first algebra, the Fock states are constructed for the generalized Fermions obeying Pauli exclusion principle. The distribution function and other thermodynamic properties such as the internal energy and entropy are derived. Another generalization of fermions from a different q-deformed algebra is investigated which deals with q-fermions not obeying the exclusion principle. Fock states are constructed for this system. The basic numbers appropriate for this system are determined as a direct consequence of the algebra. We also establish the Jackson Derivative, which is required for the q-calculus needed to describe these generalized Fermions.     

Quantum motion on a torus as a submanifold problem in a generalized Dirac’s theory of second-class constraints

A generalization of Dirac’s canonical quantization scheme for a system with second-class constraints is proposed, in which the fundamental commutation relations are constituted by all commutators between positions, momenta and Hamiltonian, so they are simultaneously quantized in a self-consistent manner, rather than by those between merely positions and momenta which leads to ambiguous forms of the Hamiltonian and the momenta. The application of the generalized scheme to the quantum motion on a torus leads to a remarkable result: the quantum theory is inconsistent if built up in an intrinsic geometric manner, whereas it becomes consistent within an extrinsic examination of the torus as a submanifold in three dimensional flat space with the use of the Cartesian coordinate system. The geometric momentum and potential are then reasonably reproduced.     

A remark on the optimal cloning of an N-level quantum system

We study quantum cloning machines (QCM) that act on an unknown N-level quantum state and make M copies. We give a formula for the maximum of the fidelity of cloning and exhibit the unitary transformations that realize this optimal fidelity. We also extend the results to treat the case of M copies from () identical N-level quantum systems. Received 21 September 1999     

Quantum rotors with regular frustration and the quantum Lifshitz point

We have discussed the zero-temperature quantum phase transition in n-component quantum rotor Hamiltonian in the presence of regular frustration in the interaction. The phase diagram consists of ferromagnetic, helical and quantum paramagnetic phase, where the ferro-para and the helical-para phase boundary meets at a multicritical point called a (d,m) quantum Lifshitz point where (d,m) indicates that the m of the d spatial dimensions incorporate frustration. We have studied the Hamiltonian in the vicinity of the quantum Lifshitz point in the spherical limit and also studied the renormalisation group flow behaviour using standard momentum space renormalisation technique (for finite n). In the spherical limit ()one finds that the helical phase does not exist in the presence of any nonvanishing quantum fluctuation for m =d though the quantum Lifshitz point exists for all d > 1+m/2, and the upper critical dimensionality is given by d _u = 3 +m/2. The scaling behaviour in the neighbourhood of a quantum Lifshitz point in d dimensions is consistent with the behaviour near the classical Lifshitz point in (d+z) dimensions. The dynamical exponent of the quantum Hamiltonian z is unity in the case of anisotropic Lifshitz point (d>m) whereas z=2 in the case of isotropic Lifshitz point (d=m). We have evaluated all the exponents using the renormalisation flow equations along-with the scaling relations near the quantum Lifshitz point. We have also obtained the exponents in the spherical limit (). It has also been shown that the exponents in the spherical model are all related to those of the corresponding Gaussian model by Fisher renormalisation. Received: 23 December 1997 / Received in final form: 6 January 1998 / Accepted: 7 January 1998     

Low frequency, low temperature properties of the spin-boson problem

Low temperature and low frequency properties of a spin-boson model are investigated within a super operator and Liouville space formulation. The leading contributions are identified with the help of projection operators projecting onto the equilibrium state. The quantities of interest are expressed in terms of weighted bath propagators and static linear and nonlinear susceptibilities. In particular the generalized Shiba relation and Wilson ratio are recovered. Received 7 July 2000 and Received in final form 8 August 2000     

Approach to the quantum evolution for underdamped, critically damped, and overdamped driven harmonic oscillators using unitary transformation

Exact solution of the Schrödinger equation is derived for underdamped, critically damped, and overdamped harmonic oscillators with a driving force. A unitary operator transforming Hamiltonian into a simple form is introduced. The transformed Hamiltonian, represented in terms of a modified frequency ω, is identical with the Hamiltonian of the standard harmonic oscillator for the underdamped oscillator, with the Hamiltonian of a free particle for the critically damped oscillator, and with the Hamiltonian of a system with a harmonic parabolic potential for the overdamped oscillator. The eigenvalues of underdamped oscillator are discrete while those of the critically damped and the overdamped oscillators are continuous.     

Time-dependent harmonic oscillator and spectral determinant on graphs

Using a path integral approach and also considerations about the time-dependent harmonic oscillator, we compute the spectral determinant of the operator () on a graph. ( is the Laplacian and V(x) is some potential defined on the graph). We recover a recent result that was obtained by constructing the Green's function on the graph. We also extend those considerations to the case when i) a magnetic field is added to the system, ii) the potential, V(x), contains repulsive peaks. Received 21 January 2000     

A singular position-dependent mass particle in an infinite potential well

An unusual singular position-dependent-mass particle in an infinite potential well is considered. The corresponding Hamiltonian is mapped through a point-canonical-transformation and an explicit correspondence between the target Hamiltonian and a Pöschl-Teller type reference Hamiltonian is obtained. New ordering ambiguity parametric setting are suggested.     

Generalized uncertainty principle and self-adjoint operators

In this work we explore the self-adjointness of the GUP-modified momentum and Hamiltonian operators over different domains. In particular, we utilize the theorem by von-Neumann for symmetric operators in order to determine whether the momentum and Hamiltonian operators are self-adjoint or not, or they have self-adjoint extensions over the given domain. In addition, a simple example of the Hamiltonian operator describing a particle in a box is given. The solutions of the boundary conditions that describe the self-adjoint extensions of the specific Hamiltonian operator are obtained.     

New formulation for squeezed fermion-pair states as a counterpart of Grassmannian evolution

A new formulation for deriving the squeezing fermion-pair operators, which manifestly shows the correspondence between pseudo-classical transformation in Grassmann number space and quantum mechanical Fermi operators, is presented. In this formulation new fermionic operator formulas regarding Bogoliubov-Valatin transformation are obtained. Received 17 February 2000     

An extended class of L-series solutions of the wave equation

We lift the constraint of a diagonal representation of the Hamiltonian by searching for square integrable bases that support an infinite tridiagonal matrix representation of the wave operator. The class of solutions obtained as such includes the discrete (for bound states) as well as the continuous (for scattering states) spectrum of the Hamiltonian. The problem translates into finding solutions of the resulting three-term recursion relation for the expansion coefficients of the wavefunction. These are written in terms of orthogonal polynomials, some of which are modified versions of known polynomials. The examples given, which are not exhaustive, include problems in one and three dimensions.     

Solution of spin-boson systems in one and two-dimensional geometry via the asymptotic iteration method

We consider solutions of the 2×2 matrix Hamiltonian of physical systems within the context of the asymptotic iteration method. Our technique is based on transformation of the associated Hamiltonian in the form of the first order coupled differential equations. We construct a general matrix Hamiltonian which includes a wide class of physical models. The systematic study presented here reproduces a number of earlier results in a natural way as well as leading to new findings. Possible generalizations of the method are also suggested.     

On Wigner functions and a damped star product in dissipative phase-space quantum mechanics

Dito and Turrubiates recently introduced an interesting model of the dissipative quantum mechanics of a damped harmonic oscillator in phase space. Its key ingredient is a non-Hermitian deformation of the Moyal star product with the damping constant as deformation parameter. We compare the Dito-Turrubiates scheme with phase-space quantum mechanics (or deformation quantization) based on other star products, and extend it to incorporate Wigner functions. The deformed (or damped) star product is related to a complex Hamiltonian, and so necessitates a modified equation of motion involving complex conjugation. We find that with this change the Wigner function satisfies the classical equation of motion. This seems appropriate since non-dissipative systems with quadratic Hamiltonians share this property.