The Aharonov-Bohm effect in the momentum space

The Schrödinger formalism of quantum mechanics is used to demonstrate the existence of the Aharonov-Bohm effect in momentum space and set-ups for experimentally demonstrating it are proposed for either free or ballistic electrons.     

Finite correction to Aharonov-Bohm scattering by a contact potential

We study the influence of a contact (or delta) potential on the Aharonov-Bohm scattering of nonrelativistic particles. In general the contact potential has no effect on the scattering as expected. However, when the magnetic flux and the strength of the contact potential take some special values, the Aharonov-Bohm scattering cross-section is manifestly changed. It is shown that these special values correspond to the simultaneous existence of two half-bound states in two adjacent angular momentum channels. Two limiting processes are presented to deal with the singularity of the contact potential and results of the same nature are obtained.     

Feynman integral treatment of the Bargmann potential

A method based on path integral formulation is given for obtaining exact solution of the s states for the Bargmann potential

     

Scattering of a relativistic scalar particle by a cusp potential

We solve the Klein–Gordon equation in the presence of a spatially one-dimensional cusp potential. The scattering solutions are obtained in terms of Whittaker functions and the condition for the existence of transmission resonances is derived. We show the dependence of the zero-reflection condition on the shape of the potential. In the low-momentum limit, transmission resonances are associated with half-bound states. We express the condition for transmission resonances in terms of the phase shifts.     

Classical and quantum pumping in closed systems

Pumping of charge (Q) in a closed ring geometry is not quantized even in the strict adiabatic limit. The deviation form exact quantization can be related to the Thouless conductance. We use the Kubo formalism as a starting point for the calculation of both the dissipative and the adiabatic contributions to Q. As an application we bring examples for classical dissipative pumping, classical adiabatic pumping, and in particular we make an explicit calculation for quantum pumping in case of the simplest pumping device, which is a three site lattice model. We make a connection with the popular S-matrix formalism which has been used to calculate pumping in open systems.     

Phase modulation of charge density wave around a short-range potential by the Aharonov-Bohm effect

The analytic result of the Aharonov-Bohm effect for its influence on the oscillation of a two-dimensional charge density around a short range potential is given. Without losing generality, the cases of δ-function and hard disc potentials were examined. Numerical calculations show that the interferences among quantum particles are greatly influenced by the nonlocal effect, which leads to the modulation of phase and amplitude of the oscillation of charge density. Since the presence of a nonlocal influence of the Aharonov-Bohm effect on charged particles is universal, the results in the specific potentials examined are expected to appear also in other general systems which may be beneficial to the study of nanotechnology.     

The laser-dressed potential binding energy of a hydrogenic impurity in a spherical quantum dot by the analytical transfer matrix method

The analytical transfer matrix method (ATMM) is efficient and accurate for understanding the nature of bound states. In this paper, it is applied to obtain the binding energy of a hydrogenic impurity placed at the center of the spherical quantum dots (QDs) in an intense laser field. Our results agree with the exact energies in Varshni [Superlattices Microstructures 30 (2001) 45]. Therefore, ATMM gives us an alternative approach to tackle the problem of impurities placed in nano-structures under intense laser fields.     

Hartman effect in presence of Aharanov-Bohm flux

The Hartman effect for a tunnelling particle implies the independence of group delay time on the opaque barrier width with superluminal velocities as a consequence. This effect is further examined on a quantum ring geometry in the presence of Aharonov-Bohm flux. We show that while tunnelling through an opaque barrier, the group delay time for given incident energy becomes independent of the barrier thickness as well as the magnitude of the flux. The Hartman effect is thereby extended beyond one dimension in the presence of Aharonov-Bohm flux.     

Quasi-exact solvability of Dirac equation with Lorentz scalar potential

We consider exact/quasi-exact solvability of Dirac equation with a Lorentz scalar potential based on factorizability of the equation. Exactly solvable and sl (2)-based quasi-exactly solvable potentials are discussed separately in Cartesian coordinates for a pure Lorentz potential depending only on one spatial dimension, and in spherical coordinates in the presence of a Dirac monopole.     

Eigenchannel method in quantum potential scattering

The eigenchannel method, generalizing the familiar phaseshift method, is formulated for scattering from a Hermitian short range potential. Scattering eigenchannels are defined as eigenstates of some generalized (weighted) operator spectral problem. Eigenvalues of that problem define eigenphaseshifts, the former being the negative of cotangents of the latter. Eigenchannel representations of generalized scattering states, transition operators, and Green operators are constructed. A variational approach to the method is also presented. The general theory is illustrated by applications to scattering of Schrödinger and Dirac particles.     

Analytic and numeric Green’s functions for a two-dimensional electron gas in an orthogonal magnetic field

We have derived closed analytic expressions for the Green’s function of an electron in a two-dimensional electron gas threaded by a uniform perpendicular magnetic field, also in the presence of a uniform electric field and of a parabolic spatial confinement. A workable and powerful numerical procedure for the calculation of the Green’s functions for a large infinitely extended quantum wire is considered exploiting a lattice model for the wire, the tight-binding representation for the corresponding matrix Green’s function, and the Peierls phase factor in the Hamiltonian hopping matrix element to account for the magnetic field. The numerical evaluation of the Green’s function has been performed by means of the decimation-renormalization method, and quite satisfactorily compared with the analytic results worked out in this paper. As an example of the versatility of the numerical and analytic tools here presented, the peculiar semilocal character of the magnetic Green’s function is studied in detail because of its basic importance in determining magneto-transport properties in mesoscopic systems.     

Two-dimensional Schrödinger Hamiltonians with effective mass in SUSY approach

The general solution of SUSY intertwining relations of first order for two-dimensional Schrödinger operators with position-dependent (effective) mass is built in terms of four arbitrary functions. The procedure of separation of variables for the constructed potentials is demonstrated in general form. The generalization for intertwining of second order is also considered. The general solution for a particular form of intertwining operator is found, its properties—symmetry, irreducibility, and separation of variables—are investigated.     

First-order intertwining operators and position-dependent mass Schrödinger equations in d dimensions

The problem of d-dimensional Schrödinger equations with a position-dependent mass is analyzed in the framework of first-order intertwining operators. With the pair (H, H₁) of intertwined Hamiltonians one can associate another pair of second-order partial differential operators (R, R₁), related to the same intertwining operator and such that H (resp. H₁) commutes with R (resp. R₁). This property is interpreted in superalgebraic terms in the context of supersymmetric quantum mechanics (SUSYQM). In the two-dimensional case, a solution to the resulting system of partial differential equations is obtained and used to build a physically relevant model depicting a particle moving in a semi-infinite layer. Such a model is solved by employing either the commutativity of H with some second-order partial differential operator L and the resulting separability of the Schrödinger equation or that of H and R together with SUSYQM and shape-invariance techniques. The relation between both approaches is also studied.     

Newton-Leibniz integration for ket-bra operators in quantum mechanics (V)—Deriving normally ordered bivariate-normal-distribution form of density operators and developing their phase space formalism

We show that Newton-Leibniz integration over Dirac’s ket-bra projection operators with continuum variables, which can be performed by the technique of integration within ordered product (IWOP) of operators [Hong-yi Fan, Hai-liang Lu, Yue Fan, Ann. Phys. 321 (2006) 480], can directly recast density operators and generalized Wigner operators into normally ordered bivariate-normal-distribution form, which has resemblance in statistics. In this way the phase space formalism of quantum mechanics can be developed. The Husimi operator, entangled Husimi operator and entangled Wigner operator for entangled particles with different masses are naturally introduced by virtue of the IWOP technique, and their physical meanings are explained.     

Quantum treatment of the time-dependent coupled oscillators under the action of a random force

In this communication we introduce the problem of time-dependent frequency converter under the action of external random force. We have assumed that the coupling parameter and the phase pump are explicitly time dependent. Using the equations of motion in the Heisenberg picture the dynamical operators are obtained, however, under a certain integrability condition. When the system is initially prepared in the even coherent states the squeezing phenomenon is discussed. The correlation function is also considered and it has been shown that the nonclassical properties are apparent and sensitive to any variation in the integrability parameter. Furthermore, the wave function in Schrödinger picture is calculated and used it to derive the wave function in the coherent states. The accurate definition of the creation and annihilation operators are also introduced and employed to diagonalize the Hamiltonian system.     

Newton-Leibniz integration for ket-bra operators in quantum mechanics (IV)—Integrations within Weyl ordered product of operators and their applications

We show that the technique of integration within normal ordering of operators [Hong-yi Fan, Hai-liang Lu, Yue Fan, Ann. Phys. 321 (2006) 480-494] applied to tackling Newton-Leibniz integration over ket-bra projection operators, can be generalized to the technique of integration within Weyl ordered product (IWWOP) of operators. The Weyl ordering symbol is introduced to find the Wigner operator’s Weyl ordering form Δ(p,q) =  δ(p − P)δ(q − Q) , and to find operators’ Weyl ordered expansion formula. A remarkable property is that Weyl ordering of operators is covariant under similarity transformation, so it has many applications in quantum statistics and signal analysis. Thus the invention of the IWWOP technique promotes the progress of Dirac’s symbolic method.     

Hamiltonian quantum dynamics with separability constraints

Schroedinger equation on a Hilbert space H, represents a linear Hamiltonian dynamical system on the space of quantum pure states, the projective Hilbert space PH. Separable states of a bipartite quantum system form a special submanifold of PH. We analyze the Hamiltonian dynamics that corresponds to the quantum system constrained on the manifold of separable states, using as an important example the system of two interacting qubits. The constraints introduce nonlinearities which render the dynamics nontrivial. We show that the qualitative properties of the constrained dynamics clearly manifest the symmetry of the qubits system. In particular, if the quantum Hamilton’s operator has not enough symmetry, the constrained dynamics is nonintegrable, and displays the typical features of a Hamiltonian dynamical system with mixed phase space. Possible physical realizations of the separability constraints are discussed.     

The scattering of the Manning-Rosen potential with centrifugal term

In this Letter the approximately analytical scattering state solutions of the l-wave Schrödinger equation for the Manning-Rosen potential are carried out by a proper approximation to the centrifugal term. The normalized radial wave functions of l-wave scattering states are presented and the calculation formula of phase shifts is derived. It is well shown that the poles of the S-matrix in the complex energy plane correspond to bound states for real poles and scattering states for complex poles in the lower half of the energy plane. We consider and verify two special cases: the l=0 and the s-wave Hulthén potential.     

Berry phase and shot noise for spin-polarized and entangled electrons

Shot noise for entangled and spin-polarized states in a four-probe geometric setup has been studied by adding two rotating magnetic fields in an incoming channel. Our results show that the noise power oscillates as the magnetic fields vary. The singlet, entangled triplet and polarized states can be distinguished by adjusting the magnetic fields. The Berry phase can be derived by measuring the shot noise power.     

Linear transformations of quantum states

This paper considers the most general linear transformation of a quantum state. We enumerate the conditions necessary to retain a physical interpretation of the transformed state: hermiticity, normalization and complete positivity. We show that these can be formulated in terms of an associated transformation introduced by Choi in 1975. We extend his treatment and display the mathematical argumentation in a manner closer to that used in traditional quantum physics. We contend that our approach displays the implications of the physical requirements in a simple and intuitive way. In addition, defining an arbitrary vector, we may derive a probability distribution over the spectrum of the associated transformation. This fixes the average of the eigenvalue independently of the vector chosen. The formal results are illustrated by a couple of examples.