Geometry,commutation relations and the quantum fictitious force

We express the commutation relation between the operators of the momentum and the radial unit vectors in D dimensions in differential and integral form. We connect this commutator with the quantum fictitious potential emerging in the radial Schr?dinger equation of an s-wave. Received: 6 August 2002 / Revised version: 30 October 2002 / Published online: 26 February 2003 RID="*" ID="*"Corresponding author. Fax: +49-731/502-3086, E-mail: markus.cirone@physik.uni-ulm.de     

Oscillatory motion in confined potential systems with dissipation in the context of the Schrödinger-Langevin-Kostin equation

The quantum dissipative motion of wave packets in confined systems with polynomial potentials is numerically investigated in the context of the Schrödinger-Langevin-Kostin equation. Oscillatory patterns are studied in detail and they confirm the validity of the correspondence principle. The transition to the stationary state is also discussed.     

Main effects of the Earth's rotation on the stationary states of ultra-cold neutrons

The relativistic corrections in the Hamiltonian for a particle in a uniformly rotating frame are discussed. They are shown to be negligible in the case of ultra-cold neutrons (UCN) in the Earth's gravity. The effect, on the energy levels of UCN, of the main term due to the Earth's rotation, i.e. the angular-momentum term, is calculated. The energy shift is found proportional to the energy level itself.     

Quantum particles from coarse grained classical probabilities in phase space

Quantum particles can be obtained from a classical probability distribution in phase space by a suitable coarse graining, whereby simultaneous classical information about position and momentum can be lost. For a suitable time evolution of the classical probabilities and choice of observables all features of a quantum particle in a potential follow from classical statistics. This includes interference, tunneling and the uncertainty relation.     

Build-up of coherence between initially-independent subsystems: The case of Bose-Einstein condensates

When initially-independent subsystems are made to contact, coherence can develop due to interaction between them. We exemplify and demonstrate this paradigm through several scenarios of two initially-independent Bose-Einstein condensates which are allowed to collide. The build-up of coherence depends strongly on time, interaction strength and other parameters of each condensate. Implications are discussed.     

Quantum wave packet revival in two-dimensional circular quantum wells with position-dependent mass

We study quantum wave packet revivals on two-dimensional infinite circular quantum wells (CQWs) and circular quantum dots with position-dependent mass (PDM) envisaging a possible experimental realization. We consider CQWs with radially varying mass, addressing particularly the cases where M(r)∝rw with w=1,2, or −2. The two PDM Hamiltonians currently allowed by theory were analyzed and we were able to construct a strong theoretical argument favoring one of them.     

Dissipation and quantization for composite systems

In the framework of 't Hooft's quantization proposal, we show how to obtain from the composite system of two classical Bateman's oscillators a quantum isotonic oscillator. In a specific range of parameters, such a system can be interpreted as a particle in an effective magnetic field, interacting through a spin-orbit interaction term. In the limit of a large separation from the interaction region one can describe the system in terms of two irreducible elementary subsystems which correspond to two independent quantum harmonic oscillators.     

Enhanced Quantum Reflection of Ultracold Atoms with Strong Interatomic Interaction

We calculate the reflection probability for ultracold alkali atoms incident on a solid surface. By considering the interatomic interaction and using the WKB method, it is shown that the repulsive interaction between atoms has the effect of increasing the reflection probability. The increasing amplitude is related with the interatomic interaction and the depth of atom-surface potential. In addition, we also perform a numerical calculation to testify the effect of the interatomic interaction, and the analytic result is proven by the numerical result.     

Tunneling in a “breathing” double well: Adiabatic and antiadiabatic limits and tunneling suppression

Tunneling in a piecewise harmonic potential coupled to a harmonic oscillator is considered by means of the path integral technique. The reduced propagator for the tunneling particle is calculated explicitly and the tunneling splitting is found in semiclassical approximation. The result holds for arbitrary values of the parameters of the system. From this the adiabatic and antiadiabatic approximations are obtained as particular cases and compared with the results obtained differently. The limit of a strong interaction is also considered. It is found that for strong interaction or equivalently for the harmonic frequency tending to zero the preexponential factor in the tunneling splitting tends to zero which results in a suppression of tunneling. Implications of this result for tunneling in a more general potential are discussed.     

Solutions of nonrelativistic Schrödinger equation from relativistic Klein-Gordon equation

The relativistic one-dimensional Klein-Gordon equation can be exactly solved for a certain class of potentials. But the nonrelativistic Schrödinger equation is not necessarily solvable for the same potentials. It may be possible to obtain approximate solutions for the inexact nonrelativistic potential from the relativistic exact solutions by systematically removing relativistic portion. We search for the possibility with the harmonic oscillator potential and the Coulomb potential, both of which can be exactly solvable nonrelativistically and relativistically. Though a rigorous algebraic approach is not deduced yet, it is found that the relativistic exact solutions can be a good starting point for obtaining the nonrelativistic solutions.     

Exact solution of the Klein-Gordon equation in the presence of a minimal length

We obtain exact solutions of the (1+1)-dimensional Klein-Gordon equation with linear vector and scalar potentials in the presence of a minimal length. Algebraic approach to the problem has also been studied.     

Repeatable procedures and maps in open quantum dynamics

Examples of repeatable procedures and maps are found in the open quantum dynamics of one qubit that interacts with another qubit. They show that a mathematical map that is repeatable can be made by a physical procedure that is not.     

Phase-Dependent Effects in Stern-Gerlach Experiments

In the frame of quantum mechanics, we consider an ensemble of spin-1/2 neutral particles passing through a Stern-Gerlach apparatus and explore how their motions depend on the initial phase difference between two internal spin states. Assuming the particles moving along y-axis, due to the initial phase difference between spin states, they not only split along the longitudinal direction （z-axis） but also separate along the lateral direction （x-axis）. The dependence of the lateral displacement on the initial phase difference reminds one of the picture of a quantum interference. This generalized interference provides an alternative approach to measuring the initial phase difference. The experimental realization with ultracold atoms or Bose-Einstein condensates is also discussed.     

Operator equations and Moyal products–metrics in quasi-Hermitian quantum mechanics

The Moyal product is used to cast the equation for the metric of a non-Hermitian Hamiltonian in the form of a differential equation. For Hamiltonians of the form p²+V(ix)$p^2+V(ix)$ with V polynomial this is an exact equation. Solving this equation in perturbation theory recovers known results. Explicit criteria for the hermiticity and positive definiteness of the metric are formulated on the functional level.     

Strong causality from weak via continuous monitoring

Repeated unbiased measurements cause a continual application of the weak causality principle, leading to an apparent arrow of time for continuously-monitored quantum systems.     

Quantum-classical correspondence of the relativistic equations

According to the Heisenberg correspondence principle, in the classical limit, quantum matrix element of a Hermitian operator reduces to the coefficient of the Fourier expansion of the corresponding classical quantity. In this article, such a quantum-classical connection is generalized to the relativistic regime. For the relativistic free particle or the charged particle moving in a constant magnetic field, it is shown that matrix elements of quantum operators go to quantities in Einstein’s special relativity in the classical limit. Especially, matrix element of the standard velocity operator in the Dirac theory reduces to the classical velocity. Meanwhile, it is shown that the classical limit of quantum expectation value is the time average of the classical variable.     

Tomographic approach to the violation of Bell's inequalities for quantum states of two qutrits

The tomographic method is employed to investigate the presence of quantum correlations in two classes of parameter-dependent states of two qutrits. The violation of some Bell's inequalities in a wide domain of the parameter space is shown. A comparison between the tomographic approach and a recent method elaborated by Wu, Poulsen, and Mølmer shows the better adequacy of the former method with respect to the latter one.