Nucleon–nucleon scattering in the light of supersymmetric quantum mechanics

By exploiting supersymmetry-inspired factorization method together with a judiciously chosen deuteron ground-state wave function, approximate higher partial wave nucleon–nucleon potentials are generated. In this context, a minor modification is also introduced to the generated potentials. The n–p scattering phase shifts are computed and analysed via the phase function method.     

Dual wavefunctions in two–dimensional quantum mechanics

It is shown that the Schrödinger equation for a large family of pairs of two–dimensional quantum potentials possess wavefunctions for which the amplitude and the phase are interchangeable, producing two different solutions which are dual to each other. This is a property of solutions with vanishing Böhm potential. These solutions can be extended to three–dimensional systems. We explicitly calculate dual solutions for physical systems, such as the repulsive harmonic oscillator and the two–dimensional hydrogen atom. These dual wavefunctions are also solutions of an analogue optical system in the eikonal limit. In this case, the potential is related to the refractive index, allowing the study of this two–dimensional dual wavefunction solutions with an optical (analogue) system.     

Measurement and “beables” in quantum mechanics

It is argued that the measurement problem reduces to the problem of modeling quasi-classical systems in a modified quantum mechanics with superselection rules. A measurement theorem is proved, demonstrating, on the basis of a principle for selecting the quantities of a system that are determinate (i.e., have values) in a given state, that after a suitable interaction between a systemS and a quasi-classical systemM, essentially only the quantity measured in the interaction and the indicator quantity ofM are determinate. The theorem justifies interpreting the noncommutative algebra of observables of a quantum mechanical system as an algebra of beables, in Bell's sense.     

The path integral on the Poincaré upper half plane and for Liouville quantum mechanics

We present a rigorous path integral treatment of free motion on the Poincaré upper half plane. The Poincaré upper half plane, as a riemannian manifold, has recently become important in string theory and in the theory of quantum chaos. The calculation is done by a time-transformation and the use of the canonical method for determining quantum corrections to the classical lagrangian. Furthermore, we shall show that the same method also works for Liouville quantum mechanics. In both cases, the energy spectrum and the normalized wavefunctions are determined.     

Unprovability of a “precognition” effect in quantum mechanics

It is shown that nonlocality gives rise to an undecidable proposition, meaning it cannot be proved true nor proved false from the usual assumptions, but is independent of them. A variation on the usual thought experiment is considered in which the observers are timelike separated, but the nonlocality fails to become a precognition effect because of this independence result.     

Poincaré's theorem and unitary transformations for classical and quantum systems

Poincaré's celebrated theorem on the nonexistence of analytical invariants of motion is extended to the case of a continuous spectrum to deal with large classical and quantum systems. It is shown that Poincaré's theorem applies to situations where there exist continuous sets of resonances. This condition is equivalent to the nonvanishing of the asymptotic collision operator as defined in modern kinetic theory. Typical examples are systems presenting relaxation processes or exhibiting unstable quantum levels. As the result of Poincaré's theorem, the unitary transformation, leading to a cyclic Hamiltonian in classical mechanics or to the diagonalization of the Hamiltonian operator in quantum mechanics, diverges. We obtain therefore a dynamical classification of large classical or quantum systems. This is of special interest for quantum systems as, historically, quantum mechanics has been formulated following closely the patterns of classical integrable systems. The well known results of Friedrichs concerning the coupling of discrete states with a continuum are recovered. However, the role of the collision operator suggests new ways of eliminating the divergence in the unitary transformation theory.     

Observations of the Ramsauer–Townsend effect in quaternionic quantum mechanics

In this article, one of the well-known effects in quantum mechanics is addressed and also the extended form of quantum mechanics which is based on quaternions is presented. In the presence of this version of quantum mechanics the Ramsauer–Townsend effect has been investigated and the existence of this phenomenon is studied according to quaternionic calculations; results are presented by graphs.     

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.     

A generalization of the Fényes — Nelson stochastic model of quantum mechanics

It is shown that the stochastic model of Fényes and Nelson can be generalized in such a way that the diffusion constant of the Markov theory becomes a free parameter. This extra freedom allows one to identify quantum mechanics with a class of Markov process with diffusion constants varying from 0 to .     

Position measurements in the de Broglie–Bohm interpretation of quantum mechanics

The de Broglie–Bohm interpretation of quantum mechanics assigns positions and trajectories to particles. We analyze the validity of a formula for the velocities of Bohmian particles which makes the analysis of these trajectories particularly simple. We apply it to particle detectors of four different types and show that the detectors of three of these types lead to “surrealistic trajectories”, i.e., leave a trace where the Bohmian particle was not present.     

Gauge covariance of the Aharonov–Bohm phase in noncommutative quantum mechanics

The gauge covariance of the wave function phase factor in noncommutative quantum mechanics (NCQM) is discussed. We show that the naive path integral formulation and an approach where one shifts the coordinates of NCQM in the presence of a background vector potential leads to the gauge non-covariance of the phase factor. Due to this fact, the Aharonov–Bohm phase in NCQM which is evaluated through the path-integral or by shifting the coordinates is neither gauge invariant nor gauge covariant. We show that the gauge covariant Aharonov–Bohm effect should be described by using the noncommutative Wilson lines, what is consistent with the noncommutative Schrödinger equation. This approach can ultimately be used for deriving an analogue of the Dirac quantization condition for the magnetic monopole.     

Scattering states of Woods–Saxon interaction in minimal length quantum mechanics

We propose a very simple approach to deal with the problems of the modified Schrödinger equation due to minimal length and thereby solve the minimal length Schrödinger equation in the presence of a non-minimal Woods–Saxon interaction. The transmission and reflection coefficients are reported as well.     

Can the breaking of time-reversal symmetry remove degeneracies in quantum mechanics?

Reduction of spatial symmetry can remove the degeneracy of energy levels in quantum mechanics. The break of time-reversal symmetry by inclusion of a dissipative environment can have a similar effect. The corresponding time-evolution of position and momentum fluctuations can be described by a nonlinear differential equation that can lead to bifurcations and, thus, splitting of energy levels.     

A new non-perturbative approach in quantum mechanics for time-independent Schr?dinger equations

A new non-perturbative approach is proposed to solve time-independent Schr?dinger equations in quantum mechanics.It is based on the homotopy analysis method(HAM)that was developed by the author in 1992 for highly nonlinear equations and has been widely applied in many fields.Unlike perturbative methods,this HAM-based approach has nothing to do with small/large physical parameters.Besides,convergent series solution can be obtained even if the disturbance is far from the known status.A nonlinear harmonic oscillator is used as an example to illustrate the validity of this approach for disturbances that might be one thousand times larger than the possible superior limit of the perturbative approach.This HAM-based approach could provide us rigorous theoretical results in quantum mechanics,which can be directly compared with experimental data.Obviously,this is of great benefit not only for improving the accuracy of experimental measurements but also for validating physical theories.     

Integer quantum Hall effect in Kekulé-patterned graphene

Y-shaped Kekulébond textures in a honeycomb lattice on a graphene-copper superlattice have recently been experimentally revealed.In this paper,the effects of such a bond modulation on the transport coefficients of Kekulé-patterned graphene are investigated in the presence of a perpendicular magnetic field.Analytical expressions are derived for the Hall and longitudinal conductivities using the Kubo formula.It is found that the Y-shaped Kekulébond texture lifts the valley degeneracy of all Landau levels except that of the zero mode,leading to additional plateaus in the Hall conductivity accompanied by a split of the corresponding peaks in the longitudinal conductivity.Consequently,the Hall conductivity is quantized as±ne²/h for n=2,4,6,8,10,...,excluding some plateaus that disappear due to the complete overlap of the Landau levels of different cones.These results also suggest that DC Hall conductivity measurements will allow us to determine the Kekulébond texture amplitude.     

Multichannel scattering amplitudes of microparticle in a quantum wire with three-dimensional δ-potentials

The penetration of the microparticle (electron) through the δ-potential barriers which are embedded into the quantum wire of cylindrical shape with the characteristic effect of dimensional quantization in the transverse direction is considered. The analytical expressions for the amplitudes of the multichannel scattering, which are valid for the barriers consisting of one and two δ-potentials located at some distance from each other are obtained. It is shown that the multichannel scattering turns into the one-dimensional, if the initial energy of the longitudinal motion of electron is not sufficient to excite the higher channels of scattering. In this case, the obtained formulas for the scattering amplitudes coincide with the well-known expressions for the one-dimensional scattering. The obtained solutions satisfy the conservation law of the number of particles, which is the indirect proof of the validity of derived formulas.