The Aharonov–Bohm effect in noncommutative quantum mechanics

The Aharonov–Bohm effect in noncommutative (NC) quantum mechanics is studied. First, by introducing a shift for the magnetic vector potential we give the Schrödinger equations in the presence of a magnetic field on NC space and NC phase space, respectively. Then, by solving the Schrödinger equations, we obtain the Aharonov–Bohm phase on NC space and NC phase space, respectively.     

Quantum projection. Integration of the open Tod quantum chain

A quantum projection method is developed on the basis of noncommutative integration of linear differential equations and the results of M. A. Ol’shanetskii and A. M. Perelomov on the integration of classical Hamiltonian systems (projection method). The method proposed makes it possible to obtain in explicit form solutions of the quantum equations whose classical analogs can be integrated by projection. Then the semisimplicity property of the symmetry algebra of the original equation is no longer a factor. The solution basis of a Schrödinger equation with the potential of an open three-particle Tod chain is constructed as a nontrivial example.     

Foundations of quantum mechanics: The Langevin equations for QM

Stochastic derivations of the Schrödinger equation are always developed on very general and abstract grounds. Thus, one is never enlightened which specific stochastic process corresponds to some particular quantum mechanical system, that is, given the physical system—expressed by the potential function, which fluctuation structure one should impose on a Langevin equation in order to arrive at results identical to those comming from the solutions of the Schrödinger equation. We show, from first principles, how to write the Langevin stochastic equations for any particular quantum system. We also show the relation between these Langevin equations and those proposed by Bohm in 1952. We present numerical simulations of the Langevin equations for some quantum mechanical problems and compare them with the usual analytic solutions to show the adequacy of our approach. The model also allows us to address important topics on the interpretation of quantum mechanics.     

The Madelung Picture as a Foundation of Geometric Quantum Theory

Despite its age, quantum theory still suffers from serious conceptual difficulties. To create clarity, mathematical physicists have been attempting to formulate quantum theory geometrically and to find a rigorous method of quantization, but this has not resolved the problem. In this article we argue that a quantum theory recursing to quantization algorithms is necessarily incomplete. To provide an alternative approach, we show that the Schrödinger equation is a consequence of three partial differential equations governing the time evolution of a given probability density. These equations, discovered by Madelung, naturally ground the Schrödinger theory in Newtonian mechanics and Kolmogorovian probability theory. A variety of far-reaching consequences for the projection postulate, the correspondence principle, the measurement problem, the uncertainty principle, and the modeling of particle creation and annihilation are immediate. We also give a speculative interpretation of the equations following Bohm, Vigier and Tsekov, by claiming that quantum mechanical behavior is possibly caused by gravitational background noise.     

Remarks on a quantum mechanical treatment of internal excitation in low energy nuclear fission

Internal excitations of the fissioning nucleus are usually described phenomenologically by friction terms. In the present paper an approach is discussed which is in principle based on a correct quantum mechanical treatment taking the projection form of the Schrödinger equation as a starting point. Considering nuclear fission as an almost adiabatic process an estimate for the friction energy is made. In this very crude estimate only 10–15% of the collective energy gain in going from the saddle to the scission point is transformed into internal excitation energy. This is in agreement with experimental data showing pronounced substructure effects which would be destroyed in the presence of a larger friction. As compared to other microscopic calculations, in the present work the total Hamiltonian is split in such a way that the only perturbation term being responsible for the deformation is essentially the Coulomb energy. By this assumption the calculation of transition probabilities to intrinsically excited states becomes rather insensitive to the exact excitation energy spectrum of the compound nucleus.     

Theory of Stochastic Schrödinger Equation in Complex Vector Space

A generalized Schrödinger equation containing correction terms to classical kinetic energy, has been derived in the complex vector space by considering an extended particle structure in stochastic electrodynamics with spin. The correction terms are obtained by considering the internal complex structure of the particle which is a consequence of stochastic average of particle oscillations in the zeropoint field. Hence, the generalised Schrödinger equation may be called stochastic Schrödinger equation. It is found that the second order correction terms are similar to corresponding relativistic corrections. When higher order correction terms are neglected, the stochastic Schrödinger equation reduces to normal Schrödinger equation. It is found that the Schrödinger equation contains an internal structure in disguise and that can be revealed in the form of internal kinetic energy. The internal kinetic energy is found to be equal to the quantum potential obtained in the Madelung fluid theory or Bohm statistical theory. In the rest frame of the particle, the stochastic Schrödinger equation reduces to a Dirac type equation and its Lorentz boost gives the Dirac equation. Finally, the relativistic Klein–Gordon equation is derived by squaring the stochastic Schrödinger equation. The theory elucidates a logical understanding of classical approach to quantum mechanical foundations.     

Spin-dependent potentials in the linearized collective Schrödinger equation for nuclear quadrupole vibrations

The linearized collective Schrödinger equation for nuclear quadrupole surface vibrations incorporates a new spin degree of freedom with a spin value of 3/2. We use this equation to describe the low energy spectrum of certain even-odd Ir nuclei which have a spin 3/2 in their ground state. For that purpose we explicitly introduce collective spin-dependent potentials which simulate the interaction of the valence nucleon with the core. The linearized Schrödinger equation is transformed into an effective Schrödinger equation with collective spin-dependent potentials. Already collective spin-orbit couplings of SO(3) and SO(5) type are sufficient to reproduce the lowest excited states of even-odd Ir nuclei.     

Derivation and application of quantum Hamilton equations of motion

Non‐relativistic quantum systems are analyzed theoretically or by numerical approaches using the Schrödinger equation. Compared to the options available to treat classical mechanical systems this is limited, both in methods and in scope. However, based on Nelson's stochastic mechanics, the mathematical structure of quantum mechanics has in some aspects been developed into a form analogous to classical analytical mechanics. We show here that finding the Nash equilibrium for a stochastic optimal control problem, which is the quantum equivalent to Hamilton's principle of least action, allows to derive two things: i) the Schrödinger equation as the Hamilton‐Jacobi‐Bellman equation of this optimal control problem and ii) a set of quantum dynamical equations which are the generalization of Hamilton's equations of motion to the quantum world. We derive their general form for the non‐stationary and the stationary case. For the harmonic oscillator, the stationary equations lead to the coherent states, and we establish a numerical procedure to solve for the ground state properties without using the Schrödinger equation.

     

Quantum Hall effect in noncommutative quantum mechanics

We study the quantum Hall (QH) effect for an electron moving in a plane whose coordinates and momenta are noncommuting under the influence of uniform external magnetic and electric fields. After solving the time independent Schrödinger equation both on a noncommutative space (NCS) and a noncommutative phase space (NCPS), we obtain the energy eigenvalues and eigenfunctions of the relevant Hamiltonian. We derive the electric current whose expectation value gives the QH effect both on a NCS and a NCPS.     

On foundation of quantum physics

Some aspects of the interpretation of quantum theory are discussed. It is emphasized that quantum theory is formulated in the Cartesian coordinate system; in other coordinates the result obtained with the help of the Hamiltonian formalism and commutator relations between “canonically conjugated” coordinate and momentum operators leads to a wrong version of quantum mechanics. The origin of time is analyzed by the example of atomic collision theory in detail; it is shown that the time-dependent Schrödinger equation is meaningless since in the high-impact-energy limit it transforms into an equation with two time-like variables. Following the Einstein-Rozen-Podolsky experiment and Bell’s inequality, the wave function is interpreted as an actual field of information in the elementary form. The concept “measurement” is also discussed.     

Non-abelian path integrals and generalised quantum mechanics in the external field of the 't Hooft-Polyakov monopole

By means of a “lagrangian” with values in the Lie algebra of some arbitrary Lie group we introduce a non-abelian path integral. For such generalised quantum mechanics we derive a non-abelian Schrödinger equation for the case of a particle in the 't Hooft-Polyakov monopole field.     

Lagrangian form of Schrödinger equation

Lagrangian formulation of quantum mechanical Schrödinger equation is developed in general and illustrated in the eigenbasis of the Hamiltonian and in the coordinate representation. The Lagrangian formulation of physically plausible quantum system results in a well defined second order equation on a real vector space. The Klein–Gordon equation for a real field is shown to be the Lagrangian form of the corresponding Schrödinger equation.     

Quantitative conditions for time evolution in terms of the von Neumann equation

The adiabatic theorem describes the time evolution of the pure state and gives an adiabatic approximate solution to the Schr ¨odinger equation by choosing a single eigenstate of the Hamiltonian as the initial state. In quantum systems, states are divided into pure states(unite vectors) and mixed states(density matrices, i.e., positive operators with trace one). Accordingly, mixed states have their own corresponding time evolution, which is described by the von Neumann equation. In this paper, we discuss the quantitative conditions for the time evolution of mixed states in terms of the von Neumann equation. First, we introduce the definitions for uniformly slowly evolving and δ-uniformly slowly evolving with respect to mixed states, then we present a necessary and sufficient condition for the Hamiltonian of the system to be uniformly slowly evolving and we obtain some upper bounds for the adiabatic approximate error. Lastly, we illustrate our results in an example.     

Quadratic and Coulomb Terms for the Spectrum of a Three-Electron Quantum Dot

We consider the Hamiltonian of a three-electron quantum dot composed of quadratic plus Coulomb terms and calculate the system’s spectra. We next apply the hyperradius to reduce the three-body Schrödinger equation into a one-variable differential equation that is solvable. To avoid the complexity, the Taylor expansion of the effective potential is enters the problem and thereby a solution is found for the eigenvalues of the corresponding three-body Schrödinger equation in terms of the Wigner parameter.     

A quasi-analytical approach to study energy levels of a two-electron quantum dot

Our approach is quasi-analytical. We consider the Hamiltonian of a two-electron quantum dot composed of quadratic plus Coulomb terms as well as a term related to the interaction with the external magnetic field. To avoid the complexity, the Taylor expansion of the effective potential is introduced into the problem and thereby a solution is found for the eigenvalues of the corresponding two-body Schrödinger equation in terms of the Wigner parameter. We have finally made a comparison with some other theoretical results.     

Quantum transport of spin-polarized carriers in quasi paramagnetic quantum wires: Green’s function formalism

In this paper, the Schrödinger equation is solved for approximation of the ground state energies and associated wave functions of carriers confined in a rectangular semiconductor (SC) quantum wire embedded in a SiO₂ matrix. The problem was treated with the effective one band Hamiltonian. The finite difference scheme was used for the discretization of 2D Schrödinger equation and LAPACK package to resolve the band matrix. The energy levels were determined and the coupling between quantum wires was investigated. The effect on energies and relative wave functions of quantum wires number, size and separation was studied. The results obtained show that the energy levels can be importantly modified and controlled by these parameters. The interaction is manifested by a reduction in energies and an increase in the peak value of the wave function of the higher energy wire. This study offers a fast and inexpensive way to check device designs and processes and can be used in diverse device applications.     

A quantum algebra approach to discrete equations on uniform lattices

A quantum algebra method for deducing the symmetries of discrete equations on uniform lattices is proposed. In principle, such a procedure can be applied to discretizations in a single coordinate (space or time) and the symmetries obtained in this way are indeed differential-difference operators. Firstly, the method is illustrated on two known examples that have been also analysed from the usual Lie symmetry approach: a uniform space lattice discretization of the (1+1) free heat-Schrödinger equation associated to a quantum Schrödinger algebra, and a discrete space (1+1) wave equation provided by a quantumso(2, 2) algebra. Furthermore, we construct a discrete space (2+1) wave equation from a new quantumso(3, 2) algebra, to show that this method is useful in higher dimensions. Time discretizations are also commented.     

A quantum model for the stock market

Beginning with several basic hypotheses of quantum mechanics, we give a new quantum model in econophysics. In this model, we define wave functions and operators of the stock market to establish the Schrödinger equation for stock price. Based on this theoretical framework, an example of a driven infinite quantum well is considered, in which we use a cosine distribution to simulate the state of stock price in equilibrium. After adding an external field into the Hamiltonian to analytically calculate the wave function, the distribution and the average value of the rate of return are shown.     

Accurate modelling of InGaN quantum wells

The internal field, the band structure and the oscillator strengths of the optical transitions of wurtzite strained InGaN quantum wells are accurately computed by a self-consistent solution of the Poisson equation and an eight-band k · p Schrödinger equation taking into account charges due to polarisation fields, doping and free carriers. The results are used to compare luminescence and gain spectra for single and triple quantum well structures and to elucidate the effect of the polarisation fields.     

On the anharmonic oscillator in quantum mechanics and in field theory

A modified Rayleigh-Schrödinger perturbation method is used to derive explicit expansions for the eigenvalues and eigensolutions of the anharmonic oscillator. We then point out the dual relationship between the anharmonic oscillator and the Schrödinger equation for a Yukawa potential. Finally we consider an application of the method to a field-theoretic Hamiltonian, since the anharmonic oscillator plays a dominant role in many field-theoretic models.