Momentum Operators Research Articles

Momentum Gauge Fields and Non-Commutative Space–Time

In this work, we present a gauge principle that starts with the momentum space representation of the position operator (x^i=iℏ∂∂pi), rather than starting with the position space representation of the momentum operator (p^i=−iℏ∂∂xi). This extension of the gauge principle can be seen as a dynamical version of Born’s reciprocity theory, which exchanges position and momentum. We discuss some simple examples with this new type of gauge theory: (i) analog solutions from ordinary gauge theory in this momentum gauge theory, (ii) Landau levels using momentum gauge fields, and (iii) the emergence of non-commutative space–times from the momentum gauge fields. We find that the non-commutative space–time parameter can be momentum dependent, and one can construct a model where space–time is commutative at low momentum, but becomes non-commutative at high momentum.

Degenerate States in Nonlinear Sigma Model with SU(2) Symmetry

Entanglement in quantum theory is a peculiar concept to scientists. With this concept we are forced to re-consider the cluster property which means that one event is irrelevant to another event when they are fully far away. In the recent works we showed that the quasi-degenerate states induce the violation of cluster property in antiferromagnets when the continuous symmetry breaks spontaneously. We expect that the violation of cluster property will be observed in other materials too, because the spontaneous symmetry breaking is found in many systems such as the high temperature superconductors and the superfluidity. In order to examine the cluster property for these materials, we studied a quantum nonlinear sigma model with U(1) symmetry in the previous work. There we showed that the model does have quasi-degenerate states. In this paper we study the quantum nonlinear sigma model with SU(2) symmetry. In our approach we first define the quantum system on the lattice and then adopt the representation where the kinetic term is diagonalized. Since we have no definition on the conjugate variable to the angle variable, we use the angular momentum operators instead for the kinetic term. In this representation we introduce the states with the fixed quantum numbers and carry out numerical calculations using quantum Monte Carlo methods and other methods. Through analytical and numerical studies, we conclude that the energy of the quasi-degenerate state is proportional to the squared total angular momentum as well as to the inverse of the lattice size.

Conversion of a beam carrying fractionnal angular momentum in High-Harmonics Generation

Exotic light fields combining non-trivial spin and angular momentum may not be eigenstates of either the spin or orbital angular momenta operators. For these fields, it is relevant to define a Generalized Angular Momentum operator of which they are eigenvectors. Their associated eigenvalues can take, depending on the case, non-integer values. We report that this new quantity is conserved via non-linear phenomena, such as High Harmonic Generation.

Bessel vortices in spin-orbit-coupled spin-1 Bose-Einstein condensates

We investigate the stationary vortex solutions in two-dimensional (2D) Rashba spin-orbit (SO) coupled spin-1 Bose-Einstein condensate (BEC). By introducing the generalized momentum operator, the linear version of the system can be solved exactly and its solutions are a set of the Bessel vortices. Based on the linear version solutions, the stationary vortex solutions of the full nonlinear system are constructed and determined entirely by the variational approximation. The results show that the variational results are in good agreement with the numerical ones. By means of the variational results, the vortex ground state phase-transition between the stationary vortex solutions, stability, and the unit Bloch vector textures are discussed in detail. The results have the potential to be realized in experiment.

The Clifford-Yang algebra, noncommutative Clifford phase spaces and the deformed quantum oscillator

We construct the novel Clifford-Yang algebra which is an extension of the Yang algebra in noncommutative phase spaces. The Clifford-Yang algebra allows us to write down the commutators of the [Formula: see text] polyvector-valued coordinates and momenta which are compatible with the Jacobi identities, the Weyl–Heisenberg algebra, and paves the way for a formulation of quantum mechanics (QM) in noncommutative Clifford spaces. We continue with a detailed study of the isotropic 3D quantum oscillator in noncommutative spaces and find the energy eigenvalues and eigenfunctions. These findings differ considerably from the ordinary quantum oscillator in commutative spaces. We find that QM in noncommutative spaces leads to very different solutions, eigenvalues, and uncertainty relations than ordinary QM in commutative spaces. The generalization of QM to noncommutative Clifford (phase) spaces is attained via the Clifford-Yang algebra. The operators are now given by the generalized angular momentum operators involving polyvector coordinates and momenta. The eigenfunctions (wave functions) are now more complicated functions of the polyvector coordinates. We conclude with some important remarks.

On Momentum Operators Given by Killing Vectors Whose Integral Curves Are Geodesics

The paper considers momentum operators on intrinsically curved manifolds. Given that momentum operators are Killing vector fields whose integral curves are geodesics, the corresponding manifold is flat or of the compact type with positive constant sectional curvature and dimensions equal to 1, 3, or 7. Explicit representations of momentum operators and the associated Casimir element are discussed for the 3-sphere S3. It is verified that the structural constants of the underlying Lie algebra are proportional to 2 ℏ/R, where R is the curvature radius of S3 and ℏ is the reduced Planck’s constant. This results in a countable energy and momentum spectrum of freely moving particles in S3. The maximal resolution of the possible momenta is given by the de Broglie wave length, λR=πR, which is identical to the diameter of the manifold. The corresponding covariant position operators are defined in terms of geodesic normal coordinates, and the associated commutator relations of position and momentum are established.

SPHERICALLY SYMMETRIC SYSTEM OF GRAVITATIONAL AND ELECTROMAGNETIC FIELDS AND THE STRUCTURE OF ITS CONFIGURATION SPACE

Geometrodynamics of charged black holes (BH) described by the system of Maxwell Einstein equations is considered. We start from a spherically symmetric metric, a reduced action, and a Lagrangian written in characteristic variables. The configuration space (CS) metric, Hamiltonian, momentum and electromagnetic constraints are constructed. The system has conservation laws of charge q and mass m. The action functional is transformed into a Jacobi-type functional in CS with a metric conformal to the CS metric. A transformation of field variables is introduced which brings the CS metric to the "Lorentzian"form. The resulting CS metric is the metric of a flat nonholonomic section of a 4-dimensional space. In the new variables, the squared momenta of the system has the Lorentz form. On this basis, quantization is considered. Thanks to the structure of the CS, the momentum operators, the DeWitt equations, and the mass and charge operators are constructed. The equations system of CBH quantum states with certain q and m is constructed. For comparison, we consider the CBH reduced model limited in the T-region. In such the simplified formulation, the T-model equations are integrated and lead to the CBH with continuous spectrum of m and q.

Restrictions Imposed by the Wave Function on the Results of Particle Momentum Measurements

Using the example of the quantum dynamics of a particle in a one-dimensional configuration space (OCS), it is shown that to know the wave function implies not only statistical restrictions on the measurement results: the integrand in the standard formula for calculating the average values of (self-adjoint) operators and the Schr\"{o}dinger equation for the modulus and phase of the wave function uniquely also define ' fields of operators' as functions of coordinate and time. A key role in establishing the physical meaning of these fields is played by the fact that the field of the kinetic energy operator contains two heterogeneous contributions: the first is determined by the field of the momentum operator, which is related only to the phase of the wave function, and the second coincides with the so-called "quantum mechanical potential", which is related only to the amplitude of the wave function. The values of these fields at each point of the OCS are considered as the average values of the corresponding observables for a pair of noninteracting particles (for a pair of systems of a one-particle ensemble). At each such point, the first contribution to the field of kinetic energy describes the kinetic energy of the center of mass of a pair of particles, and the second -- the energy of their motion relative to the center of mass. The field of the momentum operator and the field of the kinetic energy operator, taking into account the K\"{o}nig theorem, uniquely determine in the OCS two fields of particle momentum values at each point of the OCS. An analogue of the Heisenberg inequality for the deviations of both momentum fields from the field of the momentum operator is obtained.

The Quantum Mechanics Canonically Associated to Free Probability I: Free Momentum and Associated Kinetic Energy

After a short review of the quantum mechanics canonically associated with a classical real valued random variable with all moments, we begin to study the quantum mechanics canonically associated to the standard semi-circle random variable [Formula: see text], characterized by the fact that its probability distribution is the semi-circle law [Formula: see text] on [Formula: see text]. We prove that, in the identification of [Formula: see text] with the [Formula: see text]-mode interacting Fock space [Formula: see text], defined by the orthogonal polynomial gradation of [Formula: see text], [Formula: see text] is mapped into position operator and its canonically associated momentum operator [Formula: see text] into [Formula: see text] times the [Formula: see text]-Hilbert transform [Formula: see text] on [Formula: see text]. In the first part of the present paper, after briefly describing the simpler case of the [Formula: see text]-harmonic oscillator, we find an explicit expression for the action, on the [Formula: see text]-orthogonal polynomials, of the semi-circle analogue of the translation group [Formula: see text] and of the semi-circle analogue of the free evolution [Formula: see text], respectively, in terms of Bessel functions of the first kind and of confluent hyper-geometric series. These results require the solution of the inverse normal order problem on the quantum algebra canonically associated to the classical semi-circle random variable and are derived in the second part of the present paper. Since the problem to determine, with purely analytic techniques, the explicit form of the action of [Formula: see text] and [Formula: see text] on the [Formula: see text]-orthogonal polynomials is difficult, the above mentioned results show the power of the combination of these techniques with those developed within the algebraic approach to the theory of orthogonal polynomials.

The role of Pryce’s spin and coordinate operators in the theory of massive Dirac fermions

It is shown that the components of Pryce’s spin operator of Dirac’s theory are SU(2) generators of a representation carried by the space of Pauli’s spinors determining the polarization of the plane wave solutions of Dirac’s equation. These operators are conserved via Noether’s theorem such that new conserved polarization operators can be defined for various polarizations. The corresponding one-particle operators of quantum theory are derived showing how these are related to the isometry generators of the massive Dirac fermions of any polarization, including momentum-dependent ones. In this manner, the problem of separating conserved spin and orbital angular momentum operators is solved naturally. Moreover, the operator proposed by Pryce as a mass-center coordinate is studied, showing that after quantization, this becomes in fact the dipole one-particle operator. As an example, the quantities determining the principal one-particle operators are derived for the first time in a momentum-helicity basis.

A family of nonlinear Schrodinger equations and their solitons solutions

In this communication, three different forms of fractional nonlinear Schrödinger equations have been constructed based on the notion of nonlocal generalized fractional momentum operator, the fractional expansion Riccati method and the notion of Laplacian operator in fractal dimensions. Their physical properties are analyzed and their associated solitonic solutions are obtained and discussed. It was observed that the first two approaches give comparable solutions, whereas the third approach give rise to a nonlinear Schrödinger equation characterized by variable coefficients, and hence, a large family of solitons solutions may be obtained accordingly.PACS numbers: 42.65.Tg, 05.45.Df, 02.30.Tb.

Algebraic structure of Dirac Hamiltonians in non-commutative phase space

In this article we study two-dimensional Dirac Hamiltonians with non-commutativity both in coordinates and momenta from an algebraic perspective. In order to do so, we consider the graded Lie algebra generated by Hermitian bilinear forms in the non-commutative dynamical variables and the Dirac matrices in dimensions. By further defining a total angular momentum operator, we are able to express a class of Dirac Hamiltonians completely in terms of these operators. In this way, we analyze the energy spectrum of some simple models by constructing and studying the representation spaces of the unitary irreducible representations of the graded Lie algebra . As application of our results, we consider the Landau model and a fermion in a finite cylindrical well.

Formula omitted] electronic structure and strain-dependence of the crystal field splitting and inter-band transition energy of W–InN

In this paper, we study the electronic structure of wurtzite Indium Nitride (W–InN) using a k→∙π→ theory. Here π→ is the relativistic momentum operator. It is based on an eight-band model; three valence bands and one conduction band, each Kramers’ split by spin-orbit interaction. We diagonalize the k→∙π→ Hamiltonian following a two-band model, where each valence band is treated separately with the conduction band by using a suitable basis set in the presence of spin-orbit interaction. Effects of other valence bands on the dispersion of a given valence band are incorporated using second order perturbation theory. Valence band dispersions are plotted and compared with a previous calculation and our own results obtained by using a different method. Agreement is found to be satisfactory. Electronic and hole effective masses are calculated and compared with previously reported values, which are substantially different from each other. Fair agreement is noticed. Discrepancies are discussed. Effect of strain is calculated on the electronic structure by calculating the crystal field energy and transition energy for small variations of strain. Although results could not be compared with any previous calculation, due presumably to their absence in literature, trends and systematics are found to be in accord with the same found in case of other wurtzite semiconductors.

Klein–Gordon equation for electrically charged particles with new energy-momentum operator

We address the Klein–Gordon equation for a spinless charged particle in the presence of an electromagnetic (EM) field, and focus on its known shortcoming, related to the existence of solutions with a negative probability density. We disclose a principal way to overcome this shortcoming, using our recent results obtained in the analysis of quantum phase effects for charges and dipoles, which prove the need to abandon the customary definition of the momentum operator for a charged particle in an EM field through its canonical momentum, and to adopt the more general definition of this operator through the sum of mechanical and EM momenta for the system ‘charged particle in an EM field’. We show that the application of the new energy-momentum operator to the Klein–Gordon equation actually eliminates solutions with negative probability density. Some implications of the obtained results are discussed.

Position and momentum operators for a moving particle in bulk

In this paper we explore how to describe a bulk moving particle in the dual conformal field theories (CFTs). One aspect of this problem is to construct the dual state of the moving particle. On the other hand one should find the corresponding operators associated with the particle. The dynamics of the particle, i.e., the geodesic equation, can be formulated as a Hamiltonian system with canonical variables. The achievements of our paper are to construct the dual CFT states and the operators corresponding to the canonical variables. The expectation values of the operators give the expected solutions of the geodesic line, and the quantum commutators reduce to the classical Poisson brackets to leading order in the bulk gravitational coupling. Our work provides a framework to understand the geodesic equation, that is gravitational attraction, in the dual CFTs.

Schrödinger's equation as a Hamiltonian system

This article describes the concept of the time-dependent Schrödinger equation (TDSE) as a review to add a detailed understanding of the steps to formulate TDSE which can be seen from the classical mechanic's concept as a mechanical wave function. In this article, a comprehensive approach to the concepts of momentum and energy in particles will be described using operators working on wave functions. The results show that the use of momentum and energy operators can show that TDSE is a Hamiltonian system.

New Challenges for Classical and Quantum Probability

The discovery that any classical random variable with all moments gives rise to a full quantum theory (that in the Gaussian and Poisson cases coincides with the usual one) implies that a quantum–type formalism will enter into practically all applications of classical probability and statistics. The new challenge consists in finding the classical interpretation, for different types of classical contexts, of typical quantum notions such as entanglement, normal order, equilibrium states, etc. As an example, every classical symmetric random variable has a canonically associated conjugate momentum. In usual quantum mechanics (associated with Gaussian or Poisson classical random variables), the interpretation of the momentum operator was already clear to Heisenberg. How should we interpret the conjugate momentum operator associated with classical random variables outside the Gauss–Poisson class? The Introduction is intended to place in historical perspective the recent developments that are the main object of the present exposition.

Pseudo- symmetric Dirac equation: effect of a new mean spin angular momentum operator on Gilbert damping

The pseudo- symmetric Dirac equation is proposed and analyzed by using a non-unitary Foldy–Wouthuysen transformations. A new spin operator symmetric expectation value (called the mean spin operator) for an electron interacting with a time-dependent electromagnetic field is obtained. We show that spin magnetization—which is the quantity usually measured experimentally—is not described by the standard spin operator but by this new mean spin operator to properly describe magnetization dynamics in ferromagnetic materials and the corresponding equation of motion is compatible with the phenomenological model of the Landau–Lifshitz–Gilbert equation (LLG).

On the convergence of finite difference scheme for a Schrödinger type equation

In the present paper, an initial boundary value problem for the linear Schrodinger equation including the momentum operator ¨ is introduced. This problem is discretized by the finite difference method and a difference scheme is presented. Moreover, an estimate for the solution of the proposed scheme is obtained. Finally, with the help of the estimate, it is proved that the proposed scheme is unconditionally stable and convergent.

Invariant quadratic operators associated with linear canonical transformations and their eigenstates

The main purpose of this work is to identify invariant quadratic operators associated with Linear Canonical Transformations (LCTs) which could play important roles in physics. In quantum physics, LCTs are the linear transformations which keep invariant the Canonical Commutation Relations (CCRs). In this work, LCTs corresponding to a general pseudo-Euclidian space are considered and related to a phase space representation of quantum theory. Explicit calculations are firstly performed for the monodimensional case to identify the corresponding LCT-invariant quadratic operators then multidimensional generalizations of the obtained results are deduced. The eigenstates of these operators are also identified. A first kind of LCT-invariant operator is a second order polynomial of the coordinates and momenta operators. The coefficients of this polynomial depend on the mean values and the statistical variances-covariances of the coordinates and momenta operators themselves. It is shown that these statistical variances-covariances can be related with thermodynamic variables. In this context, new quantum corrections to the ideal gas state equation are deduced from correction to the Hamiltonian operator of non-relativistic free quantum particles that is suggested by LCT-covariance. Two other LCT-invariant quadratic operators, which can be considered as the number operators of some quasiparticles, are also identified: the first one is a number operator of bosonic type quasiparticles and the second one corresponds to fermionic type. This fermionic LCT-invariant quadratic operator is directly related to a spin representation of LCTs. It is shown explicitly, in the case of a relativistic pentadimensional theory, that the eigenstates of this operator can be considered as basic quantum states of elementary fermions. A classification of the fundamental fermions, compatible with the Standard Model of particle physics, is established from a classification of these states.