Momentum Operators Research Articles

Scattering in Terms of Bohmian Conditional Wave Functions for Scenarios with Non-Commuting Energy and Momentum Operators

Without access to the full quantum state, modeling quantum transport in mesoscopic systems requires dealing with a limited number of degrees of freedom. In this work, we analyze the possibility of modeling the perturbation induced by non-simulated degrees of freedom on the simulated ones as a transition between single-particle pure states. First, we show that Bohmian conditional wave functions (BCWFs) allow for a rigorous discussion of the dynamics of electrons inside open quantum systems in terms of single-particle time-dependent pure states, either under Markovian or non-Markovian conditions. Second, we discuss the practical application of the method for modeling light–matter interaction phenomena in a resonant tunneling device, where a single photon interacts with a single electron. Third, we emphasize the importance of interpreting such a scattering mechanism as a transition between initial and final single-particle BCWF with well-defined central energies (rather than with well-defined central momenta).

Noncommutative coordinate picture of the quantum phase space

We illustrate an isomorphic representation of the observable algebra for quantum mechanics in terms of the functions on the projective Hilbert space, and its Hilbert space analog, with a noncommutative product in terms of explicit coordinates and discuss the physical and dynamical picture. The isomorphism is then used as a base for the translation of the differential symplectic geometry of the infinite dimensional manifolds onto the observable algebra as a noncommutative geometry. Hence, we obtain the latter from the physical theory itself. We have essentially an extended formalism of the Schr̎odinger versus Heisenberg picture which we describe mathematically as like a coordinate map from the phase space, for which we have presented argument to be seen as the quantum model of the physical space, to the noncommutative geometry coordinated by the six position and momentum operators. The observable algebra is taken essentially as an algebra of formal functions on the latter operators. The work formulates the intuitive idea that the noncommutative geometry can be seen as an alternative, noncommutative coordinate, picture of familiar quantum phase space, at least so long as the symplectic geometry is concerned.

Linear Canonical Transformations in relativistic quantum physics

Linear Canonical Transformations (LCTs) are known in signal processing and optics as the generalization of certain useful integral transforms. In quantum theory, they can be identified as the linear transformations which keep invariant the canonical commutation relations characterizing the coordinates and momenta operators. In this work, the possibility of considering LCTs to be the elements of a symmetry group for relativistic quantum physics is studied using the principle of covariance. It is established that Lorentz transformations and multidimensional Fourier transforms are particular cases of LCTs and some of the main symmetry groups currently considered in relativistic theories can be obtained from the contractions of LCTs groups. It is also shown that a link can be established between a spinorial representation of LCTs and some properties of elementary fermions. This link leads to a classification which suggests the existence of sterile neutrinos and the possibility of describing a generation of fermions with a single field. Some possible applications of the obtained results are discussed. These results may, in particular, help in the establishment of a unified theory of fundamental interactions. Intuitively, LCTs correspond to linear combinations of energy-momentum and spacetime compatible with the principle of covariance.

Global Well-Posedness, Blow-Up and Stability of Standing Waves for Supercritical NLS with Rotation

We consider the focusing mass supercritical nonlinear Schrödinger equation with rotation $$\begin{aligned} iu_{t}=-\frac{1}{2}\Delta u+\frac{1}{2}V(x)u-|u|^{p-1}u+L_{\Omega }u,\quad (x,t)\in {\mathbb {R}}^{N}\times {\mathbb {R}}, \end{aligned}$$where \(N=2\) or 3 and V(x) is an anisotropic harmonic potential. Here \(L_{\Omega }\) is the quantum mechanical angular momentum operator. We establish conditions for global existence and blow-up in the energy space. Moreover, we prove strong instability of standing waves under certain conditions on the rotation and the frequency of the wave. Finally, we construct orbitally stable standing waves solutions by considering a suitable local minimization problem. Those results are obtained for nonlinearities which are \(L^{2}\)-supercritical.

Quantum Mechanics from the Energy Circulation Theory — Wave Function Showing an Energy Location in the 3D Real Space

The Schrödinger equation is one of the cores in quantum mechanics, but bears a contradiction. In the process to obtain the energy and momentum operators, the relation [Formula: see text] is used for [Formula: see text]. However, when they are applied to the Hamiltonian equation, the kinetic energy is set as [Formula: see text]. Based on the Energy Circulation Theory, we examine in this paper the quantization of motions of a particle. We clarify in which situation and for what energy we can use the relation [Formula: see text]. We derive a wave equation de novo to provide wave functions representing a concrete motion and energy distribution of a particle. The Schrödinger equation has a similar form by chance but the mass in our new equation is that of energy quantum expressed by [Formula: see text], which is common for any energies of any particles and decided only by the moving speed. A solution shows an energy location in the 3D real space even if it is expressed in complex. When a motion of a particle gets in circle, its circular frequency becomes quantized. In an atom, an electron circulates around the hidden dimensional axis, and the circulation can further rotate. We propose the quantization conditions for the electron orbiting, and derive the wave functions in concrete for S and P orbitals, which are different from current perceptions. We also demonstrate that the uncertainty principle is not valid for a motion of a particle.

The appearance of particle tracks in detectors

The phenomenon that a quantum particle propagating in a detector, such as a Wilson cloud chamber, leaves a track close to a classical trajectory is analyzed. We introduce an idealized quantum-mechanical model of a charged particle that is periodically illuminated by pulses of laser light resulting in repeated indirect measurements of the approximate position of the particle. For this model we present a mathematically rigorous analysis of the appearance of particle tracks, assuming that the Hamiltonian of the particle is quadratic in the position- and momentum operators, as for a freely moving particle or a harmonic oscillator.

Research on a new high-order generalized uncertainty principle in quantum system

Considering that curved space-time in coordinate or momentum representation should be curved, this paper proposes a new high-order generalized uncertainty principle by modifying the coordinate and momentum operator simultaneously, which could give a self-consistent phenomenological explanation for the existence of the minimum observable length. Moreover, using the proposed new high-order generalized uncertainty principle, we can deduce not only the minimum observable length, but also the allowable maximum observable momentum. More importantly, we further use the new generalized uncertainty principle to discuss the functional analysis of the position operator and maximum localization states, the compare the results with the existing literature. Finally, the harmonic oscillator is further discussed as an example.

New research based on the new high-order generalized uncertainty principle for Klein–Gordon equation

A new high-order generalized uncertainty principle is proposed in this paper, which can modify the coordinate operator and the momentum operator simultaneously. Afterwards, the Klein–Gordon equation with linear scalar and vector potential is investigated in the context of new principle and whose corresponding exact analytical solutions are further derived. In addition, the above results are also verified in a pure algebraic fashion by using the property of shape invariance symmetry. Moreover, the influence of the minimum length parameter β on the energy spectrum of the Klein–Gordon equation is also discussed in detail in this article.

Time-dependent momentum expectation values from different quantum probability and flux densities.

Based on the Ehrenfest theorem, the time-dependent expectation value of a momentum operator can be evaluated equivalently in two ways. The integrals appearing in the expressions are taken over two different functions. In one case, the integrand is the quantum mechanical flux density j̲, and in the other, a different quantity j̲̃ appears, which also has the units of a flux density. The quantum flux density j̲ is related to the probability density ρ via the continuity equation, and j̲̃ may as well be used to define a density ρ̃ that fulfills a continuity equation. Employing a model for the coupled dynamics of an electron and a proton, we document the properties of the densities and flux densities. It is shown that although the mean momentum derived from the two quantities is identical, the various functions exhibit a very different coordinate and time-dependence. In particular, it is found that the flux density j̲̃ directly monitors temporal changes in the probability density, and the density ρ̃ carries information about wave packet dispersion occurring in different spatial directions.

Quantum phases for electric charges and electric (magnetic) dipoles: physical meaning and implication

We analyse the physical meaning of quantum phase effects for point-like charges and electric (magnetic) dipoles in an electromagnetic (EM) field. At present, there are known eight effects of such a kind: four of them (the magnetic and electric Aharonov – Bohm phases for electrons, the Aharonov – Casher phase for a moving magnetic dipole and the He – McKellar – Wilkens phase for a moving electric dipole) had been disclosed in 20th century, while four new quantum phases had recently been found by our team (A. L. Kholmetskii, O. V. Missevitch, T. Yarman). In our analysis of physical meaning of these phases, we adopt that a quantum phase for a dipole represents a superposition of quantum phases for each charge, composing the dipole. In this way, we demonstrate the failure of the Schrödinger equation for a charged particle in an EM field to describe new quantum phase effects, when the standard definition of the momentum operator is used. We further show that a consistent description of quantum phase effects for moving particles is achieved under appropriate re-definition of this operator, where the canonical momentum of particle in EM field is replaced by the interactional EM field momentum. Some implications of this result are discussed.

Superfluids in polymer quantum mechanics

In this work, we analyze the corrections obtained on a homogeneous one-dimensional Bose gas within the high densities limit by means of the non-regular representation of quantum mechanics introduced by the polymer quantization scheme. Thus, starting from the Bogoliubov formalism, we analyze the ground expectation value of the polymer momentum operator in terms of semiclassical states in order to obtain an analytic expression for the ground state energy of the N-body system, which allows us to solve the pathological behavior commonly associated with the one-dimensional Bose–Einstein condensation through the introduction of finite size effects characterized by the contribution of the polymer corrections. We also discuss the speed of sound in our polymer version of the Bose gas and the corresponding relative shift induced by the introduction of a minimum length parameter. Finally, we investigate the emergent superfluid behavior in our polymer model by implementing an appropriate Landau’s criterion. In this case, we are able to consequently analyze the changes in the critical velocity which defines the limit between the superfluid-condensate regions, thus deducing that the polymer length acts as a kind of pseudo-potential which induces a dissipationless flow associated with the superfluid phase even in the absence of self-interactions.

On the Asymptotic Dynamics of 2-D Magnetic Quantum Systems

In this work we provide results on the long time localisation in space (dynamical localisation) of certain two-dimensional magnetic quantum systems. The underlying Hamiltonian may have the form $H=H_0+W$, where $H_0$ is rotationally symmetric, has dense point spectrum, and $W$ is a perturbation that breaks the rotational symmetry. In the latter case, we also give estimates for the growth of the angular momentum operator in time.

Algebraic approach to the Dunkl–Coulomb problem and Dunkl oscillator in arbitrary dimensions

The Dunkl–Coulomb and the Dunkl oscillator models in arbitrary space-dimensions are introduced. These models are shown to be maximally superintegrable and exactly solvable. The energy spectrum and the wave functions of both systems are obtained using different realizations of the Lie algebra $$\text{ so }(1,2).$$ The N-dimensional Dunkl oscillator admits separation of variables in both Cartesian and polar coordinates and the corresponding separated solutions are expressed in terms of the generalized Hermite, Laguerre and Gegenbauer polynomials. The N-dimensional Dunkl–Coulomb Hamiltonian admits separation of variables in polar coordinates and the separated wave functions are expressed in terms of the generalized Laguerre and Gegenbauer polynomials. The symmetry operators generalizing the Runge–Lenz vector operator are given. Together with the Dunkl angular momentum operators and reflection operators they generate the symmetry algebra of the N-dimensional Dunkl–Coulomb Hamiltonian which is a deformation of $$\text{ so }(N+1)$$ by reflections for bound states and is a deformation of $$\text{ so }(N,1)$$ by reflections for positive energy states. The symmetry operators of the N-dimensional Dunkl oscillator are obtained by the Schwinger construction and generate its invariance algebra which is a deformation of $${\textit{su}}(N)$$ by reflections.

Consequences of minimal length discretization on line element, metric tensor, and geodesic equation

When minimal length uncertainty emerging from a generalized uncertainty principle (GUP) is thoughtfully implemented, it is of great interest to consider its impacts on gravitational Einstein field equations (gEFEs) and to try to assess consequential modifications in metric manifesting properties of quantum geometry due to quantum gravity. GUP takes into account the gravitational impacts on the noncommutation relations of length (distance) and momentum operators or time and energy operators and so on. On the other hand, gEFE relates classical geometry or general relativity gravity to the energy–momentum tensors, that is, proposing quantum equations of state. Despite the technical difficulties, we intend to insert GUP into the metric tensor so that the line element and the geodesic equation in flat and curved space are accordingly modified. The latter apparently encompasses acceleration, jerk, and snap (jounce) of a particle in the quasi‐quantized gravitational field. Finite higher orders of acceleration apparently manifest phenomena such as accelerating expansion and transitions between different radii of curvature and so on.

P-wave two-particle bound and scattering states in a finite volume including QED

The mass shifts for two-fermion bound and scattering P-wave states subject to the long-range interactions due to QED in the non-relativistic regime are derived. Introducing a short range force coupling the spinless fermions to one unit of angular momentum in the framework of pionless EFT, we first calculate both perturbatively and non-perturbatively the Coulomb corrections to fermion–fermion scattering in the continuum and infinite volume context. Motivated by the research on particle–antiparticle bound states, we extend the results to fermions of identical mass and opposite charge. Second, we transpose the system onto a cubic box with periodic boundary conditions and we calculate the finite volume corrections to the energy of the lowest bound and unbound T_1^{-} eigenstates. In particular, power law corrections proportional to the fine structure constant and resembling the recent results for S-wave states are found. Higher order contributions in alpha are neglected, since the gapped nature of the momentum operator in the finite-volume environnement allows for a perturbative treatment of the QED interactions.

Compton scattering of keV photons at helium atom near the He$$^+$$(1s) threshold for small momentum transfer

We study Compton scattering at helium atom exposed to an electromagnetic field with a central frequency of 80 a.u. ( $$\sim 2.18$$ keV). We consider the situation where the incident light and scattered light are polarized along the same direction with small relative propagation angle $$\beta $$ . The energy of the emitted electron ranges from 0.2 to 2.1 a.u. ( $$\sim 5.44$$ –57.14 eV). The approach is based on previous works on stimulated Compton scattering (see Bachau et al. Phys Rev Lett 112:073001, 2014). We consider a field intensity of $$3.51\times 10^{16}$$ W/cm $$^2$$ , where stimulated Compton scattering can be treated in perturbative regime. In lowest order perturbation theory, the process results from the contribution of $$\mathbf{A\cdot \mathbf{P}}$$ in second order and $$\mathbf{A}^2$$ in first order ( $$\mathbf{P}$$ is the electron momentum operator and $$\mathbf{A}$$ the vector potential of the field); both terms induce two-photon transitions. The Compton matrix element $$|{{\mathcal {M}}}_{fg}|^2$$ is extracted numerically resolving the time-dependent Schrodinger equation, and in perturbation theory, emphasis is put on the calculation of the second-order amplitude associated with $$\mathbf{A\cdot \mathbf{P}}$$ . We investigate the cases of relative propagation angle $$\beta =0$$ and 10 degrees. The photoelectron energy distributions are dominated by the nondipole term $$\mathbf{A}^2$$ ; they increase by orders of magnitude when $$\beta $$ grows from 0 to 10 degrees. At $$\beta =0$$ degree, both $$\mathbf{A\cdot \mathbf{P}}$$ and $$\mathbf{A}^2$$ are at play and we show that the electrons are emitted in the (forward) direction of the momentum transfer $${{\mathcal {Q}}}$$ and to a lesser extent in the backward direction. When $$\beta $$ increases, the nondipole contribution $$\mathbf{A}^2$$ tends to dominate and the forward/backward asymmetry vanishes.

Dirac Delta Function from Closure Relation of Orthonormal Basis and its Use in Expanding Analytic Functions

One of revealing and widely used concepts in Physics and mathematics is the Dirac delta function. The Dirac delta function is a distribution on real lines which is zero everywhere except at a single point, where it is infinite. Dirac delta function has vital role in solving inhomogeneous differential equations. In addition, the Dirac delta functions can be used to explore harmonic information’s imbedded in the physical signals, various forms of Dirac delta function and can be constructed from the closure relation of orthonormal basis functions of functional space. Among many special functions, we have chosen the set of eigen functions of the Hamiltonian operator of harmonic oscillator and angular momentum operators for orthonormal basis. The closure relation of orthonormal functions used to construct the generator of Dirac delta function which is used to expand analytic functions log(x + 2),exp(-x2) and x within the valid region of arguments.

Group Theoretical Approach to Pseudo-Hermitian Quantum Mechanics with Lorentz Covariance and c → ∞ Limit

We present the formulation of a version of Lorentz covariant quantum mechanics based on a group theoretical construction from a Heisenberg–Weyl symmetry with position and momentum operators transforming as Minkowski four-vectors. The basic representation is identified as a coherent state representation, essentially an irreducible component of the regular representation, with the matching representation of an extension of the group C*-algebra giving the algebra of observables. The key feature is that it is not unitary but pseudo-unitary, exactly in the same sense as the Minkowski spacetime representation. The language of pseudo-Hermitian quantum mechanics is adopted for a clear illustration of the aspect, with a metric operator obtained as really the manifestation of the Minkowski metric on the space of the state vectors. Explicit wavefunction description is given without any restriction of the variable domains, yet with a finite integral inner product. The associated covariant harmonic oscillator Fock state basis has all the standard properties in exact analog to those of a harmonic oscillator with Euclidean position and momentum operators. Galilean limit and the classical limit are retrieved rigorously through appropriate symmetry contractions of the algebra and its representation, including the dynamics described through the symmetry of the phase space.

Angular momentum quantum backflow in the noncommutative plane

We study the quantum backflow problem in the noncommutative plane. In particular, we have considered a charged particle with and without an oscillator interaction with noncommuting momentum operators and examined angular momentum backflow in each case and how they differ from each other. We also propose a probability associated with the occurence of angular momentum backflow and investigate whether or not the probability depends on a physical parameter, namely the magnetic field.

Improving the Accuracy of Quasiclassical Mapping Hamiltonian Methods by Treating the Window Function Width as an Adjustable Parameter.

Mapping Hamiltonian methods for simulating electronically nonadiabatic molecular dynamics are based on representing the electronic population and coherence operators in terms of isomorphic mapping operators, which are given in terms of the auxiliary position and momentum operators. Adding a quasiclassical approximation then makes it possible to treat those auxiliary coordinates and momenta, as well as the nuclear coordinates and momenta, as classical-like phase-space variables. Within such quasiclassical mapping Hamiltonian methods, the initial sampling of the auxiliary coordinates and momenta and the calculation of expectation values of electronic observables at a later time are based on window functions whose functional form differ from one method to another. However, different methods also differ with respect to the way in which they treat the window width. More specifically, while the window width is treated as an adjustable parameter within the symmetrical quasiclassical (SQC) method, this has not been the case for methods based on the linearized semiclasscial (LSC) approximation. In the present study, we investigate the effect that turning the window width into an adjustable parameter within LSC-based methods has on their accuracy compared to SQC. The analysis is performed in the context of the spin-boson and Fenna-Matthews-Olson (FMO) complex benchmark models. We find that treating the window width in LSC-based methods as an adjustable parameter can make their accuracy comparable to that of the SQC method.