Wave Functions In Momentum Space Research Articles

Old Game, New Rules: Rethinking the Form of Physics

We investigate the modeling capabilities of sets of coupled classical harmonic oscillators (CHO) in the form of a modeling game. The application of the simple but restrictive rules of the game lead to conditions for an isomorphism between Lie-algebras and real Clifford algebras. We show that the correlations between two coupled classical oscillators find their natural description in the Dirac algebra and allow to model aspects of special relativity, inertial motion, electromagnetism and quantum phenomena including spin in one go. The algebraic properties of Hamiltonian motion of low-dimensional systems can generally be related to certain types of interactions and hence to the dimensionality of emergent space-times. We describe the intrinsic connection between phase space volumes of a 2-dimensional oscillator and the Dirac algebra. In this version of a phase space interpretation of quantum mechanics the (components of the) spinor wavefunction in momentum space are abstract canonical coordinates, and the integrals over the squared wave function represents second moments in phase space. The wave function in ordinary space-time can be obtained via Fourier transformation. Within this modeling game, 3+1-dimensional space-time is interpreted as a structural property of electromagnetic interaction. A generalization selects a series of Clifford algebras of specific dimensions with similar properties, specifically also 10- and 26-dimensional real Clifford algebras.

Scaling Limit Analysis of Borromean Halos

The analysis of the core recoil momentum distribution of neutron-rich isotopes of light exotic nuclei is performed within a model of the halo nuclei described by a core and two neutrons dominated by the $s-$wave channel. We adopt the renormalized three-body model with a zero-range force, that accounts for the universal Efimov physics. This model is applicable to nuclei with large two-neutron halos compared to the core size, and a neutron-core scattering length larger than the interaction range. The halo wave function in momentum space is obtained by using as inputs the two-neutron separation energy and the energies of the singlet neutron-neutron and neutron-core virtual states. Within our model, we obtain the momentum probability densities for the Borromean exotic nuclei Lithium-11 ($^{11}$Li), Berylium-14 ($^{14}$Be) and Carbon-22 ($^{22}$C). A fair reproduction of the experimental data was obtained in the case of the core recoil momentum distribution of $^{11}$Li and $^{14}$Be, without free parameters. By extending the model to $^{22}$C, the combined analysis of the core momentum distribution and matter radius suggest (i) a $^{21}$C virtual state well below 1 MeV; (ii) an overestimation of the extracted matter $^{22}$C radius; and (iii) a two-neutron separation energy between 100 and 400 keV.

Core momentum distribution in two-neutron halo nuclei

The core momentum distribution of a weakly-bound neutron-neutron-core exotic nucleus is computed within a renormalized zero-range three-body model, with interactions in the s-wave channel. The halo wave-function in momentum space is obtained by using as inputs the two-body scattering lengths and the two-neutron separation energy. The core momentum densities are computed for $^{11}$Li, $^{14}$Be $^{20}$C and $^{22}$C. The model describes the experimental data for $^{11}$Li, $^{14}$Be and to some extend $^{20}$C. The recoil momentum distribution of the $^{20}$C from the breakup of $^{22}$C nucleus is computed for different two-neutron separation energies, and from the comparison with recent experimental data the two-neutron separation energy is estimated in the range $100\lesssim S_{2n}\lesssim 400$ KeV. The recoil momentum distribution depends weakly on the neutron-$^{20}$C scattering length, while the matter radius is strongly sensitive to it. The expected universality of the momentum distribution width is verified by also considering excited states for the system.

Momentum distributions in light halo nuclei and structure constraints

The core recoil momentum distribution of neutron-rich isotopes of light exotic nuclei is studied within a three-body model, where the nuclei are described by a core and two neutrons, with interactions dominated by the s-wave channel. In our framework, the two-body subsystems should have large scattering lengths in comparison with the interaction range allowing to use a three-body model with a zero-range force. The ground-state halo wave functions in momentum space are obtained by using as inputs the two-neutron separation energy and the energies of the singlet neutron-neutron and neutron-core virtual states. Within our model, we obtain the momentum probability densities for the Borromean exotic nuclei 11 Li and 22 C. In the case of the core recoil momentum distribution of 11Li, a fair reproduction of the experimental data was obtained, without free parameters, considering only the two-body low-energies. By analysing the obtained core momentum distribution in face of recent experimental data, we verify that such data are constraining the 22 C two-neutron separation energy to a value between 100 and 400 keV.

Dimensional Regularization of the Spatial wave function for a singular contact interaction in the Relativistic Schrodinger Equation

Based on our previous work in PRD 89, 125023 (2014), we stress here (for the first time) the regularization of the spatial wave function for the δ-contact interaction within the relativistic Schrodinger equation. The D-dimensional inverse Fourier transform has been utilized to map the momentum-space wave function to the spatial one. To regularize the logarithmic blow up of the wave function as x → 0, we employed the dimensional regularization technique. We assert that although the technique has been used here for the bound state only, the form of the scattering states in momentum space assures the reliability of the same technique to regularize the spatial scattering wave functions.

Basis of symmetric polynomials for many-boson light-front wave functions.

We provide an algorithm for the construction of orthonormal multivariate polynomials that are symmetric with respect to the interchange of any two coordinates on the unit hypercube and are constrained to the hyperplane where the sum of the coordinates is one. These polynomials form a basis for the expansion of bosonic light-front momentum-space wave functions, as functions of longitudinal momentum, where momentum conservation guarantees that the fractions are on the interval [0,1] and sum to one. This generalizes earlier work on three-boson wave functions to wave functions for arbitrarily many identical bosons. A simple application in two-dimensional ϕ(4) theory illustrates the use of these polynomials.

Coulomb wave functions in momentum space

An algorithm to calculate non-relativistic partial-wave Coulomb functions in momentum space is presented. The arguments are the Sommerfeld parameter η, the angular momentum l, the asymptotic momentum q and the ‘running’ momentum p, where both momenta are real. Since the partial-wave Coulomb functions exhibit singular behavior when p→q, different representations of the Legendre functions of the 2nd kind need to be implemented in computing the functions for the values of p close to the singularity and far away from it. The code for the momentum-space Coulomb wave functions is applicable for values of |η| in the range of 10−1 to 10, and thus is particularly suited for momentum space calculations of nuclear reactions. Program SummaryProgram title: libcwfnCatalogue identifier: AEUQ_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEUQ_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 864503No. of bytes in distributed program, including test data, etc.: 7178021Distribution format: tar.gzProgramming language: Fortran 90, Fortran 77, Python, make (GNU Make dialect), GNU Bash shell interpreter (available as /bin/bash).Computer: Apple Powermac (Intel Xeon), ASUS K53U (AMD E-350 (Dual Core)), DELL Precision T3500 (Intel Xeon), NERSC Carver (Intel Nehalem Quad Core).Operating system: Linux, Windows (using Cygwin).RAM: less than 512 MbytesClassification: 17.8, 17.13, 17.16.Nature of problem:The calculation of partial wave Coulomb functions with integer l and all other arguments real.Solution method:Computing the value of the function using explicit formulae and algorithms.Running time:Less than 10−3 s.

Stable and efficient momentum-space solutions of the time-dependent Schrödinger equation for one-dimensional atoms in strong laser fields

One-dimensional model systems have a particular role in strong-field physics when gaining physical insight by computing data over a large range of parameters, or when performing numerous time propagations within, e.g., optimal control theory. Here we derive a scheme that removes a singularity in the one-dimensional Schrödinger equation in momentum space for a particle in the commonly used soft-core Coulomb potential. By using this scheme we develop two numerical approaches to the time-dependent Schrödinger equation in momentum space. The first approach employs the expansion of the momentum-space wave function over the eigenstates of the field-free Hamiltonian, and it is shown to be more efficient for laser parameters usual in strong field physics. The second approach employs the Crank–Nicolson scheme or the method of lines for time-propagation. The both methods are readily applicable for large-scale numerical simulations in one-dimensional model systems.

Coulomb problem in momentum space without screening

Background: The repulsive Coulomb force poses severe challenges when solving the three-body problem for ($d,p$) reactions on intermediate mass and heavy nuclei. Recently, a new approach based on the Coulomb-distorted basis in momentum space was proposed.Purpose: In this work, we demonstrate the feasibility of using the Coulomb-distorted basis in momentum space for calculating matrix elements expected in a wide range of nuclear reactions.Method: We discuss the analytic forms of the Coulomb wave function in momentum space. We analyze the singularities in the Coulomb-distorted form factors and the required regularization techniques. Employing a separable interaction derived from a realistic nucleon-nucleus optical potential, we compute and study the Coulomb-distorted form factors for a wide range of cases, including charge, angular momentum, and energy dependence. We also investigate in detail the precision of our calculations.Results: The Coulomb-distorted form factors differ significantly from the nuclear form factors except for the very highest momenta. Typically, the structure of the form factor is shifted away from zero momentum due to the Coulomb interaction. Unlike the Yamaguchi forms typically used in three-body methods, our realistic form factors have a short high-momentum tail, which allows for a safe and efficient truncation of the momentum grid.Conclusions: Our results show that the Coulomb-distorted basis can be effectively implemented.

Path integral of a relativistic spinning particle in (1+1) dimension with vector and scalar linear potentials in the presence of a minimal length

Using the path integral formalism in (1+1) dimension of the energy–momentum space representation, we calculate the Green function of relativistic spinning particle subjected to the action of combined vector and scalar linear potentials in the framework of deformed Heisenberg algebra which distinguish to the appearance of nonzero minimum position uncertainty given by [Formula: see text] where β is the deformation parameter of the modified Heisenberg uncertainty relation [Formula: see text]. We adapt the space–time transformation method to evaluate quantum corrections and this latter are dependent on the α-point discretization interval. The exact bound states spectrum and the corresponding momentum space wave function are obtained.

Contact parameters in two dimensions for general three-body systems

We study the two-dimensional three-body problem in the general case of three distinguishable particles interacting through zero-range potentials. The Faddeev decomposition is used to write the momentum-space wave function. We show that the large-momentum asymptotic spectator function has the same functional form as derived previously for three identical particles. We derive the analytic relations between the three different Faddeev components for the three distinguishable particles. We investigate the one-body momentum distributions both analytically and numerically and analyze the tail of the distributions to obtain two- and three-body contact parameters. We specialize from the general cases to the examples of two identical, interacting or non-interacting, particles. We find that the two-body contact parameter is not a universal constant in the general case and show that the universality is recovered when a subsystem is composed of two identical non-interacting particles. We also show that the three-body contact parameter is negligible in the case of one non-interacting subsystem compared to the situation where all the subsystems are bound. As an example, we present the results for mixtures of lithium with two cesium or two potassium atoms, which are systems of current experimental interest.

Saturation effects in exclusive ρ T,L meson electroproduction

Abstract We use recent results for the $ \gamma_L^{*}\to {\rho_L} $ and $ \gamma_T^{*}\to {\rho_T} $ impact factors, computed in the impact parameter representation within the collinear factorization scheme, to get predictions for the polarized cross-sections σ T and σ L of the diffractive leptoproduction of the ρ meson at high energy. In this approach the helicity amplitude is a convolution of the scattering amplitude of a color dipole with a target, together with the virtual gamma wave function and with the first moments of the ρ meson wave function (in the transverse momentum space), given by the distribution amplitudes up to twist 3 for the $ \gamma_T^{*}\to {\rho_T} $ impact factor and up to twist 2 for the $ \gamma_L^{*}\to {\rho_L} $ impact factor. Combining these results with recent dipole models fitted to DIS data, which include saturation effects, we show that the predictions are in good agreement with HERA data for photon virtuality (Q 2) larger than typically 5 GeV2, without free parameters and with a weak dependence on the choice of the factorization scale, i.e. the shape of the DAs, for both longitudinally and transversely polarized ρ meson. For lower values of Q 2, the inclusion of saturation effects is not enough to provide a good description of HERA data. We believe that it is a signal of a need for higher twist contributions in the ρ meson DAs. We also analyze the radial distributions of dipoles between the initial γ * and the final ρ meson states.

Spectral Minimum and Giant Enhancement in Photoelectron Spectra from Xenon Atoms Driven by Intense Midinfrared Laser Fields

Our theoretical study shows that the spectral minimum and the giant enhancement structures observed in the high harmonic spectra also exist in the photoelectron spectra from driven Xe atoms. They are attributed to the inherent property of the radial part of the wave function of the Xe 5p subshell in momentum space. The spectral minimum is caused by the nodal point in the modulus of the radial wave function in momentum space, and the giant enhancement reflects the increase in magnitude of the modulus of the wave function. To observe these structures, midinfrared lasers of about 0.2 PW/cm(2) intensity are preferred. Employing circularly polarized laser light is suggested for exhibiting these structures in photoelectron spectra.

Coulombic Transformation in Momentum Space

We studied the Coulombic transformation of potential in momentum space. The Coulombic transformation is defined as a unitary transformation in momentum space, which is equivalent of the Coulomb–Fourier transformation in coordinate space. The analytic continuation scheme avoids the difficulty which is occurred from the singularity of the Coulomb wave function in momentum space. We adopted the point method to perform the analytic continuation. The validity of the new scheme is checked by comparing with the analytic solution for the Malfliet-Tjon potential. Numerical calculation of the integration was done by separating into four intervals. We demonstrate the high accuracy of our calculation.

The biharmonic oscillator and asymmetric linear potentials: from classical trajectories to momentum-space probability densities in the extreme quantum limit

The biharmonic oscillator and the asymmetric linear well are two confining power-law-type potentials for which complete bound-state solutions are possible in both classical and quantum mechanics. We examine these problems in detail, beginning with studies of their trajectories in position and momentum space, evaluation of the classical probability densities for both x and p, and calculation of the corresponding quantum-mechanical solutions which give |ψn(x)|2 and |ϕn(p)|2 for comparison to their classical counterparts in the classically allowed regions. We then focus on the behavior of ϕn(p) for large momenta, motivated by recent studies of the very direct connections between the continuity behavior of V(x) and the large-p limit of the momentum-space wavefunction.

Polarizability of screened Coulomb potential

We have obtained the co-ordinate space wavefunction by the inverse Fourier transform of the momentum space wavefunction for the screened Coulomb potential, $$ V\left( r \right)\; = \; - {\text{e}}^{{{ - }\mu {\text{r}}}} /{\text{r}} $$ , problem. The dipole polarizability αd is evaluated in the closure approximation. In this approximation $$ \alpha_{\text{d}} \; = \;\frac{2}{3}\frac{{\langle r^{2} \rangle }}{\Updelta } $$ , where ∆ is the mean excitation energy which is related to the information entropy via Bethe theory for the stopping cross-section for a moving point charge interacting with a bound quantum mechanical system. We find that αd increases with μ. Near the Coulomb limit, we find $$ \alpha_{\text{d}} \left( \mu \right)\; = \,\alpha_{\text{H}} \left( {1\; + \;2.9245 \mu^{2} } \right), $$ where α H is the polarizability of hydrogen atom. Results have been compared with the recent theoretical works.

Berezinskii-Kosterlitz-Thouless transition of two-dimensional Bose gases in a synthetic magnetic field

We study the Berezinskii-Kosterlitz-Thouless transition of two-dimensional Bose gases in a synthetic magnetic field using the standard Metropolis Monte Carlo method. The system is described by the frustrated XY model and the critical temperature is calculated though the absence of central peak of the wave function in momentum space, which can be directly measured by the time-of-flight absorbing imaging in cold atoms experiments. The results of our work show agreement with former studies on superconducting Josephson arrays.

Uncertainty Relation for Photons

The uncertainty relation for the photons in three dimensions that overcomes the difficulties caused by the nonexistence of the photon position operator is derived in quantum electrodynamics. The photon energy density plays the role of the probability density in configuration space. It is shown that the measure of the spatial extension based on the energy distribution in space leads to an inequality that is a natural counterpart of the standard Heisenberg relation. The equation satisfied by the photon wave function in momentum space which saturates the uncertainty relations has the form of the Schrödinger equation in coordinate space in the presence of electric and magnetic charges.

Influence of thed-state component of the deuteron wave function on the application of the Trojan horse method

The Trojan horse method allows one to extract the cross section of a binary reaction by properly selecting the quasi-free component of a suitable surrogate reaction. A crucial point of its application is the discrimination of the quasi-free contribution to the total reaction yield, requiring the study of the experimental momentum distribution of the adopted Trojan horse nucleus. Deuteron has been largely used as Trojan horse nucleus in $p$- and $n$-induced reactions and the extracted experimental momentum distribution compared with the square of the Hulth\'en wave function in momentum space, which accounts for the $s$-wave component of the deuteron wave function. Here, the contribution of the $d$-wave has been evaluated, turning out to be negligibly small, thus corroborating the assumptions made so far in Trojan horse experiments where deuteron has been used.

Gauge independent theory applied to a model of atomic ionization by an intense laser pulse

An explicitly gauge invariant strong field theory is introduced and tested using a model laser-atom interaction. The theory relies on a power series in the target potential. Transitions amplitudes are obtained by using a corresponding series in the momentum space wave function. We demonstrate that this approach is explicitly gauge invariant to all orders. A well know 1D delta function potential model is used to test the convergence of the series in the evaluation of total ionization probabilities and ionization spectra. Actually, the convergence is verified when both, the perturbation as well as the order of the expansion, are increased.