Wave Functions In Space Research Articles

Exponential wave-packet spreading via self-interaction time modulation

The time-periodic modulation of the self-interaction of a Bose--Einstein condensate or a nonlinear optics system has been recognized as an exciting tool to explore interesting physics that was previously unavailable. This tool is exploited here to examine the exotic dynamics of a nonlinear system described by the Gross--Pitaevskii equation. We observe three remarkable and closely related dynamical phenomena, exponentially localized profile of wave functions in momentum space with localization length exponentially increasing in time, exponential wave-packet spreading, and exponential sensitivity to initial conditions. A hybrid quantum-classical theory is developed to partly explain these findings. Time-periodic self-interaction modulation is seen to be a robust method to achieve superfast spreading and induce genuine chaos even in the absence of any external potential.

Microscopic effective reaction theory for deuteron-induced reactions

The microscopic effective reaction theory is applied to deuteron-induced reactions. A reaction model-space characterized by a $p+n+{\rm A}$ three-body model is adopted, where A is the target nucleus, and the nucleon-target potential is described by a microscopic folding model based on an effective nucleon-nucleon interaction in nuclear medium and a one-body nuclear density of A. The three-body scattering wave function in the model space is obtained with the continuum-discretized coupled-channels method (CDCC), and the eikonal reaction theory (ERT), an extension of CDCC, is applied to the calculation of neutron removal cross sections. Elastic scattering cross sections of deuteron on $^{58}$Ni and $^{208}$Pb target nuclei at several energies are compared with experimental data. The total reaction cross sections and the neutron removal cross sections at 56 MeV on 14 target nuclei are calculated and compared with experimental values.

Charge and longitudinal momentum distributions in transverse coordinate space

We investigate the charge distributions for the $u$ and $d$ quarks in transverse coordinate space in a light-front quark-diquark model for the nucleons using the overlaps of the wave functions constructed from the soft-wall AdS/QCD prediction. We have also obtained the charge distributions for proton and neutron in transverse coordinate space and compared it with the distributions obtained in impact-parameter space. Further, we study the longitudinal momentum distributions using the wave functions in the transverse coordinate space. We have also shown the explicit fermionic and bosonic contributions for different struck $u$ and $d$ quarks.

Counting Majorana bound states using complex momenta

Recently, the connection between Majorana fermions bound to defects in arbitrary dimensions, and complex momentum roots of the vanishing determinant of the corresponding bulk Bogoliubov-de Gennes (BdG) Hamiltonian, has been established (EPL, 2015, $\textbf{110}$, 67005). Based on this understanding, a formula has been proposed to count the number ($n$) of the zero energy Majorana bound states, which is related to the topological phase of the system. In this paper, we provide a proof of the counting formula and we apply this formula to a variety of 1d and 2d models belonging to the classes BDI, DIII and D. We show that we can successfully chart out the topological phase diagrams. Studying these examples also enables us to explicitly observe the correspondence between these complex momentum solutions in the Fourier space, and the localized Majorana fermion wavefunctions in the position space. Finally, we corroborate the fact that for systems with a chiral symmetry, these solutions are the so-called "exceptional points", where two or more eigenvalues of the complexified Hamiltonian coalesce.

Monopole excitation of the Hoyle state and linear-chain state inC12

Background: Two new $J^\pi=0^+$ states are recently observed around a few MeV above the Hoyle state (the second $0^+$ state in 12C). The characters of them are only poorly discussed in theory and are still mysterious. Purpose: I give for the first time a comprehensive understanding of the structures of the $0^+$ states by analyzing their wave functions, and discuss relationship with the Hoyle state, similarities, and differences between the states. Method: I extend a microscopic $\alpha$-cluster model called Tohsaki-Horiuchi-Schuck-R\"opke (THSR) wave function so as to incorporate $2\alpha+\alpha$ asymmetric configuration explicitly. The so-called $r^2$-constraint method to effectively eliminate spurious continuum components is also used. Results: The $0_3^+$ state is shown to have a very large squared overlap with a single configuration of the extended THSR wave function in an orthogonal space to the Hoyle state as well as to the ground state. The $0_4^+$ state has a maximal squared overlap with a single extended THSR wave function with an extremely prolately-deformed shape. Conclusions: The $0_3^+$ state appears as a family of the Hoyle state to have a higher nodal structure in the internal motions of the $3\alpha$ clusters, due to the orthogonalization to the Hoyle state. The $0_4^+$ state dominantly has a linear-chain structure, where the $3\alpha$ clusters move freely in a non-localized way, like a one-dimensional gas of the $3\alpha$ clusters.

New analytical forms of wave function in coordinate space and tensor polarization of deuteron

The coefficients of new analytical forms for the deuteron wave function (DWF) in coordinate space for NijmI, NijmII, Nijm93, Reid93 and Argonne v18 potentials have been numerically calculated. The obtained wave functions do not contain any superfluous knots. The designed parameters of the deuteron are in good agreement with the experimental and theoretical data. The tensor polarization [Formula: see text] calculated based on the wave functions is proportionate to the earlier published results.

Second-Order Perturbation Theory for Generalized Active Space Self-Consistent-Field Wave Functions.

A multireference second-order perturbation theory approach based on the generalized active space self-consistent-field (GASSCF) wave function is presented. Compared with the complete active space (CAS) and restricted active space (RAS) wave functions, GAS wave functions are more flexible and can employ larger active spaces and/or different truncations of the configuration interaction expansion. With GASSCF, one can explore chemical systems that are not affordable with either CASSCF or RASSCF. Perturbation theory to second order on top of GAS wave functions (GASPT2) has been implemented to recover the remaining electron correlation. The method has been benchmarked by computing the chromium dimer ground-state potential energy curve. These calculations show that GASPT2 gives results similar to CASPT2 even with a configuration interaction expansion much smaller than the corresponding CAS expansion.

Bethe–Salpeter bound-state structure in Minkowski space

The quantitative investigation of the scalar Bethe–Salpeter equation in Minkowski space, within the ladder-approximation framework, is extended to include the excited states. This study has been carried out for an interacting system composed by two massive bosons exchanging a massive scalar, by adopting (i) the Nakanishi integral representation of the Bethe–Salpeter amplitude, and (ii) the formally exact projection onto the null plane. Our analysis, on one hand, confirms the reliability of the method already applied to the ground state and, on the other one, extends the investigation from the valence distribution in momentum space to the corresponding quantity in the impact-parameter space, pointing out some relevant features, like (i) the equivalence between Minkowski and Euclidean transverse-momentum amplitudes, and (ii) the leading exponential fall-off of the valence wave function in the impact-parameter space.

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.

Replacing the Singlet Spinor of the EPR-B Experiment in the Configuration Space with Two Single-Particle Spinors in Physical Space

Recently, for spinless non-relativistic particles, Norsen, Marian and Oriols show that in the de Broglie-Bohm interpretation it is possible to replace the wave function in the configuration space by single-particle wave functions in physical space. In this paper, we show that this replacment of the wave function in the configuration space by single-particle functions in the 3D-space is also possible for particles with spin, in particular for the particles of the EPR-B experiment, the Bohm version of the Einstein-Podolsky-Rosen experiment.

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.

Effect of barrier width on the exciton states in coupled quantum wells in an applied electric field

The effect of barrier width on the exciton states in coupled quantum wells has been theoretically studied using an efficient approach. By solving the Schrödinger equation in real space, the electron and hole energies and wave functions are calculated in the presence of an applied electric field. It is found that with zero electric field the energy splitting of the doublets is large in the case of thin barrier width. However for thicker barrier width, the Stark effect is stronger at large electric field. The exciton states, binding energies and oscillator strength are also calculated as a function of electric field. There is shown the direct-to-indirect crossover of the exciton ground state at approximately F=10 kV/cm and F=5 kV/cm for Lb=2 nm and Lb=4 nm respectively, corresponding to the dramatic decrease of its binding energy and oscillator strength. This direct-to-indirect crossover happens at lower electric field for thicker barrier width. We have also studied the light–matter coupling and calculated the DX, IX and CM components of the polariton states as a function of electric field.

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.

Hilbert space theory of classical electrodynamics

Classical electrodynamics is reformulated in terms of wave functions in the classical phase space of electrodynamics, following the Koopman–von Neumann–Sudarshan prescription for classical mechanics on Hilbert spaces sans the superselection rule which prohibits interference effects in classical mechanics. This is accomplished by transforming from a set of commuting observables in one Hilbert space to another set of commuting observables in a larger Hilbert space. This is necessary to clarify the theoretical basis of the much recent work on quantum-like features exhibited by classical optics. Furthermore, following Bondar et al, Phys. Rev. A 88, 052108 (2013), it is pointed out that quantum processes that preserve the positivity or nonpositivity of the Wigner function can be implemented by classical optics. This may be useful in interpreting quantum information processing in terms of classical optics.

A Density Functional Theory and ab Initio Investigation of the Oxidation Reaction of CO by IO Radicals.

To get an insight into the possible reactivity between iodine oxides and CO, a first step was to study the thermochemical properties and kinetic parameters of the reaction between IO and CO using theoretical chemistry tools. All stationary points involved were optimized using the Becke's three-parameter hybrid exchange functional coupled with the Lee-Yang-Parr nonlocal correlation functional (B3LYP) and the Møller-Plesset second-order perturbation theory (MP2). Single-point energy calculations were performed using the coupled cluster theory with the iterative inclusion of singles and doubles and the perturbative estimation for triple excitations (CCSD(T)) and the aug-cc-pVnZ (n = T, Q, and 5) basis sets on geometries previously optimized at the aug-cc-pVTZ level. The energetics was then recalculated using the one-component DK-CCSD(T) approach with the relativistic ANO basis sets. The spin-orbit coupling for the iodine containing species was calculated a posteriori using the restricted active space state interaction method in conjunction with the multiconfigurational perturbation theory (CASPT2/RASSI) employing the complete active space (CASSCF) wave function as the reference. The CCSD(T) energies were also corrected for BSSE for molecular complexes and refined with the extrapolation to CBS limit while the DK-CCSD(T) values were refined with the extrapolation to FCI. The exploration of the potential energy surface revealed a two-steps mechanism with a trans and a cis pathway. The rate constants for the direct and complex mechanism were computed as a function of temperature (250-2500 K) using the canonical transition state theory. The three-parameter Arrhenius expressions obtained for the direct and indirect mechanism at the DK-CCSD(T)-cf level of theory is 1.49 × 10(-17) × T(1.77) exp(-47.4 (kJ mol(-1))/RT).

Models of spontaneous wave function collapse: what they are, and how they can be tested

There are few proposals, which explicitly allow for (experimentally testable) deviations from standard quantum theory. Collapse models are among the most-widely studied proposals of this kind. The Schrödinger equation is modified by including nonlinear and stochastic terms, which describe the collapse of the wave function in space. These spontaneous collapses are rare for microscopic systems, hence their quantum properties are left almost unaltered. On the other hand, collapses become more and more frequent, the larger the object, to the point that macroscopic superpositions are rapidly suppressed. The main features of collapse models will be reviewed. An update of the most promising experimental tests will be presented.

Asymptotic behavior of the wave function of three particles in a continuum

We study the wave function of a system of three particles in a continuum. The Faddeev equations are used to explicitly identify the singularities of the wave function in the momentum space. We obtain the asymptotic behavior of the wave function in the configuration space by calculating the asymptotic behavior of the Fourier transform of the wave function in the momentum space. Our attention is focused on configurations in which two particles are at a relatively small distance from each other while the third particle is significantly remote from the center of mass of the pair. We show that the coordinate asymptotic form of the wave function for such a configuration contains scattered waves of a new type in addition to the standard terms. We use the obtained exact data concerning the coordinate asymptotic form of the wave function to critically analyze the multiplicative ansatz used in several works to describe systems of three particles in a continuum.

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.

Quasi-Feynman formulas – a method of obtaining the evolution operator for the Schrödinger equation

For a densely defined self-adjoint operator H in Hilbert space F the operator exp⁡(−itH) is the evolution operator for the Schrödinger equation iψt′=Hψ, i.e. if ψ(0,x)=ψ0(x) then ψ(t,x)=(exp⁡(−itH)ψ0)(x) for x∈Q. The space F here is the space of wave functions ψ defined on an abstract space Q, the configuration space of a quantum system, and H is the Hamiltonian of the system. In this paper the operator exp⁡(−itH) for all real values of t is expressed in terms of the family of self-adjoint bounded operators S(t), t≥0, which is Chernoff-tangent to the operator −H. One can take S(t)=exp⁡(−tH), or use other, simple families S that are listed in the paper. The main theorem is proven on the level of semigroups of bounded operators in F so it can be used in a wider context due to its generality. Two examples of application are provided.

Tensor Network Renormalization Yields the Multiscale Entanglement Renormalization Ansatz

We show how to build a multiscale entanglement renormalization ansatz (MERA) representation of the ground state of a many-body Hamiltonian H by applying the recently proposed tensor network renormalization [G. Evenbly and G. Vidal, Phys. Rev. Lett. 115, 180405 (2015)] to the Euclidean time evolution operator e(-βH) for infinite β. This approach bypasses the costly energy minimization of previous MERA algorithms and, when applied to finite inverse temperature β, produces a MERA representation of a thermal Gibbs state. Our construction endows tensor network renormalization with a renormalization group flow in the space of wave functions and Hamiltonians (and not merely in the more abstract space of tensors) and extends the MERA formalism to classical statistical systems.