Exact Wave Functions Research Articles

Scattering of plane compressional waves by a variable thickness circular lined tunnel

The determination of the dynamic stress concentration factor (DSCF) associated to a circular lined tunnel with variable thickness liner incident upon by compressional P-waves is of particular interest. Due to the lack of symmetry, the treatment of such a problem via the traditional wave-function expansion method is not possible. To circumvent this dilemma it is proposed to employ the dynamic equivalent inclusion method (DEIM). As it will be shown, this method which is based on the notions of the eigenstress and eigenbody-force fields provides an accurate solution for the problem under consideration. In the simple special case where a tunnel with a concentric liner is subjected to plane compressional P-waves the exact wave-function expansion solution given in the literature is recovered. The effect of liner thickness variability on the static and dynamic stress concentration along the interior surface of the liner and along the liner-matrix interface is studied in detail through numerical examples. The maximum value of the DSCF for the concrete and steel liners with positive and negative values of eccentricity is obtained and compared with those of the corresponding concentric cases. It is observed that the eccentricity of the liner results in different behaviors of the stress concentration for the hard (steel) and soft (concrete) liners.

Boson star normal modes

Boson stars are gravitationally bound objects that arise in ultralight dark matter models and form in the centers of galactic halos or axion miniclusters. We systematically study the excitations of a boson star, taking into account the mixing between positive and negative frequencies introduced by gravity. We show that the spectrum contains zero-energy modes in the monopole and dipole sectors resulting from spontaneous symmetry breaking by the boson star background. We analyze the general properties of the eigenmodes and derive their orthogonality and completeness conditions which have non-standard form due to the positive-negative frequency mixing. The eigenvalue problem is solved numerically for the first few energy levels in different multipole sectors and the results are compared to the solutions of the Schrödinger equation in fixed boson star gravitational potential. The two solutions differ significantly for the lowest modes, but get close for higher levels. We further confirm the normal mode spectrum in 3D wave simulations where we inject perturbations with different multipoles. As an application of the normal mode solutions, we compute the matrix element entering the evaporation rate of a boson star immersed in a hot axion gas. The computation combines the use of exact wavefunctions for the low-lying bound states and of the Schrödinger approximation for the high-energy excitations.

Ordinary Muon Capture on 136Ba: Comparative Study Using the Shell Model and pnQRPA

In this work, we present a study of ordinary muon capture (OMC) on 136Ba, the daughter nucleus of 136Xe double beta decay (DBD). The OMC rates at low-lying nuclear states (below 1 MeV of excitation energy) in 136Cs are assessed by using both the interacting shell model (ISM) and proton–neutron quasiparticle random-phase approximation (pnQRPA). We also add chiral two-body (2BC) meson-exchange currents and use an exact Dirac wave function for the captured s-orbital muon. OMC can be viewed as a complementary probe of the wave functions in 136Cs, the intermediate nucleus of the 136Xe DBD. At the same time, OMC can be considered a powerful probe of the effective values of weak axial-type couplings in a 100 MeV momentum exchange region, which is relevant for neutrinoless DBD. The present work represents the first attempt to compare the ISM and pnQRPA results for OMC on a heavy nucleus while also including the exact muon wave function and the 2BC. The sensitivity estimates of the current and future neutrinoless DBD experiments will clearly benefit from future OMC measurements taken using OMC calculations similar to the one presented here.

Quench dynamics of anyon Tonks–Girardeau gases

In this paper, we investigate the quench dynamics of strongly interacting anyons triggered by the quench of harmonic trap frequency. The exact time-dependent many-body wavefunction is obtained by combing anyon-fermion mapping method with the time-dependent single-particle wavefunction. The density profiles, momentum distribution, and the reduced one-body density matrix are evaluated for different statistical parameters. It is shown that the evolution behaviours dependent on statistical parameter are displayed in the momentum distribution and the reduced one-body density matrix rather than in the density profiles in real space. As the harmonic trap is turned off suddenly, the momentum distributions exhibit the symmetric fermion-like behaviour in the long-time evolution. As the trap frequency is quenched, the momentum distribution exhibits an asymmetry breath mode during the evolution. The reduced one-body density matrix shows the dynamical symmetry broken and reproduced behaviour.

Hyper-Rayleigh scattering optical activity: Theory, symmetry considerations, and quantum chemistry applications.

This work reports on the first computational quantum-chemistry implementation of the hyper-Rayleigh scattering optical activity (HRS-OA), a nonlinear chiroptical phenomenon. First, from the basics of the theory, which is based on quantum electrodynamics, and focusing on the electric dipole, magnetic-dipole, and electric-quadrupole interactions, the equations for the simulation of the differential scattering ratios of HRS-OA are re-derived. Then, for the first time, computations of HRS-OA quantities are presented and analyzed. They have been enacted on a prototypical chiral organic molecule (methyloxirane) at the time-dependent density functional theory level using a broad range of atomic orbital basis sets. In particular, (i) we analyze the basis set convergence, demonstrating that converged results require basis sets with both diffuse and polarization functions, (ii) we discuss the relative amplitudes of the five contributions to the differential scattering ratios, and (iii) we study the effects of origin-dependence and derived the expression of the tensor shifts and we prove the origin-independence of the theory for exact wavefunctions. Our computations show the ability of HRS-OA as a nonlinear chiroptical method, able to distinguish between the enantiomers of the same chiral molecule.

Correlation functions and polaron-molecule crossover in one-dimensional attractive Fermi gases

We study the correlation functions in a one-dimensional attractive Fermi polaron system of spin-1/2 Fermi gas with one flipped spin as an impurity immersed in the fully polarized spin-up media. By means of the exact Bethe ansatz wave function with string solutions for the momentum, we manage to explicitly deduce the asymptotic form of the correlation functions in the infinite attractive limit and show that the system undergoes a polaron-molecule crossover when the attraction grows. The correlation functions in the infinite attractive limit can be decomposed into the free fermion term with the number of particles reduced by one and the molecular term indicated by the emergence of a sharp peak or dip in the central area of the correlation. These analytical results are checked by the numerical Monte Carlo methods for multiple integrals. The calculation is further extended to the excited states, i.e., the super-Tonks-Girardeau gas, and there is no peak or dip found in the vicinity of the correlation center. Tan contact is found to be proportional to the cubic of the interaction strength and the total energy is verified to satisfy the Tan adiabatic theorem in the strong attraction regime.

Relativistic energies and information entropy of the inversely quadratic Hellmann potential

The solutions to the Dirac equation are obtained in the spin and pseudospin symmetry limits are presented using the parametric Nikiforov-Uvarov (pNU) method with the inversely quadratic Hellman (IQH) potential. In the exact non-relativistic spin symmetry limit, the energy and wave function of the IQH potential are obtained and used to investigate the Shannon information entropy of the system. Numerical results of the relativistic energies of the spin and pseudospin symmetry limits of the Dirac equation with the IQH potential are presented and observed to exhibit degeneracy. Also, the results of the Shannon entropy for six states (n = 0, 1, 2, 3, 4, 5) show that the momentum space wave function and probability density are better localized than the position space wave function. Also, the Bialynicki-Birula and Mycielski (BBM) inequality is verified for the system. Our results are found to be consistent with those previously reported in the literature.

Correlation in momentum space of Tonks–Girardeau gas

In momentum space, we investigate the correlation properties of the ground state of Tonks–Girardeau gases. With Bose–Fermi mapping method, the exact ground state wavefunction in coordinate space can be obtained based on the wavefunction of spin-polarized Fermions. By Fourier transformation we obtain the ground state wavefunction in momentum space, and therefore the pair correlation and the reduced one-body density matrix (ROBDM) in momentum space, whose diagonal part is the momentum distribution. The ROBDM in momentum space is the Fourier transformation of the ROBDM in coordinate space and the pair correlation in momentum space is the Fourier transformation of the reduced two-body density matrix in coordinate space. The correlations in momentum space display larger values only in small momentum region and vanish in most other regions. The lowest natural orbital and occupation distribution in momentum space are also obtained.

Entanglement bipartitioning and tree tensor networks

AbstractWe propose the entanglement bipartitioning approach to design an optimal network structure of the tree tensor network (TTN) for quantum many-body systems. Given an exact ground-state wavefunction, we perform sequential bipartitioning of spin-cluster nodes so as to minimize the mutual information or the maximum loss of the entanglement entropy associated with the branch to be bipartitioned. We demonstrate that entanglement bipartitioning of up to 16 sites gives rise to nontrivial tree network structures for S = 1/2 Heisenberg models in one and two dimensions. The resulting TTNs enable us to obtain better variational energies, compared with standard TTNs such as the uniform matrix product state and perfect binary tree tensor network.

N-body correlation of Tonks–Girardeau gas

For the well-known exponential complexity it is a giant challenge to calculate the correlation function for general many-body wave function. We investigate the ground state $n$th-order correlation functions of the Tonks-Girardeau (TG) gases. Basing on the wavefunction of free fermions and Bose-Fermi mapping method we obtain the exact ground state wavefunction of TG gases. Utilizing the properties of Vandermonde determinant and Toeplitz matrix, the $n$th-order correlation function is formulated as $(N-n)$-order Toeplitz determinant, whose element is the integral dependent on 2$(N-n)$ sign functions and can be computed analytically. By reducing the integral on domain $[0,2\pi]$ into the summation of the integral on several independent domains, we obtain the explicit form of the Toeplitz matrix element ultimately. As the applications we deduce the concise formula of the reduced two-body density matrix and discuss its properties. The corresponding natural orbitals and their occupation distribution are plotted. Furthermore, we give a concise formula of the reduced three-body density matrix and discuss its properties. It is shown that in the successive second measurements, atoms appear in the regions where atoms populate with the maximum probability in the first measurement.

Time-Dependent Expectation Values from Integral Equations for Quantum Flux and Probability Densities.

We compare the calculation of time-dependent quantum expectation values performed in different ways. In one case, they are obtained from an integral over a function of the probability density, and in the other case, the integral is over a function of the probability flux density. The two kinds of coordinate-dependent integrands are very different in their appearance, but integration yields identical results, if the exact wave function enters into the computation. This can be different, if one applies approximations to the wave function. For illustration, we treat one- and two-dimensional dynamics in coupled electron-nuclear systems. Using the adiabatic expansion of the total wave function, the expectation values are decomposed into different contributions. This allows us to discuss the validity of the Born-Oppenheimer (BO) approximation applied to the calculation of the expectation values from probability density- and flux density- integrals. Choosing force- and torque operators as examples, we illustrate the different spatiotemporal characteristics of the various integrands.

Testing the scalar sector of the standard-model extension with neutron gravity experiments

In the present study we analyse, within the scalar sector of the standard-model extension framework, the influence of a spontaneous Lorentz symmetry breaking on gravitational quantum states of ultracold neutrons. The model is framed according to the laboratory conditions of the recent high-sensitivity GRANIT and qBounce experiments. The high-precision data achieved in such experiments allow us to set bounds on the symmetry breaking parameters of the model. The effective Hamiltonian governing the neutron’s motion along the axis of free fall is derived explicitly. It describes a particle in a gravitational field with an effective gravitational constant controlled non-trivially by the Lorentz-violating parameters. In particular, using the exact wave functions and the energy spectrum, we evaluate both the heights associated with the quantum states and the transition frequencies between neighborhoring quantum states. By comparing our theoretical results with those reported in the GRANIT and the qBounce experiments, upper bounds on the Lorentz-violating parameters are determined. We also consider for the first time the gravity-induced interference pattern in a COW-type experiment to test Lorentz–invariance. In this case, an upper bound for the parameters is established as well.

Charged Particle Attraction and Repulsion Explained: A Detailed Mechanism Based on Particle Wave-Functions

The phenomenon of electrical attraction and repulsion between charged particles is well known, and described mathematically by Coulomb’s Law, yet until now there has been no explanation for why this occurs. There has been no mechanistic explanation that reveals what causes the charged particles to accelerate, either towards or away from each other. This paper gives a detailed explanation of the phenomena of electrical attraction and repulsion based on my previous work that determined the exact wave-function solutions for both the Electron and the Positron. It is revealed that the effects are caused by wave interactions between the particles’ wave-functions that result in Electromagnetic reflections of parts of these wave-functions, causing a change in the particles’ momenta.

Quantum phase transition in the one-dimensional Dicke-Hubbard model with coupled qubits

We study the ground state phase diagram of a one-dimensional two qubits Dicke-Hubbard model with XY qubit-qubit interaction. We use a numerical method combing the cluster mean-field theory and the matrix product state(MPS) to obtain the exact wave function of the ground state. When counter-rotating wave terms(CRTs) in the qubit-cavity coupling are neglected, we observe a rich phase diagram including a quantum phase transition between the Mott-insulating phase and the superfluid phase. This phase transition can be either the first-order or the second-order type depending on whether the total angular momentum changes across the phase diagram. Moreover, we observe two quantum triple points, at which three different phases coexist, with both positive and negative XY interactions. By further considering the effect of CRTs, we find that the main feature in the previous phase diagram, including the existence of quantum triple points, is retained. We also show that CRTs extremely demolish the non-local correlations in the coherent phase.

Exact solution for SU(2)-symmetry-breaking bosonic mixtures at strong interactions

We study the equilibrium properties of a one-dimensional mixture of two Tonks-Girardeau gases on a ring geometry in the limit of strongly-repulsive inter-species interactions. We derive the exact many-body wavefunction and compare it to the $SU(2)$ solution where intra- and inter-species interactions are also diverging but equal. We focus on the role of the $SU(2)$-symmetry breaking on the behaviour of the large- and short-distance correlations by studying the zero-momentum occupation number and the Tan's contact from the asymptotic behavior of the momentum distribution. Although the symmetry is only weakly broken, it has important consequences on spin correlations in the system as the reduction by a factor of two of the zero-momentum occupation number with respect to the $SU(2)$ case in the thermodynamic limit and the decrease of the Tan's contact.

Secondary Kinetic Peak in the Kohn-Sham Potential and Its Connection to the Response Step.

We consider a prototypical 1D model Hamiltonian for a stretched heteronuclear molecule and construct individual components of the corresponding KS potential, namely, the kinetic, the N - 1, and the conditional potentials. These components show very special features, such as peaks and steps, in regions where the density is drastically low. Some of these features are quite well-known, whereas others, such as a secondary peak in the kinetic potential or a second bump in the conditional potential, are less or not known at all. We discuss these features building on the analytical model treated in Giarrusso et al. J. Chem. Theory Comput. 2018, 14, 4151. In particular, we provide an explanation for the underlying mechanism which determines the appearance of both peaks in the kinetic potential and elucidate why these peaks delineate the region over which the plateau structure, due to the N - 1 potential, stretches. We assess the validity of the Heitler-London Ansatz at large but finite internuclear distance, showing that, if optimal orbitals are used, this model is an excellent approximation to the exact wave function. Notably, we find that the second natural orbital presents an extra node very far out on the side of the more electronegative atom.

Wave function of the Universe as a sum over eventually inflating universes

We consider a proposal to define the wave function of the Universe as a sum over spacetimes that eventually inflate. In the minisuperspace model, we explicitly show that a simple family of initial conditions, parametrized by a positive real number $a_0$, can be imposed to realise this prescription. The resulting wave function is found to be proportional to the Hartle-Hawking wave function and its dependence on $a_0$ is only through an overall phase factor. Motivated by this observation, we ask whether it is possible to analytically extend $a_0$ to an extended region $\bar{\mathcal{D}}$ in complex $a_0-$plane, while retaining the Hartle-Hawking form of the wave function. We use the condition for convergence of path integral and a recent theorem due to Kontsevich and Segal, further extended by Witten, to explicitly find $\bar{\mathcal{D}}$. Interestingly, a special point on the boundary of $\bar{\mathcal{D}}$ recovers the exact no-boundary wave function. Following that, we show that our prescription leads to a family of quantum states for the perturbations, which give rise to scale-invariant power spectra. If we demand, as an extra ingredient to our prescription, a matching condition at the "no-boundary point" in $\bar{\mathcal{D}}$, we converge on a unique quantum state for the perturbations.

True gravitational atoms: Spherical cloud of dilatonic black holes

Black hole as elementary particle is a fairly glamorous idea. For ordinary black holes, a surrounding particle inevitably penetrates into the interior of the hole since the center of the hole is an infinite potential well. For the first time we demonstrate that an extreme dilatonic black hole in spherical symmetry perfectly behaves as an atom, in the sense that its surrounding cloud of particles are completely stable. Thus we reach a spherical cloud of dilatonic black hole. We find exact wave functions of the cloud for arbitrary gravitational fine structure constant $\mu M$, and clear the underlying physical nature of the stability. Through careful studies of the exact wave function, we find the spectrum of this system. We discuss the physical meaning of this discovery especially from considerations of entropy, and the resultant possibility to explore quantization of gravitational waves from coming observations.

Systematically Improvable Generalization of Self-Interaction Corrected Density Functional Theory.

Perdew-Zunger self-interaction correction (PZSIC) reintroduces an exact constraint to approximate density functional theory (DFT). However, PZSIC can paradoxically degrade performance, and standard DFT approximations (with or without PZSIC) are not systematically improvable. We use the adiabatic projection formalism (Janesko, B. G. J. Chem. Phys. 2022, 156, 014111, https://doi.org/10.1063/5.0076144) to derive PZSIC in terms of a reference system experiencing only electron self-interaction. Generalization to a "self-and-some-others" interaction introduces correlation into the reference system, systematically bridging from PZSIC to exact wave function theory without the double counting of correlation. Minimal active spaces accurately treat nearly one electron, near-equilibrium, and strongly correlated model systems at modest computational expense.

Variational polaron equations applied to the anisotropic Fröhlich model

Starting from recent advances in the first-principles modeling of polarons, variational polaron equations in the strong-coupling adiabatic approximation are formulated in Bloch space. In this framework, polaron formation energy as well as individual electron, phonon and electron-phonon contributions are obtained. We suggest an efficient gradient-based optimization algorithm and apply these equations to the generalized Fr\"ohlich model with anisotropic non-degenerate electronic bands, both in two- and three-dimensional cases. The effect of the divergence of Fr\"ohlich electron-phonon matrix elements at $\Gamma$-point is treated analytically, improving the convergence with respect to the sampling in reciprocal space. The whole methodology is validated by obtaining the known asymptotic solution of the standard Fr\"ohlich model in isotropic scenario and also by comparing our results with the Gaussian ansatz approach, showing the difference between the numerically exact and Gaussian trial wavefunctions. Additionally, decomposition of the energy into individual terms allows one to recover the Pekar's 1:2:3:4 theorem, which is shown to be valid even in the anisotropic case. We expect that the improvements in the formalism and numerical implementation will be applicable beyond the large polaron hypothesis inherent to Fr\"ohlich model.