Wave Functions In Space Research Articles

Probability distribution for the quantum universe

We determine the inner product on the Hilbert space of wavefunctions of the universe by imposing the Hermiticity of the quantum Hamiltonian in the context of the minisuperspace model. The corresponding quantum probability density reproduces successfully the classical probability distribution in the ħ → 0 limit, for closed universes filled with a perfect fluid of index w. When −1/3 < w ≤ 1, the wavefunction is normalizable and the quantum probability density becomes vanishingly small at the big bang/big crunch singularities, at least at the semiclassical level. Quantum expectation values of physical geometrical quantities, which diverge classically at the singularities, are shown to be finite.

Momentum-Space Decoherence of Distinguishable and Identical Particles in the Caldeira–Leggett Formalism

In this work, momentum-space decoherence using minimum and nonminimum-uncertainty-product (stretched) Gaussian wave packets in the framework of Caldeira–Leggett formalism and under the presence of a linear potential is studied. As a dimensionless measure of decoherence, purity, a quantity appearing in the definition of the linear entropy, is studied taking into account the role of the stretching parameter. Special emphasis is on the open dynamics of the well-known cat states and bosons and fermions compared to distinguishable particles. For the cat state, while the stretching parameter speeds up the decoherence, the external linear potential strength does not affect the decoherence time; only the interference pattern is shifted. Furthermore, the interference pattern is not observed for minimum-uncertainty-product-Gaussian wave packets in the momentum space. Concerning bosons and fermions, the question we have addressed is how the symmetry of the wave functions of indistinguishable particles is manifested in the decoherence process, which is understood here as the loss of being indistinguishable due to the gradual emergence of classical statistics with time. We have observed that the initial bunching and anti-bunching character of bosons and fermions, respectively, in the momentum space are not preserved as a function of the environmental parameters, temperature, and damping constant. However, fermionic distributions are slightly broader than the distinguishable ones and these similar to the bosonic distributions. This general behavior could be interpreted as a residual reminder of the symmetry of the wave functions in the momentum space for this open dynamics.

Theory of Nonradiative Energy Transfer between Two Optical Ions Using the Proper Adiabatic Approximation

Using adiabatic approximation of polyatomic molecules and methods of molecular quantum electrodynamics, we have derived the Förster-Dexter theory of energy transfer between two optical ions. Although we have assumed a single configurational coordinate, the theory can be easily formulated for multiple normal coordinate system. This approach leads smoothly to the energy transfer rates originally derived by Dexter1 for electric dipole-electric dipole interaction without any additional assumptions regarding the normalization of the wavefunctions in the energy space. Using the transfer rates thus derived, and the band shape functions, Wp , a new function F(T) has been proposed to study the temperature dependence of energy transfer rates due to the overlap of the emission and absorption line shapes of the donor and acceptor ions, respectively. This function has been studied for various scenarios of the relative alignment of the zero phonon lines of these ions in order to understand the participation of phonons in the energy transfer process. We have also compared this quantum mechanically derived F(T) function with that derived using semiclassical theory. The agreements are very good at high temperatures. D. L. Dexter, J. Chem. Phys. 21, 836(1953)

Bound and scattering states for supersingular potentials

Supersingular potentials are defined as potentials involving derivatives of the delta function of order higher than one. In this work we solve the Schrödinger equation for supersingular potentials. For the bound states, we calculate their energy and the corresponding wave functions (when they exist). For the scattering state we calculate the reflection and transmission coefficients in addition to the wave function of these states. We study the case of the delta double prime potential in details, showing that we may have none, one or two bound states, depending on the relations between the values of the coupling constants. In general the wave function in the position space has discontinuity at the point of support of the supersingular potential.

Quantum dynamics of a nucleon in the Fermi accelerator

The quantum dynamics of a particle in a one-dimensional box with an oscillating wall (the Fermi accelerator) is investigated. The model is applied to the motion of a single nucleon in the mean-field potential of a heavy atomic nucleus whose surface vibrates. By directly solving the time-dependent Schrödinger equation, both the state of the particle and its mean-energy are studied. The effects of the frequency of the wall oscillation on the nucleon’s energy are addressed. Its energy oscillates in phase with the moving wall for all frequencies, showing no chaotic behaviour. There is a large initial peak of the nucleon’s energy as the particle adjusts to the sudden change in the size of the box and a varying relaxation time as it plateaus towards lower energy and a partial equilibrium. Small oscillations in energy continue, since there cannot be a true equilibrium while the wall is moving. The quantum coherence between the different parts of the nucleon’s wave-function in real space is very much preserved. This research lays the foundation for future investigations into quantum tunnelling in the Fermi accelerator.

Two-body wave functions and compositeness from scattering amplitudes. II. Application to the physical N* and Δ* resonances

The meson-baryon molecular components for the ${N}^{*}$ and ${\mathrm{\ensuremath{\Delta}}}^{*}$ resonances are investigated in terms of the compositeness, which is defined as the norm of the two-body wave function from the meson-baryon scattering amplitudes. The scattering amplitudes are constructed in a $\ensuremath{\pi}N\text{\ensuremath{-}}\ensuremath{\eta}N\text{\ensuremath{-}}\ensuremath{\sigma}N\text{\ensuremath{-}}\ensuremath{\rho}N\text{\ensuremath{-}}\ensuremath{\pi}\mathrm{\ensuremath{\Delta}}$ coupled-channels problem in a meson exchange model together with several bare ${N}^{*}$ and ${\mathrm{\ensuremath{\Delta}}}^{*}$ states, and parameters are fitted so as to reproduce the on-shell $\ensuremath{\pi}N$ partial wave amplitudes up to the center-of-mass energy $1.9\phantom{\rule{0.16em}{0ex}}\mathrm{GeV}$ with the orbital angular momentum $L\ensuremath{\le}2$. As a result, the Roper resonance $N(1440)$ is found to be dominated by the $\ensuremath{\pi}N$ and $\ensuremath{\sigma}N$ molecular components while the bare-state contribution is small. The squared wave functions in coordinate space imply that both in the $\ensuremath{\pi}N$ and $\ensuremath{\sigma}N$ channels the separation between the meson and baryon is about more than $1\phantom{\rule{0.16em}{0ex}}\mathrm{fm}$ for the $N(1440)$ resonance. On the other hand, dominant meson-baryon molecular components are not observed in any other ${N}^{*}$ and ${\mathrm{\ensuremath{\Delta}}}^{*}$ resonances in the present model, although they have some fractions of the meson-baryon clouds.

Making squeezed-coherent states concrete by determining their wavefunction

With the successes of the laser interferometer gravitational-wave observatory, we anticipate increased interest in working with squeezed states in the undergraduate and graduate quantum-mechanics classroom. Because squeezed-coherent states are minimum uncertainty states, their wavefunctions in position and momentum space must be Gaussians. But this result is rarely discussed in treatments of squeezed states in quantum textbooks or quantum optics textbooks. In this work, we show three different ways to construct the wavefunction for squeezed-coherent states: (i) a differential equation-based approach; (ii) an approach that uses an expansion in terms of the simple-harmonic oscillator wavefunctions; and (iii) a fully operator-based approach. We do this to illustrate that the concept of the wavefunction can be introduced no matter what methodology an instructor wishes to use. We hope that working with the wavefunction will help demystify the concept of a squeezed-coherent state.

Understanding the impact of correlation on bond length alternation in polyenes

This article analyzes the electronic factors governing bond length alternation (BLA) in linear polyenes. The impact of the various effects is illustrated on small all-trans polyenes, namely butadiene, hexatriene and octatetraene prototype molecules. It is well known that self-consistent-field single determinant treatments overestimate the bond length alternation and the paper aims to identify physical effects of correlation which correct this defect. The question is addressed using an orthogonal valence bond-type formalism in which the wave function is expressed in terms of strongly localized bonding and antibonding molecular orbitals. This paper shows that dynamic polarization effects of π orbitals accounted for in the full-π complete active space wave function significantly reduce bond alternation. These effects are brought by single excitations applied on the inter-bond charge transfer determinants. The dynamic polarization of σ bonds, of either CC or CH character, is analyzed afterward by either enlarging the active space or by adding the 1hole-1particle excitations. It is shown that these effects also decrease the BLA and increase the coefficients of the charge transfer determinants. Moreover, the relation with dynamic polarization of ligand-to-metal and metal-to-ligand charge transfer (LMCT and MLCT, respectively) components in magnetic transition-metal compounds is discussed.

Topologically protected optical signal processing using parity–time-symmetric oscillation quenching

AbstractThe concept of topology is universally observed in various physical objects when the objects can be described by geometric structures. Although a representative example is the knotted geometry of wavefunctions in reciprocal space for quantum Hall family and topological insulators, topological states have also been defined for other physical quantities, such as topologically distinct Fermi surfaces and enhanced lattice degrees of freedom in hyperbolic geometry. Here, we investigate a different class of topological states – topological geometry of dynamical state trajectories – in non-Hermitian and nonlinear optical dynamics, revealing topologically protected oscillation quenching mechanisms determined by parity–time (PT) symmetry. For coupled systems composed of nonlinear gain and loss elements, we classify the topology of equilibria separately for unbroken and broken PT symmetry, which result in distinct oscillation quenching mechanisms: amplitude death and oscillation death. We then show that these PT-symmetric quenching mechanisms lead to immunity against temporal perturbations, enabling the applications of topologically protected laser modulation and rectification. The observed connection between the topological geometry of dynamical states, oscillation quenching phenomena in dynamical systems theory, and PT symmetry provides a powerful toolkit for noise-immune signal processing.

Two-dimensional boson oscillator under a magnetic field in the presence of a minimal length in the non-commutative space

Minimal length in non-commutative space of a two-dimensional Klein-Gordon oscillator is investigated and illustrates the wave functions in the momentum space. The eigensolutions are found and the system is mapping to the well-known Schrodinger equation in a Pöschl-Teller potential.

Order-by-disorder from bond-dependent exchange and intensity signature of nodal quasiparticles in a honeycomb cobaltate

Recent theoretical proposals have argued that cobaltates with edge-sharing octahedral coordination can have significant bond-dependent exchange couplings thus offering a platform in 3d ions for such physics beyond the much-explored realisations in 4d and 5d materials. Here we present high-resolution inelastic neutron scattering data within the magnetically ordered phase of the stacked honeycomb magnet CoTiO3 revealing the presence of a finite energy gap and demonstrate that this implies the presence of bond-dependent anisotropic couplings. We also show through an extensive theoretical analysis that the gap further implies the existence of a quantum order-by-disorder mechanism that, in this material, crucially involves virtual crystal field fluctuations. Our data also provide an experimental observation of a universal winding of the scattering intensity in angular scans around linear band-touching points for both magnons and dispersive spin-orbit excitons, which is directly related to the non-trivial topology of the quasiparticle wavefunction in momentum space near nodal points.

Theory of Nonradiative Energy Transfer between Two Optical Ions Using Proper Adiabatic Approximation

Using the adiabatic approximation of polyatomic molecules and methods of molecular quantum electrodynamics, we have derived the Förster-Dexter theory of energy transfer between two optical ions. Although we have assumed a single configurational coordinate, the theory can be easily formulated for a multiple normal coordinate system. This approach leads smoothly to the energy transfer rates originally derived by Dexter1 for electric dipole-electric dipole interaction without any additional assumptions regarding the normalization of the wavefunctions in the energy space. Using the transfer rates thus derived, and the band shape functions, W p , a new function F(T) has been proposed to study the temperature dependence of energy transfer rates due to the overlap of the emission and absorption lineshapes of the donor and acceptor ions, respectively. This function has been studied for various scenarios of the relative alignment of the zero phonon lines of these ions in order to understand the participation of phonons in the energy transfer process. We have also compared this quantum mechanically derived F(T) function with that derived using semiclassical theory. The agreements are very good at high temperatures.

Particle on the sphere: group-theoretic quantization in the presence of a magnetic monopole

The problem of quantizing a particle on a two-sphere has been treated by numerous approaches, including Isham’s global method based on unitary representations of a symplectic symmetry group that acts transitively on the phase space. Here we reconsider this simple model using Isham’s scheme, enriched by a magnetic flux through the sphere via a modification of the symplectic form. To maintain complete generality we construct the Hilbert space directly from the symmetry algebra, which is manifestly gauge-invariant, using ladder operators. In this way, we recover algebraically the complete classification of quantizations, and the corresponding energy spectra for the particle. The famous Dirac quantization condition for the monopole charge follows from the requirement that the classical and quantum Casimir invariants match. In an appendix we explain the relation between this approach and the more common one that assumes from the outset a Hilbert space of wave functions that are sections of a nontrivial line bundle over the sphere, and show how the Casimir invariants of the algebra determine the bundle topology.

Quantum Chaos Driven by Long-Range Waveguide-Mediated Interactions.

We study theoretically quantum states of a pair of photons interacting with a finite periodic array of two-level atoms in a waveguide. Our calculation reveals two-polariton eigenstates that have a highly irregular wave function in real space. This indicates the Bethe ansatz breakdown and the onset of quantum chaos, in stark contrast to the conventional integrable problem of two interacting bosons in a box. We identify the long-range waveguide-mediated coupling between the atoms as the key ingredient of chaos and nonintegrability. Our results provide new insights in the interplay between order, chaos, and localization in many-body quantum systems and can be tested in state-of-the-art setups of waveguide quantum electrodynamics.

Multiconfiguration Pair-Density Functional Theory for Transition Metal Silicide Bond Dissociation Energies, Bond Lengths, and State Orderings.

Transition metal silicides are promising materials for improved electronic devices, and this motivates achieving a better understanding of transition metal bonds to silicon. Here we model the ground and excited state bond dissociations of VSi, NbSi, and TaSi using a complete active space (CAS) wave function and a separated-pair (SP) wave function combined with two post-self-consistent field techniques: complete active space with perturbation theory at second order and multiconfiguration pair-density functional theory. The SP approximation is a multiconfiguration self-consistent field method with a selection of configurations based on generalized valence bond theory without the perfect pairing approximation. For both CAS and SP, the active-space composition corresponds to the nominal correlated-participating-orbital scheme. The ground state and low-lying excited states are explored to predict the state ordering for each molecule, and potential energy curves are calculated for the ground state to compare to experiment. The experimental bond dissociation energies of the three diatomic molecules are predicted with eight on-top pair-density functionals with a typical error of 0.2 eV for a CAS wave function and a typical error of 0.3 eV for the SP approximation. We also provide a survey of the accuracy achieved by the SP and extended separated-pair approximations for a broader set of 25 transition metal–ligand bond dissociation energies.

Lewis Structures from Open Quantum Systems Natural Orbitals: Real Space Adaptive Natural Density Partitioning.

Building chemical models from state-of-the-art electronic structure calculations is not an easy task, since the high-dimensional information contained in the wave function needs to be compressed and read in terms of the accepted chemical language. We have already shown (Phys. Chem. Chem. Phys.2018, 20, 2136830095829) how to access Lewis structures from general wave functions in real space by reformulating the adaptive natural density partitioning (AdNDP) method proposed by Zubarev and Boldyrev (Phys. Chem. Chem. Phys.2008, 10, 520718728862). This provides intuitive Lewis descriptions from fully orbital invariant position space descriptors but depends on not immediately accessible higher order cumulant density matrices. By using an open quantum systems (OQS) perspective, we here show that the rigorously defined OQS fragment natural orbitals can be used to build a consistent real space adaptive natural density partitioning based only on spatial information and the system’s one-particle density matrix. We show that this rs-AdNDP approach is a cheap, efficient, and robust technique that immerses electron counting arguments fully in the real space realm.

Experimental measurement of the intrinsic excitonic wave function.

An exciton, a two-body composite quasiparticle formed of an electron and hole, is a fundamental optical excitation in condensed matter systems. Since its discovery nearly a century ago, a measurement of the excitonic wave function has remained beyond experimental reach. Here, we directly image the excitonic wave function in reciprocal space by measuring the momentum distribution of electrons photoemitted from excitons in monolayer tungsten diselenide. By transforming to real space, we obtain a visual of the distribution of the electron around the hole in an exciton. Further, by also resolving the energy coordinate, we confirm the elusive theoretical prediction that the photoemitted electron exhibits an inverted energy-momentum dispersion relationship reflecting the valence band where the partner hole remains, rather than that of conduction band states of the electron.

Studying Inelastic Electron Scattering of Magnetic Form Factors M1 for 48Ca Nucleus Using Skyrme Interaction

The M1 inelastic scattering form factors from 48Ca nucleus have been investigated through nuclear shell theory. The nuclear effective two body interaction fpbm of Richter has been utilized to generate the two body wave vectors with Harmonic oscillator wave function as a single particle wave function in Fp shell model space. Core polarization effect has been used to include the discarded space (core+ higher configuration) via model space with realistic interaction of Skyrme Sly5 to couple the model space active particles with the pair (particle-hole) excited in energy of 2ћω. Calculated results has been compared to available experimental data.

The basis of the semiclassical description of electron transport in solids

The theoretical account of electron transport in crystalline solids is often taught to graduate students through a semiclassical equation, which is a classical kinetic equation amended with elements of quantum physics in a non-systematic way. This blending of classical and quantal elements accounts for many experimental features of electron transport, save for a few ones observed over short distances. For students acquainted with solid-state quantum mechanics, it is preferable to derive the semiclassical equation from quantum dynamics so as to pinpoint the shortcomings and approximations underlying the semiclassical picture of transport. This paper explores the pathway leading from quantum dynamics to semiclassical kinetics within a simplified model not exceeding the proficiencies of a graduate student in physics or electronic engineering. In this model an electron moves in a crystal under the joint influences of the periodic crystal potential, an external electrostatic field and a set of lattice imperfections haphazardly distributed in the crystal volume. The first influence is treated exactly by means of standard energy-band theory and the other two are handled perturbatively. Following a procedure already tested in free-space issues, we replace Schrödinger’s wave function in position space with a Wigner function defined in the phase (position–pseudomomentum) space of semiclassical kinetics. The dynamical equation on the Wigner function is worked out and approximated so as to retrieve the semiclassical equation. The errors incurred in making use of the latter are assessed quantitatively. It is explained how a time-irreversible evolution equation—governing a phase-space occupancy—arises from reversible Schrödinger dynamics—governing a complex-valued probability amplitude. Besides clarifying the foundation of the semiclassical picture of electron transport and providing an explicit quantum-coherence correction to that picture, this paper may help students in general physics to better grasp, in a concrete issue, the interplay between classical and quantum concepts.

Fractional-quantum-Hall-effect (FQHE) in 1D Hubbard models

We study the quantum self-organization of interacting particles in one-dimensional(1D) many-body systems, modeled via Hubbard chains with short-range interactions between the particles. We show the emergence of 1D states with density-wave and clustering order, related to topology, at odd denominator fillings that appear also in the fractional-quantum-Hall-effect (FQHE), which is a 2D electronic system with Coulomb interactions between the electrons and a perpendicular magnetic field. For our analysis we use an effective topological measure applied on the real space wavefunction of the system, the Euler characteristic describing the clustering of the interacting particles. The source of the observed effect is the spatial constraints imposed by the interaction between the particles. In overall, we demonstrate a simple mechanism to reproduce many of the effects appearing in the FQHE, without requiring a Coulomb interaction between the particles or the application of an external magnetic field.