Wave Function In Position Space Research Articles

Macroscopic quantum response to gravitational waves

We study the excitation of a one-electron quantum cyclotron bygravitational waves. The electron in such as a penning trap is prepared tobe at the lowest Landau level, which has an infinitedegeneracy parameterized by the spread of the wave function in position space. We find that the excitation rate from the ground state to the first excited state is enhanced by the size of the electron wave function: an electron with a larger wave function feels gravitational waves more. As a consequence, we derive a good sensitivity to gravitational waves at a macroscopic one-electron quantum cyclotron.

High-accuracy calculation of nonrelativistic Compton profile for H-like ions

In collision processes such as Compton scattering, radiative electron capture (REC) and resonant transfer and excitation (RTE) and so on, the shape of their cross sections in the impulse approximation can be determined by the so-called Compton profile (CP), which reflects the initial distribution in the momentum space for bound electrons in atomic and molecular targets. In this paper, the pseudospectral-fitting (PSF) method is proposed to obtain the radial wave function in position space numerically, which is proved equivalent to the analytical solution, and then the radial wave function in momentum space can be calculated by the series method analytically in order to avoid the numerical difficulties in highly oscillatory integrals. Finally, taking the ns(n=1−10) orbitals of hydrogenic ions as examples, high-accuracy Compton profile has also been calculated at an arbitrary momentum with enough significant digits by using the Gauss–Legendre quadrature.

Incorporating the Coulomb potential into a finite, unitary perturbation theory

We have constructed a perturbation theory to treat interactions that can include the Coulomb interaction, describing a physical problem that is often encountered in nuclear physics. The Coulomb part is not treated perturbatively; the exact solutions are employed. The method is an extension of the results presented in the work of Hoffmann [J. Math. Phys. 62, 032105 (2021)]. It is designed to calculate phase shifts directly rather than the full form of the wavefunctions in position space. We present formulas that allow for the calculation of the phase shifts to second order in the perturbation. The phase shift results to second order, for a short-range potential, were compared with the exact solution, where we found an error of third order in the coupling strength. A different model, meant as a simple approximation of nuclear scattering of a proton on helium-4 and including a Coulomb potential and a spherical well, was constructed to test the theory. The wavepacket scattering formalism of Hoffmann [J. Phys. B: At., Mol. Opt. Phys. 50, 215302 (2017)], known to give everywhere finite results, was employed. We found physically acceptable results and a cross section of the correct order of magnitude.

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.

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.

Analysis of Compton profile through information theory in H-like atoms inside impenetrable sphere

Confinement of atoms inside various cavities has been studied for nearly eight decades. However, the Compton profile (CP) for such systems has not yet been investigated. Here we construct the CP for a H atom radially confined inside a hard spherical enclosure, as well as in a free condition. Some exact analytical relations for the CP’s of circular or nodeless states of free atoms is presented. By means of a scaling idea, this has been further extended to the study of a H-like atom trapped inside an impenetrable cavity. The accuracy of these constructed CPs has been confirmed by computing various momentum moments. Apart from that, several information theoretical measures, like Shannon entropy (S) and Onicescu energy (E) have been exploited to characterize these profiles. Exact closed-form expressions are derived for S and E using the ground state CP in free H-like atoms. A detailed study reveals that, increase in confinement inhibits the rate of dissipation of kinetic energy. At a fixed ℓ, this rate diminishes with a rise in n. However, at a certain n, this rate accelerates with progress in ℓ. Similar analysis on the respective free counterpart displays an exactly opposite trend as that in a confined system. However, in both free and confined environments, the CP generally gets broadened with rise in Z. Representative calculations are done numerically for low-lying states of the confined systems, taking two forms of position-space wave functions: (a) exact (b) highly accurate eigenfunctions through a generalized pseudospectral method. In essence, CPs are reported for confined H atoms (and isoelectronic series) and investigated adopting an information-theoretic framework.

Rising and lowering operator approach to the problem of a charged particle in a uniform magnetic field

In this paper, we analyse the quantization of cyclotron orbits for a 2D non-relativistic and spinless charged particle under the influence of a static and uniform magnetic field, the Landau problem. We follow the algebraic approach of rising and lowering operators. The Hamiltonian is expressed in terms of the usual rising and lowering operators of an isotropic quantum oscillator. Through a Bogoliubov transformation, we show that the Hamiltonian may be reduced to the one of a quantum oscillator that accounts for the formation of infinitely degenerate Landau levels. The explicit form of the position-space wave functions of Fock–Darwin states for a vanishing parabolic confinement potential is worked out by applying the techniques of ladder operators and the induction method. To explore the internal dynamics of the electron states, we compute the Fock–Darwin states’ density currents expressing them in terms of the ladder operators. In the supplementary material, we provide the reader with Mathematica algorithm that can help perform some of the calculations exposed in the paper. The charged particle in a magnetic field is used to explore some of the fundamental concepts of quantum mechanics as commuting operators, simultaneous eigenstates, degeneracy, rising and lowering operators, etc. This approach can be used as supplementary material in undergraduate final-year courses like quantum mechanics or solid state physics.

Fisher information of radial wavefunctions for relativistic hydrogenic atoms

We calculate Fisher information and the inverse participation ratio normalised by the electric binding energy for the relativistic hydrogenic atoms based on the radial wavefunctions in position space. Fisher information rapidly decreases as the atomic number increases, but it increases again for heavy atoms for S1/2 and P1/2 states due to the relativistic effect. We consider not only the ground state but also excited states by numerically evaluating the generalised Laguerre polynomials in the exact Dirac’s relativistic radial wavefunctions. This study contrasts with the previous studies of the relativistic many-electron systems (i.e., ground-state neutral atoms), where numerical densities were employed.

Topological boundaries and bulk wavefunctions in the Su–Schreiffer–Heeger model

Working in the context of the Su–Schreiffer–Heeger model, the effect of topological boundaries on the structure and properties of bulk position-space wavefunctions is studied for a particle undergoing a quantum walk in a one-dimensional lattice. In particular, we consider what happens when the wavefunction reaches a boundary at which the Hamiltonian changes suddenly from one topological phase to another and construct an exact solution for the wavefunction on both sides of the boundary. The reflection and transmission coefficients at the boundary are calculated as a function of the system’s hopping parameters, and it is shown that for some parameter ranges the transmission coefficient can be made very small. Therefore, it is possible to arrange a high degree of bulk wavefunction localization within each topological region, a fact that has information processing applications. This ‘topologically-assisted’ suppression of transitions, although not of direct topological origin itself, exists because of the presence of an abrupt change in the properties of the Hamiltonian at the topological boundary. We give a quantitative examination of the reflection and transmission coefficients of incident waves at the boundary between regions of different winding number.

Spectroscopy and decay properties of charmonium

The mass spectra of charmonium are investigated using a Coulomb plus linear (Cornell) potential. Gaussian wave functions in position space as well as in momentum space are employed to calculate the expectation values of potential and kinetic energy respectively. Various experimental states (X(4660)(53S1), X(3872)(23P1), X(3900)(21P1), X(3915)(23P0) and X(4274)(33P1) etc.) are assigned as charmonium states. We also study the Regge trajectories, pseudoscalar and vector decay constants, electric and magnetic dipole transition rates, and annihilation decay widths for charmonium states.

On some non-Gaussian wave packets

Abstract We expand on a previous study by offering a generalized wave function associated with the parabolic cylinder function and a connection with a two-particle position-space wave function. We also provide an explicit formula for a wave function associated with a recent work by the present author and M. Wolf.

Long-range nonlocality in six-point string scattering: Simulation of black hole infallers

We set up a tree-level six point scattering process in which two strings are separated longitudinally such that they could only interact directly via a non-local spreading effect such as that predicted by light cone gauge calculations and the Gross-Mende saddle point. One string, the `detector', is produced at a finite time with energy $E$ by an auxiliary $2\to 2$ sub-process, with kinematics such that it has sufficient resolution to detect the longitudinal spreading of an additional incoming string, the `source'. We test this hypothesis in a gauge-invariant S-matrix calculation convolved with an appropriate wavepacket peaked at a separation $X$ between the central trajectories of the source and produced detector. The amplitude exhibits support for scattering at the predicted longitudinal separation $X\sim\alpha' E$, in sharp contrast to the analogous quantum field theory amplitude (whose support manifestly traces out a tail of the position-space wavefunction). The effect arises in a regime in which the string amplitude is not obtained as a convergent sum of such QFT amplitudes, and has larger amplitude than similar QFT models (with the same auxiliary four point amplitude). In a linear dilaton background, the amplitude depends on the string coupling as expected if the scattering is not simply occuring on the wavepacket tail in string theory. This manifests the scale of longitudinal spreading in a gauge-invariant S-matrix amplitude, in a calculable process with significant amplitude. It simulates a key feature of the dynamics of time-translated horizon infallers.

Single-slit focusing and its representations

We illustrate the phenomenon of the focusing of a freely propagating rectangular wave packet using three different tools: (1) the time-dependent wave function in position space, (2) the Wigner phase-space approach, and (3) an experiment using laser light.

Horizon wave function for single localized particles: GUP and quantum black-hole decay

A localised particle in Quantum Mechanics is described by a wave packet in position space, regardless of its energy. However, from the point of view of General Relativity, if the particle's energy density exceeds a certain threshold, it should be a black hole. In order to combine these two pictures, we introduce a horizon wave-function determined by the particle wave-function in position space, which eventually yields the probability that the particle is a black hole. The existence of a minimum mass for black holes naturally follows, albeit not in the form of a sharp value around the Planck scale, but rather like a vanishing probability that a particle much lighter than the Planck mass be a black hole. We also show that our construction entails an effective Generalised Uncertainty Principle (GUP), simply obtained by adding the uncertainties coming from the two wave-functions associated to a particle. Finally, the decay of microscopic (quantum) black holes is also described in agreement with what the GUP predicts.

Precision Measurement of the Position-Space Wave Functions of Gravitationally Bound Ultracold Neutrons

Gravity is the most familiar force at our natural length scale. However, it is still exotic from the view point of particle physics. The first experimental study of quantum effects under gravity was performed using a cold neutron beam in 1975. Following this, an investigation of gravitationally bound quantum states using ultracold neutrons was started in 2002. This quantum bound system is now well understood, and one can use it as a tunable tool to probe gravity. In this paper, we review a recent measurement of position-space wave functions of such gravitationally bound states and discuss issues related to this analysis, such as neutron loss models in a thin neutron guide, the formulation of phase space quantum mechanics, and UCN position sensitive detectors. The quantum modulation of neutron bound states measured in this experiment shows good agreement with the prediction from quantum mechanics.

EXACT GROUND STATES OF CORRELATED ELECTRONS ON PENTAGON CHAINS

We construct a class of exact ground states for correlated electrons on pentagon chains in the high density region and discuss their physical properties. In this procedure the Hamiltonian is first cast in a positive semidefinite form using composite operators as a linear combination of creation operators acting on the sites of finite blocks. In the same step, the interaction is also transformed to obtain terms which require for their minimum eigenvalue zero at least one electron on each site. The transformed Hamiltonian matches the original Hamiltonian through a nonlinear system of equations whose solutions place the deduced ground states in restricted regions of the parameter space. In the second step, nonlocal product wave functions in position space are constructed. They are proven to be unique ground states which describe non-saturated ferromagnetic and correlated half metallic states. These solutions emerge when the strength of the Hubbard interaction Ui is site-dependent inside the unit cell. In the deduced phases, the interactions tune the bare dispersive band structure such to develop an effective upper flat band. We show that this band flattening effect emerges for a broader class of chains and is not restricted to pentagon chains. For the characterization of the deduced solutions, uniqueness proofs, exact ground state expectation values for long-range hopping amplitudes and correlation functions are also calculated. The study of physical reasons which lead to the appearance of ferromagnetism has revealed a new mechanism for the emergence of an ordered phase, described here in detail. This works as follows: starting from a completely dispersive bare band structure, the interactions quench the kinetic energy, hence the ordered phase is obtained solely by a drastic decrease of the interaction energy. Since Ui are site dependent, this determinative decrease is obtained by a redistribution of the double occupancy di such to attain small di where the on-site Coulomb repulsion Ui is high, and vice versa. The kinetic energy quench leads to the upper effective flat band, whose role is to enhance by its degeneracy the switching to the ordered phase dictated and stabilized by the interactions present. It is shown that for this phenomenon to occur, a given degree of complexity is needed for the chain, and the mechanism becomes inactive when the Ui interactions are homogeneous, or are missing from the ground state wave function.

Less than perfect quantum wavefunctions in momentum-space: How ϕ(p) senses disturbances in the force

We develop a systematic approach to determine the large |p| behavior of the momentum-space wave function, ϕ(p), of a one-dimensional quantum system for which the position-space wave function, ψ(x), has a discontinuous derivative at any order. We find that if the kth derivative of the potential energy function for the system has a discontinuity, there is a corresponding discontinuity in ψ(k+2)(x) at the same point. This discontinuity leads directly to a power-law tail in the momentum-space wave function proportional to 1/pk+3. A number of familiar pedagogical examples are examined in this context, leading to a general derivation of the result.

Geometric Quantization, Parallel Transport and the Fourier Transform

In quantum mechanics, the momentum space and position space wave functions are related by the Fourier transform. We investigate how the Fourier transform arises in the context of geometric quantization. We consider a Hilbert space bundle \(\mathcal{H}\) over the space \(\mathcal{J}\) of compatible complex structures on a symplectic vector space. This bundle is equipped with a projectively flat connection. We show that parallel transport along a geodesic in the bundle \(\mathcal{H} \to \mathcal{J}\) is a rescaled orthogonal projection or Bogoliubov transformation. We then construct the kernel for the integral parallel transport operator. Finally, by extending geodesics to the boundary (for which the metaplectic correction is essential), we obtain the Segal-Bargmann and Fourier transforms as parallel transport in suitable limits.

Exact ground states of the periodic Anderson model inD=3dimensions

We construct a class of exact ground states of three-dimensional periodic Anderson models (PAMs) -- including the conventional PAM -- on regular Bravais lattices at and above 3/4 filling, and discuss their physical properties. In general, the f electrons can have a (weak) dispersion, and the hopping and the non-local hybridization of the d and f electrons extend over the unit cell. The construction is performed in two steps. First the Hamiltonian is cast into positive semi-definite form using composite operators in combination with coupled non-linear matching conditions. This may be achieved in several ways, thus leading to solutions in different regions of the phase diagram. In a second step, a non-local product wave function in position space is constructed which allows one to identify various stability regions corresponding to insulating and conducting states. The compressibility of the insulating state is shown to diverge at the boundary of its stability regime. The metallic phase is a non-Fermi liquid with one dispersing and one flat band. This state is also an exact ground state of the conventional PAM and has the following properties: (i) it is non-magnetic with spin-spin correlations disappearing in the thermodynamic limit, (ii) density-density correlations are short-ranged, and (iii) the momentum distributions of the interacting electrons are analytic functions, i.e., have no discontinuities even in their derivatives. The stability regions of the ground states extend through a large region of parameter space, e.g., from weak to strong on-site interaction U. Exact itinerant, ferromagnetic ground states are found at and below 1/4 filling.

Analytic Results for Gaussian Wave Packets in Four Model Systems: II. Autocorrelation Functions

The autocorrelation function, A(t), measures the overlap (in Hilbert space) of a time-dependent quantum mechanical wave function, psi(x,t), with its initial value, psi(x,0). It finds extensive use in the theoretical analysis and experimental measurement of such phenomena as quantum wave packet revivals. We evaluate explicit expressions for the autocorrelation function for time-dependent Gaussian solutions of the Schrodinger equation corresponding to the cases of a free particle, a particle undergoing uniform acceleration, a particle in a harmonic oscillator potential, and a system corresponding to an unstable equilibrium (the so-called `inverted' oscillator.) We emphasize the importance of momentum-space methods where such calculations are often more straightforwardly realized, as well as stressing their role in providing complementary information to results obtained using position-space wavefunctions.