Single-particle Schrodinger Equation Research Articles

An analogy between effective potential representing prime numbers and Schwarzschild black hole

The effective two body interaction is derived using the single particle Schrödinger equation for the prime numbers with probability density based on the prime counting function (which is the number of primes below a certain x). Though the speculative connection of primes to Schwarzschild spacetime or black hole has been there for a long time, here for the first time our derived potential serendipitously shows the unique similarity with the effective potential derived from general theory of relativity (GTR) for Schwarzschild spacetime. Also, we report the existence of ISCO and photon sphere in prime potential and compare with GTR. It is found that when the prime numbers are represented by a quantum system, the two-body interactions mimic the effective general relativistic gravitational interaction in black hole with the four-acceleration diverging. We also establish an analogy between Einstein's static universe and the universe of primes. The question is: “Who ordered that?”

Path integral approach to electron scattering in classical electromagnetic potential**Project supported by the National Natural Science Foundation of China (Grant Nos. 11374360, 11405266, and 11505285) and the National Basic Research Program of China (Grant No. 2013CBA01504).

As is known to all, the electron scattering in classical electromagnetic potential is one of the most widespread applications of quantum theory. Nevertheless, many discussions about electron scattering are based upon single-particle Schrodinger equation or Dirac equation in quantum mechanics rather than the method of quantum field theory. In this paper, by using the path integral approach of quantum field theory, we perturbatively evaluate the scattering amplitude up to the second order for the electron scattering by the classical electromagnetic potential. The results we derive are convenient to apply to all sorts of potential forms. Furthermore, by means of the obtained results, we give explicit calculations for the one-dimensional electric potential.

In-situheight engineering of InGaAs / GaAs quantum dots by chemical beam epitaxy

This work reports on a chemical beam epitaxy growth study of InGaAs/GaAs quantum dots (QDs) engineered using an in-situ indium-flush technique. The emission energy of these structures has been selectively tuned over 225 meV by varying the dot height from 7 to 2 nm. A blueshift of the photoluminescence (PL) emission peak and a decrease of the intersublevel spacing energy are observed when the dot height is reduced. Numerical investigations of the influence of dot structural parameters on their electronic structure have been carried out by solving the single-particle one-band effective mass Schrodinger equation in cylindrical coordinates, for lens-shaped QDs. The correlation between numerical calculations and PL results is used to better describe the influence of the In-flush technique on both the dot height and the dot composition.

A novel method for determining the mean-field directly from the single particle matter density

We present a novel method, using the single particle Schrodinger equation, to determine the central potential directly from the single particle matter density and its first and second derivatives. As an example, we consider the experimental data for the charge density difference between the isotones 206Pb - 205Tl that corresponds to the shell model 3s1/2 proton orbit, and determine the corresponding single particle potential (mean-field). We also present results of least-square fits to parametrized single particle potentials.

Electron spectrum of a single-wall carbon nanotube in the framework of the nonlinear Schrödinger equation

The electron spectrum of a single-wall carbon metal nanotube is analyzed numerically. The interaction of a free electron with atomic ions and bound electrons is approximated by an attractive delta-function potential in the single-particle Schrodinger equation. The interaction of an electron with other free electrons is presented by the Hartree nonlinear repulsive short-range potential.

Properties of bottomonium in a semi-relativistic model

Using a semi-relativistic potential model we investigate the spectra and decays of the bottomonium (bb̄) system. The Hamiltonian of our model consists of a relativistic kinetic energy term, a vector Coulomb-like potential and a scalar confining potential. Using this Hamiltonian, we obtain a spinless wave equation, which is then reduced to the form of a single particle Schrodinger equation. The spin dependent potentials are introduced as a perturbation. The three-dimensional harmonic oscillator wave function is employed as a trial wave function and the bb̄ mass spectrum is obtained by the variational method. The model parameters and the wave function that reproduce the the bb̄ spectrum are then used to investigate some of their decay properties. The results obtained are then compared with the experimental data and with the predictions of other theoretical models.

Numerical calculation of strong-field laser-atom interaction: An approach with perfect reflection-free radiation boundary conditions

The time-dependent, single-particle Schrodinger equation with a finite-range potential is solved numerically on a three-dimensional spherical domain. In order to correctly account for outgoing waves, perfect reflection-free radiation boundary conditions are used on the surface of a sphere. These are computationally most effective if the particle wavefunction is expanded in the set of spherical harmonics and computations are performed in the Kramers-Henneberger accelerated frame. The method allows one to solve the full ionization dynamics in intense laser fields within a small region of atomic dimensions.

Eigenvalue problem for bundle of operators and the sturm expansion method in quantum mechanics

Different variants of the generalized formulation of the eigenvalue problem related to a single-particle Schrodinger equation are considered. The nature of the spectrum of problems for a bundle of operators containing the Hamiltonian of the system is analyzed in relation to the form of the weighting operator. For rather general conditions, corresponding systems of eigenfunctions provide diagonal representations of the Green’s function and provide of a rather fast convergence of expansions over intermediate states. Cases of potentials with Coulomb asymptotics and short-range potentials are considered.

A many-particle quantum-trajectory approach for modeling electron transport and its correlations in nanoscale devices

Electron transport in mesoscopic systems is analyzed in terms of quantum (Bohm) trajectories associated to wave-function solutions of a many-particle (effective-mass) Schrodinger equation. Many-particle Bohm trajectories can be computed from single-particle Schrodinger equations. As an example, electron correlations for a triple-barrier tunneling system with electron-electron interactions are computed. Simulated noise results for interacting electrons that tunnels through triple barriers are presented. The approach opens a new path for studying electron transport and quantum noise in nanoscale systems, beyond the “Fermi liquid” paradigm.

The Single-Determinant Approximation with a Local Potential for Excited States

The specific features of the calculations of the electronic structure in the approximation of a local exchange potential that is identical for all the electrons involved are considered. An optimized effective potential method is proposed for calculating the energies of excited electronic states of the same symmetry. A single-particle Schro dinger equation is derived for an excited state whose orbitals are described by a single-determinant wave function orthogonal to the ground state. The equations determining the local potential for excited states are obtained within the variational approach. The solution to these equations is analyzed in the framework of the parameterized representation of the effective potential. The efficiency of the proposed method is demonstrated by calculating the energies of three excited states of the same symmetry for a HeH molecule. The difference between the results obtained by the Hartree-Fock method and the method proposed in this paper is equal, on average, to 0.05%. A comparison with the results obtained from precise calculations based on the configuration interaction method shows that the accuracy in determining the energy of the excited states by the optimized effective potential method is comparable to the accuracy in calculating the energy of the ground state.

Time-dependent optimized effective potential.

Given an expression for the quantum mechanical action of an N -electron system as a functional of N time-dependent spin orbitals, we present a method of constructing the variationally best local time-dependent single-particle potentials which, when inserted in time-dependent singleparticle Schrodinger equations for the spin-up and spin-down electrons yield orbitals that make stationary. We also propose a simplification of this scheme leading to a time-dependent generalization of the static optimized effective potentials recently introduced by Krieger, Li, and Iafrate [Phys. Lett. A 146, 256 (1990)]. Owing to rapid experimental progress in the field of laser physics, ultrashort laser pulses of very high intensity have become available in recent years. The electric field produced in such pulses can reach the strength of the electric field caused by atomic nuclei. If an atomic system is placed in the focus of such a laser pulse, one observes a wealth of new phenomena [1] which cannot be explained by perturbation theory. The nonperturbative quantum mechanical description of interacting particles moving in a very strong time-dependent external field therefore has become a prominent problem of theoretical physics. Since its rigorous foundation by Runge and Gross [2], time-dependent density functional theory (TDDFT) [3 ‐ 5] is available as a method to deal with time-dependent many-particle problems of this kind. The central statement of TDDFT is that the time-dependent density of a system of interacting particles moving in an external potential can be calculated, in principle exactly, from a set of time-dependent single-particle equations which can be viewed as the time-dependent counterpart of the Kohn-Sham scheme. These singleparticle equations involve an exchange-correlation potential, , which is a functional of and has to be approximated in practice. An extension of TDDFT to spin-polarized systems has been proposed by Liu and Vosko [6]. Neglecting magnetic effects associated with the orbital motion of the electrons, they consider external time-dependent potentials acting on electrons with spin only. In this case, two different corresponding to the two spin orientations are needed which are functionals of the spin densities . Again, the key problem is to obtain good approximations of . To date, only a rather crude adiabatic approximation is available which adopts the functional form of the static exchangecorrelation potential. The purpose of this paper is to introduce a different approach to the construction of

A simplified self-consistent model of charge control in quasi-square quantum well HFETs

A new, approximate model for determining the one-dimensional charge control characteristics of quasi-square quantum well channel heterojunction field-effect transistors is presented. The model involves solving self-consistently the one-dimensional forms of Poisson's equation and the single-particle Schrodinger equation in which the potential variation inside the quantum well is approximated by a ramp potential. It is shown that this approximation models more closely the effect of including an explicit exchange and correlation potential than the Hartree potential alone. Finally the effect of including quantum confinement effects in calculating the charge control characteristics is discussed.

Common nonlinearities in different approaches to nuclear physics

It is pointed out that for 1D (3D spherically symmetric systems) the static versions of the single-particle Schrodinger equations of Hartree and Thomas-Fermi methods and of nuclear hydrodynamics are (for two- and three-body Skyrme forces) equivalent to each other having the same nonlinearities,\( \sim \psi ^3 + \psi ^5 \). At the level of two-body forces the same holds for their relation to the cubic nonlinear Schrodinger equations arising from the application of inverse methods to bound-state problems. It is speculated that this nonlinearity,\( \sim \psi ^3 \), is characteristic for nuclear physics.

Time dependent many body potential scattering and quantum well tunneling currents

Scattering is analyzed nonperturbatively for time dependent one dimensional independent particle models representing quantum well structures in applied electric fields. A compact electron tunneling current expression is given from which the Tsu-Esaki result appears as the first term in a Neumann or Fredholm series expansion. Results are recast in terms of S-matrices obtainable from the single particle Schrodinger equation. Generalized unitarity relations are presented for the S-matrices.

Triton and alpha-particle cluster states in43Sc

A cluster-model calculation of the properties of triton and alpha-particle cluster states in 43Sc is presented. For the triton cluster states, the 40Ca core is assumed to be in its ground state. An effective, local, cluster-core potential containing central and spin-orbit terms, obtained from a simple folding procedure, is used to generate the cluster-state energies and wavefunctions from a single-particle Schrodinger equation. For the alpha-particle cluster states, the 39K core is assumed to be in its ground or first excited state, which is treated as a d3/2 or s1/2 hole in an inert 40Ca core. In addition to a central folded potential, obtained in the same way as for the triton cluster states, a tensor force between cluster and core is calculated microscopically, using an alpha-nucleon scattering potential. This produces a significant splitting of the ground-state band of alpha-particle cluster states into quadruplets, and also mixes members of the bands based on the two different core states. In both triton and alpha-particle cases an orthogonality condition is used to reject redundant states. A comparison of the theoretically calculated spectra and electromagnetic properties of these cluster states with the equivalent measured quantities of some likely experimental counterparts casts doubt on the existence of the triton cluster states. There is good agreement for the alpha-particle cluster states, which appear to be related to similar states in 40Ca and 44Ti, and contain some members with high spins.

Bivariational methods applied to Schrodinger's and Dirac's equations

A class of bivariational functionals is derived whose stationary points are pairs of solutions of the single-particle Schrodinger equation (or Dirac equation, respectively) subject to so-called 'complementary boundary conditions'. The formulation of the boundary value problem is sufficiently general to include matching conditions and Bloch conditions as well as scattering conditions. It is shown how bivariational translational techniques may be applied to problems with three- and two-dimensional translational symmetry (calculation of complex band structures and propagation matrices, scattering problems).

The dynamics of potential-bags and tunneling

Within a simple one-dimensional nonlinear field-theoretical model based on the single-particle Schrodinger equation with the mean fieldU, the dynamics of colliding potential-bags and of quantum mechanical tunneling are studied.

The effective quasiparticle potential in molecules and solids

Using many-body Green function theory it can be shown that there is a single-particle Schrodinger equation containing an effective potential, whose eigenvalues give the quasiparticle excitations, a...

On the theory of the work function

Starting from a definition of the work function in terms of total energies of the electron system it is shown that this work function can be obtained from the single-particle Schrodinger equations in the density-functional formalism in the way suggested by the Sommerfeld model. It is also shown that the change of the equilibrium ion positions accompanying the ionization of the crystal has no influence on the work function. A comparison is made with Koopmans' theorem. The use of several potentials for computations of the work function is critically investigated.

On the single-particle Schrodinger fluid

The coherent state basis is employed to obtain, in the semiclassical limit, the Euler equation of hydrodynamics corresponding to the single-particle Schrodinger equation. This may also correspond to a collective vibrational mode of the system. By introducing unobserved modes, the concept of viscosity (and the Navier-Stokes equation) is extracted. The motivation for the study stems from the physics that underlies the increasing use of macro-concepts and of hydrodynamical ideas in nuclear physics, particularly in the description of heavy-ion reactions and of fission and fusion processes.