Analytical Wave Functions Research Articles

Two-colour three-photon transitions in a hydrogen atom embedded in Debye plasmas

We sketch two-colour three-photon bound–bound transitions in hydrogen atoms embedded in weakly coupled plasmas. In the present context, analytical wavefunctions are used for initial and final states along with the pseudostate summation technique for intermediate states. The analytical wavefunctions are obtained from the modified wavefunctions for the screening Coulomb potential (Debye model) using the Ritz variation method. The resonance enhancement frequencies are varied with the Debye length. The results reveal plasma effects on the three-photon bound–bound transition amplitudes for the plasma-imbedded hydrogen atom.

IMPROVED ARBITRARY l-STATE SOLUTIONS OF THE HULTHÉN POTENTIAL

We developed a new and simple approximation scheme for centrifugal term. Using the new approximate formula for 1/r2 we derived approximately analytical solutions to the radial Schrödinger equation of the Hulthén potential with arbitrary l-states. Normalized analytical wave-functions are also obtained. Some energy eigenvalues are numerically calculated and compared with those obtained by C. S. Jia et al. and other methods such as the asymptotic iteration, the supersymmetry, the numerical integration methods and a Mathematica program, schroedinger, by W. Lucha and F. F. Schöberl.

Two-dimensional dipolar scattering

We characterize the long-range dipolar scattering in two dimensions. We use the analytic zero energy wave function including the dipolar interaction; this solution yields universal dipolar scattering properties in the threshold regime. We also study the semiclassical dipolar scattering and find universal dipolar scattering for this energy regime. For both energy regimes, we discuss the validity of the universality and give physical examples of the scattering.

Interaction-tuned compressible-to-incompressible phase transitions in quantum Hall systems

We analyze transitions between quantum Hall ground states at prominent filling factors $\ensuremath{\nu}$ in the spherical geometry by tuning the width parameter of the Zhang-Das Sarma interaction potential. We find that incompressible ground states evolve adiabatically under this tuning, whereas the compressible ones are driven through a first-order phase transition. Overlap calculations show that the resulting phase is increasingly well described by appropriate analytic model wave functions (Laughlin, Moore-Read, Read-Rezayi). This scenario is shared by both odd $(\ensuremath{\nu}=1/3,1/5,3/5,7/3,11/5,13/5)$ and even denominator states $(\ensuremath{\nu}=1/2,1/4,5/2,9/4)$. In particular, the Fermi-liquid-like state at $\ensuremath{\nu}=1/2$ gives way, at large enough value of the width parameter, to an incompressible state identified as the Moore-Read Pfaffian on the basis of its entanglement spectrum.

Application of variational method for three-color three-photon transitions in hydrogen atom implanted in Debye plasmas

Multiphoton processes, where transparency appears, have long fascinated physicists. Plasma screening effects are investigated on three-color three-photon bound-bound transitions in hydrogen atom embedded in Debye plasmas; where photons are linearly and circularly polarized, two left circular and one right circular. All possible combinations of frequency and polarization are considered. Analytical wave functions are used for initial and final states along with the pseudostate wave functions for intermediate states. The analytical wave functions are obtained from the modified wave functions for screening Coulomb potential (Debye model) using Ritz variational method. Here, we have found three-photon transparency for lower values of Debye length. This type of phenomenon occurs due to energy level shifting in the presence of Debye plasma environments. The description of resonance enhancements and three-photon transparency is reported in the present context along with the region of resonance enhancements and three-photon transparency.

Geometric phases and squeezed quantum states of relic gravitons

In this work, we investigate the quantum effects of relic gravitons from a Schrödinger-picture point of view. By considering the gravity-wave equations in the Friedmann–Robertson–Walker cosmological background, we reduce the problem to that of a generalized time-dependent harmonic oscillator and find the corresponding exact analytic wave functions with the help of the dynamical invariant operator method. Afterward, we derive the geometric phases and squeezed quantum states for this system. We also evaluate the quantum fluctuations and the uncertainty product for each mode of the quantized field.

The scattering states of the generalized Hulthén potential with an improved new approximate scheme for the centrifugal term

This paper finds the approximate analytical scattering state solutions of the arbitrary 1-wave Schrödinger equation for the generalized Hulthén potential by taking an improved new approximate scheme for the centrifugal term. The normalized analytical radial wave functions of the l-wave Schrödinger equation for the generalized Hulthén potential are presented and the corresponding calculation formula of phase shifts is derived. Some useful figures are plotted to show the improved accuracy of the obtained results and two special cases for the standard Hulthén potential and Woods–Saxon potential are also studied briefly.

An analytic and parameter-free wavefunction for studying the stability of three-body systems

An analytic wavefunction is proposed for the ground state of general atomic three-body systems in which two light particles are negatively charged and the third (heavy) is positively charged. By construction the wavefunction (i) has the same analytical form for all systems; (ii) is parameter-free; (iii) is nodeless; (iv) satisfies all two-particle cusp conditions; and (v) yields reasonable ground state energies for several three-body systems, including the prediction of a bound state for H− , D− , T− and Mu− . Simple polynomial fits are provided for certain important subcases, allowing for a rapid estimate of the ground state energy and of the stability of three-body systems.

Some applications of atomic analytical wave functions

Calculations of certain atomic quantities were carried out, using our accurate analytical self-consistent-field (SCF) wave functions for the ground and excited states. The total weighted electric dipole oscillator strength for certain optical multiplets of Al, Cr3+, Cr, and Mn3+, and Fe3+, was calculated with the determinantal wave functions, while comparing various approximations. Orientational calculations of the electric quadrupole multiplet strength were carried out for some transitions in Cr3+, Cr, and Fe3+. Our SCF wave functions were also used for the calculation of the estimates of the electronic density at the nucleus, considered in the Mossbauer isomer shift studies.

THE APPROXIMATE ANALYTICAL SOLUTIONS OF THE KLEIN–GORDON EQUATION WITH THE SECOND PÖSCHL–TELLER LIKE POTENTIAL FOR NONZERO ANGULAR MOMENTUM

The approximate analytical bound and scattering state solutions of the arbitrary l-wave Klein–Gordon equation for the second Pöschl–Teller like potential are carried out by a new approximation to the centrifugal term. The analytical radial wave functions of the l-wave Klein–Gordon equation with the second Pöschl–Teller like potential are presented and the corresponding energy equations for bound states and phase shifts for scattering states are derived. It is well shown that the poles of S-matrix in the complex energy plane correspond to bound states for real poles and scattering states for complex poles in the lower half of the energy plane. Some numerical results are calculated to show the improved accuracy of our results and the special case for s-wave is also studied briefly.

11Li structure from experimental data

An analytical wave function for the 11Li ground state is constructed in a three-body Li 9 + n + n approach. Its parameters are fixed from the fit to the set of most accurate experimental data and tested on all other available data. The obtained 11Li ground state wave function allows to describe all considered experimental data within their error bars simultaneously. The main contributions to the 11Li ground state wave function are found to come from ( s 1 / 2 ) 2 (37%) and ( p 1 / 2 ) 2 (47%) components.

Possible Verification of Tilted Anisotropic Dirac Cone in α-(BEDT-TTF)2I3 Using Interlayer Magnetoresistance

It is proposed that the presence of a tilted and anisotropic Dirac cone can be verified using the interlayer magnetoresistance in the layered Dirac fermion system, which is realized in quasi-two-dimensional organic compound α-(BEDT-TTF) 2 I 3 . Theoretical formula for the interlayer magnetoresistance is derived using the analytic Landau level wave functions and assuming local tunneling of electrons. It is shown that the resistivity as a function of the azimuthal angle of the magnetic field takes the maximum in the direction of the tilt if anisotropy of the Fermi velocity of the Dirac cone is small. The procedure is described to determine the parameters of the tilt and anisotropy.

The scattering states of the l-wave Schrödinger equation with the second Pöschl–Teller-like potential

The approximately analytical scattering state solutions of the l-wave Schrödinger equation with the second Pöschl–Teller-like potential are carried out by taking a new approximation scheme to the centrifugal term. The normalized analytical radial wave functions are presented and the corresponding calculation formula of phase shifts is derived. The energy levels of the bound states are also obtained from the poles of the scattering amplitude. Some numerical results are calculated to show the improved accuracy of our results and the s-wave case is studied briefly.

The Relativistic Scattering States of the Hulthén Potential with an Improved New Approximate Scheme to the Centrifugal Term

The approximately analytical scattering state solutions of the l-wave Klein-Gordon equation with the unequal scalar and vector Hulthen potentials are carried out by an improved new approximate scheme to the centrifugal term. The normalized analytical radial wave functions of the l-wave Klein-Gordon equation with the mixed Hulthen potentials are presented and the corresponding calculation formula of phase shifts is derived. It is well shown that the energy levels of the continuum states reduce to those of the bound states at the poles of the scattering amplitude. Some useful figures are plotted to show the improved accuracy of our results and two special cases for s-wave (l=0) and for l=0 and equal scalar and vector Hulthen potentials are also studied briefly.

THE RELATIVISTIC BOUND AND SCATTERING STATES OF THE ECKART POTENTIAL WITH A PROPER NEW APPROXIMATE SCHEME FOR THE CENTRIFUGAL TERM

The approximately analytical bound and scattering state solutions of the arbitrary l wave Klein–Gordon equation for mixed Eckart potentials are obtained through a proper new approximation to the centrifugal term. The normalized analytical radial wave functions of the l wave Klein–Gordon equation with the mixed Eckart potentials are presented and the corresponding energy equations for bound states and phase shifts for scattering states are derived. It is shown that the energy levels of the continuum states reduce to those of the bound states at the poles of the scattering amplitude. Two special cases — for the s wave and for l = 0 and β = 0 — are also studied, briefly.

The relativistic bound and scattering states of the Manning-Rosen potential with an improved new approximate scheme to the centrifugal term

Abstract The approximately analytical bound and scattering state solutions of the arbitrary l-wave Klein-Gordon equation for the mixed Manning-Rosen potentials are carried out by an improved new approximation to the centrifugal term. The normalized analytical radial wave functions of the l-wave Klein-Gordon equation with the mixed Manning-Rosen potentials are presented and the corresponding energy equations for bound states and phase shifts for scattering states are derived. It is shown that the energy levels of the continuum states, reduce to the bound states of those at the poles of the scattering amplitude. Some useful figures are plotted to show the improved accuracy of our results and the special case for wave is studied briefly.

Approximate description of decaying quasi-stationary state

A simple model of the decaying quantum state composed of a one-dimensional electron moving in the potential with the quantum dot separated from the innite half-axis by the ‐-function is studied. The quasibound quantum dot state properties calculated by means of approximate methods (complex energy technique and local density of states method) are compared with results obtained with the aid of an exact analytical non-stationary wave function. The physical meaning of the emitted electron wave function peculiarities is discussed.

Multipole expansions: Magnetic and electric fields generated by electrons bound in spin–orbit eigenstates

A formalism is presented that enables the calculation of atomic charge and current densities in Russel-Saunders states of free atoms and ions in terms of scalar and vector spherical harmonics. The electric and magnetic fields generated by the multipole components of charge and current distributions are easily calculated. They are found to have the same multipolarity as their generating sources and Maxwell's equations are satisfied component by component. Calculations of these fields for hydrogenlike atoms and specific transition-metal and rare-earth ions are presented using realistic analytic radial wave functions. (C) 2008 Elsevier B.V. All rights reserved.

The relativistic bound states of the hyperbolical potential with the centrifugal term

The approximately analytical bound state solutions of the arbitrary l-wave Klein–Gordon and arbitrary k-state Dirac equations for the mixed hyperbolical potentials are carried out by taking a proper approximate expansion of the centrifugal term. The analytical radial wavefunctions of the l-wave Klein–Gordon and k-state Dirac equations for the mixed hyperbolical potentials are presented and the corresponding energy equations are derived. Two special cases of the Klein–Gordon equation are discussed briefly.

Solution to the Schrödinger Equation for the Time-Dependent Potential

In this work, the Schrodinger equation with the time-dependent potential \(V(z,\hat{p},t)=g_{1}(t)z+g_{2}(t)\hat{p}+g_{3}(t)\) has been solved by the method of time-space transformation in 1+1 dimensions. The corresponding analytical wave function to Schrodinger equation is obtained. In addition, the discussion of solutions to particular cases has been made.