WKB approach to calculating the lifetime of quasistationary states

A novel WKB approach to calculating the lifetime of quasistationary states in the potential wells of the form V(x)=P(x)-muQ(x), where P(x) is the radial part of the potential for the spherically symmetric harmonic oscillator or the hydrogen atom and Q(x) is a polynomial, is suggested. In this approach, the usual explicit procedure of the asymptotic matching of the perturbative and WKB wave functions is avoided and a simple formula for the imaginary part of the energy is found. The leading and the first correction terms for the imaginary part of the energy and the related lifetime are analytically calculated.     

Coulomb Sturmians in spheroidal coordinates and their application for diatomic molecular calculations

Coulomb Sturmians are obtained in prolate spheroidal coordinates by separation of variables in the Schrödinger equation and direct solution of the appropriate one-dimensional equations. Molecular orbitals are expressed as linear combinations of the introduced Coulomb Sturmians and some low-lying energy terms and corresponding wave functions are calculated for one-electron diatomic molecules. It is shown that similarity of the one- and two-centre orbitals in spheroidal coordinates, combined with completeness and good convergence properties of Coulomb Sturmians, substantially speeds up convergence and makes the calculated results closer to the exact ones. Application of the elaborated calculating scheme for diatomic many-electron molecules is discussed.     

Quantum model of the prolate spheroidal Thomson hydrogen atom

In a uniformly charged prolate spheroidal Thomson hydrogen atom the electron states have been investigated. It has been shown from the mathematical point of view that the problem is equivalent to a spheroidal hydrogen atom in a parabolic potential with the cylindrical symmetry. In the framework of adiabatic approximation, the energy of ground state has been calculated. Comparison with the case of uncharged spheroidal quantum dot has been made, and the analytical form of wave function of electron has been also obtained.     

Scaled Schrödinger equation and the exact wave function

We propose the scaled Schr?dinger equation and the related principles, and construct a general method of calculating the exact wave functions of atoms and molecules in analytical forms. The nuclear and electron singularity problems no longer occur. Test applications to hydrogen atom, helium atom, and hydrogen molecule are satisfactory, implying a high potentiality of the proposed method.     

n维氢原子的散射态

研究了n维氢原子的散射态性质．给出了精确的按“k/2π标度”归一化的散射态的精确解波函数及相移表达式，讨论了相移的解析性质，获得了束缚-连续跃迁矩阵元的解析计算公式．普通氢原子(n=3)散射态的有关结果作为特例包含在本文的一般结论之中． 关键词： n维氢原子 散射态 精确解 相移 束缚-连续跃迁矩阵元     

Electron-impact ionization of atomic hydrogen: dynamical variational treatment

A simple and straightforward calculating scheme is proposed for electron-impact single and multiple ionization of atoms. The method is based on the application of the Hulthén-Kohn dynamical variational principle. An effective charge seen by the scattered electron is determined for a certain type of trial wave functions mathematically in a rigorous way excluding any empirical assumptions. Validity of the elaborated approach is assessed by calculating triply differential cross section (TDCS) for electron-impact ionization of hydrogen. It is shown that, inclusion of the effective charge into the calculation reduces height of a “binary peak” in comparison with the first Born approximation result. The height of a “recoil peak” depends on the sign of the effective charge. The calculated TDCS are compared with the available experimental data and with the results of sophisticated theories and agreement is found.     

通过计算氢原子的玻尔半径,加深对量子力学的理解

通过玻尔理论基本假设、不确定关系、薛定谔方程以及波函数性质4种方法求氢原子的玻尔半径,理解了玻尔理论基本假设的半经典、半量子;不确定关系正是微观粒子波动性的必然结果;氢原子中电子遵从薛定谔方程,方程的解析即为波函数,是概率波.     

Solving ground eigenvalue and eigenfunction of spheroidal wave equation at low frequency by supersymmetric quantum mechanics method

The spheroidal wave functions are found to have extensive applications in many branches of physics and mathematics. We use the perturbation method in supersymmetric quantum mechanics to obtain the analytic ground eigenvalue and the ground eigenfunction of the angular spheroidal wave equation at low frequency in a series form. Using this approach, the numerical determinations of the ground eigenvalue and the ground eigenfunction for small complex frequencies are also obtained.     

Adiabatic representation for a hydrogen atom photoionization in a uniform magnetic field

A new effective method of calculating wave functions of discrete and continuous spectra of a hydrogen atom in a strong magnetic field is developed on the basis of the adiabatic approach to parametric eigenvalue problems in spherical coordinates. The two-dimensional spectral problem for the Schrödinger equation at a fixed magnetic quantum number and parity is reduced to a spectral parametric problem for a one-dimensional angular equation and a finite set of ordinary second-order radial differential equations. The results are in good agreement with the photoionization calculations by other authors and have a true threshold behavior.     

Acoustic radiation impedance of rectangular pistons on prolate spheroids

The self and mutual radiation impedances for rectangular piston(s) arbitrarily located on a rigid prolate spheroidal baffle are formulated. The pistons are assumed to vibrate with uniform normal velocity and the solution is expressed in terms of a modal series representation in spheroidal eigenfunctions. The prolate spheroidal wave functions are obtained using computer programs that have been recently developed to provide accurate values of the wave functions at high frequencies. Results for the normalized self and mutual radiation resistance and reactance are presented over a wide frequency range for different piston sizes and spheroid shapes.     

Photo-Ionization of Hydrogen Atom in a Circularly Polarized Standing Electromagnetic Wave

Applying time-independent non-perturbative formalism to the photo-ionization of hydrogen atom immersed in a strong circularly polarized standing electromagnetic wave, we calculate the shift of energy levels and the distortion of wave functions for the hydrogen atom, the ionization cross section induced by the standing wave, and the angular distribution of photoelectrons and obtain some interesting results.     

正交非均匀电场中氢原子的高激发态

用数学软件Mathematica研究正交非均匀电场中氢原子两个高激发态(n=4和5)的能级和波函数.讨论能级分裂,并绘制零级近似下氢原子的电子概率角分布图.     

Collision of Rydberg atom A** with atom B in the ground electronic state. Optical potential

A method for calculating the complex optical potential of slowly colliding Rydberg atom A^** and neutral atom B in the ground electronic state is suggested. The method is based on the asymptotic approach and the theory of multichannel quantum defects, which uses the formalism of renormalized Lippmann-Schwinger equations. The potential is introduced as the 〈q|V _opt|q〉 matrix element of the optical interaction operator, for which the integral equation is derived, and is calculated in the basis set of free particle wave functions |q〉. Fairly simple equations for the shift and broadening of the ionic term are obtained, and the principal characteristics of these equations are analyzed. By way of illustration, the optical potential of the Na^**(nl)+B systems, where B is a rare gas atom, is calculated.     

Modern methods for calculating photoionization and electron-impact ionization of two-electron atoms and molecules

A number of recently developed methods for calculating multifold differential cross sections for photoionization and electron impact ionization of atoms and molecules with two active electrons are reviewed. The methods are based on unique approaches to the calculation of three-body Coulomb wave functions. The exterior scaling method and the driven Schrödinger equation formalism are considered. The effectiveness of the time-dependent approaches to the scattering problem, such as the paraxial approximation and the time-dependent scaling, is demonstrated. A novel numerical method is formulated, which has been developed by the authors to solve the six-dimensional Schrödinger equation for an atom with two active electrons on the basis of the Chang-Fano transformation and the discrete variable representation. The threshold behavior manifested by the angular distributions of the two-electron photoionization of a negative hydrogen ion and a helium atom and the multifold differential cross sections for the electronimpact ionization of hydrogen and nitrogen molecules are analyzed on the basis of numerical simulations. The Wannier law for the angular distribution of double ionization is demonstrated to be incorrect even at very low energies.     

Ground-state calculations of confined hydrogen molecule H2 using variational Monte Carlo method

The variational Monte Carlo method is used to evaluate the ground-state energy of a confined hydrogen molecule H₂. Accordingly, we considered the.me case of hydrogen molecule confined by a hard prolate spheroidal cavity when the nuclear positions are clamped at the foci (on-focus case). Also, the case of off-focus nuclei in which the two nuclei are not clamped to the foci is studied. This case provides flexibility for the treatment of the molecular properties by selecting an arbitrary size and shape for the confining spheroidal box. A simple chemical analysis concerning the catalytic role of enzyme is investigated. An accurate trial wave function depending on many variational parameters is used for this purpose. The obtained results for the case of clamped foci exhibit good accuracy compared with the high precision variational data presented previously. In the case of off-focus nuclei, an improvement is obtained with respect to the most recent uncorrelated results existing in the literature.     

二维氢原子的相对论修正

求解了二维氢原子的Ｄｉｒａｃ方程,得到了精确的相对论能级与波函数,并详细讨论了非相对论极限。     

Scattering of shaped beam by an arbitrarily oriented spheroid having layers with non-confocal boundaries

An approach to express an incident shaped beam with respect to an arbitrarily oriented spheroidal particle having layers with non-confocal boundaries is presented. To overcome the difficulty of non-confocal boundary conditions connected with different spheroidal coordinate systems, a theoretical procedure is developed to deal with the non-confocal boundary conditions by virtue of a transformation for vector wave functions. The unknown coefficients of scattered and internal electromagnetic fields are determined by solving a system of linear equations derived from the boundary conditions and relations between the spheroidal vector wave functions and spherical ones. Numerical results of the normalized scattering cross section for a two-layered non-confocal prolate spheroid are evaluated. PACS 42.25.Fx; 42.25.Bs     

First order corrections to the wave function of a hydrogen atom in an external electromagnetic field: A sturmian expansion

The linear response of a hydrogen atom in an external electromagnetic field is considered. Expressions for the lowest order corrections to discrete and continuum wave functions are derived with the help of a sturmian expansion of the Coulomb Green function.     

Multiphoton ionization of the hydrogen atom by a circularly polarized electromagnetic field

This paper examines the multiphoton ionization of the ground state of the hydrogen atom in the field of a circularly polarized intense electromagnetic wave. To describe the states of photoelectrons, quasiclassical wave functions are introduced that partially allow for the effect of an intense electromagnetic wave and that of the Coulomb potential. Expressions are derived for the angular and energy distributions of photoelectrons with energies much lower than the ionization potential of an unperturbed atom. It is found that, due to allowance for the Coulomb potential in the wave function of the final electron states, the transition probability near the ionization threshold tends to a finite value. In addition, the well-known selection rules for multiphoton transitions in a circularly polarized electromagnetic field are derived in a natural way. Finally, the results are compared with those obtained in the Keldysh-Faisal-Reiss approximation. Zh. éksp. Teor. Fiz. 116, 807–820 (September 1999)     

Study of correlation effects during reemission of ultrashort electromagnetic pulses by the helium atom

The calculation of the energy of the ground state of atoms is the criterion for the correctness of compact analytical atomic wave functions. However, such a test is in fact static. We studied the degree of taking into account electron correlations in various simple analytical wave functions in the dynamic process of reemission by the helium atom of ultrashort pulses of an electromagnetic field. Direct guidelines are given on the use of particular analytical wave functions, which can be helpful in simple calculations and estimations of dynamic processes.