Asymptotic methods for Rydberg transitions (original) (raw)
Related papers
Journal of Quantitative Spectroscopy and Radiative Transfer, 1994
The rotational dependence of dipole moments M$" = (YuJJlfl Y,,J) in infrared transitions (tJcto'J') is considered. Analytic theoretical expressions of the Herman-Wallis coefficients for the infrared transitions are investigated by using an unconventional approach of the Rayleigh-Schrijdinger perturbation theory. The rotational factor G$" = M$'/Mg is expressed, in terms of the Bouanich-Herman-Wallis coefficients, by G,,.(m) = 1 + C'm+D'm2+E'm'+... with m = [J'(J' + 1)-J(J + 1)]/2 and J'-J = f 1. The coefficients C', D', and E' are given in terms of the intergrals (@,lfl@,,) and (O,j@,,) where @ stands for the pure-vibrational wavefunction Y', or for one of its successive perturbative corrections go, 9",. .. These expressions are valid for any potential, whether empirical or of the RKR type, and for any operatorf of the form x = (rr,)/r,, x', exp(ax) and others. Numerical application of the present formulation in terms of the Dunham potential in the ground state of CO (for the transitions v = 0, 10, 20 and v'-v < 4) and the ground state of HCI (for the transitions v = 0 and Av < 7) shows the good accuracy.
Regardless of the theory used for understanding atomic construction, equations that define atomic spectra must include as a derivative the Rydberg equation describing hydrogen and other single-electron ions. The following work product is evidence that defining all elemental spectra is achievable by merely changing some assumptions. The first assumption is that using integers in calculating spectra is incorrect. The second is that spin values for electrons in atoms begin at a ground state level and increase in spin-1 increments due to adding photons of spin-1. The third assumption is that photons are dual spin-1/2 composites. Using spin-1/2 as a basis produces a set of Rydberg-style equations providing spectra for all elements and their many ionic forms. The presented examples, hydrogen through neon, plus phosphorus, a spin-1/2 nucleus, reveal the equation behavior for elements of varying character. Unfortunately, determining which spectral lines dominate in multiple electron atoms is not straightforward without a solid theoretical underpinning. Current theory is lacking or it would be able to provide a similar solution. The term ‘preliminary’ in the title indicates that more than one equation solution exists for multi-electron ions. The equation sets provided represent the most logical and reasonable solutions. Although dominant line prediction is not yet forthcoming, an explanation for photon production using a combination of equation components is proffered. The enclosed information should lead to a full explanation once solutions for all elements are determined and examined in detail and theory further develops.
2012
Symbolic-numeric solving of the boundary value problem for the Schrödinger equation in cylindrical coordinates is given. This problem describes the impurity states of a quantum wire or a hydrogen-like atom in a strong homogeneous magnetic field. It is solved by applying the Kantorovich method that reduces the problem to the boundaryvalue problem for a set of ordinary differential equations with respect to the longitudinal variables. The effective potentials of these equations are given by integrals over the transverse variable. The integrands are products of the transverse basis functions depending on the longitudinal variable as a parameter and their first derivatives. To solve the problem at high magnetic quantum numbers |m| and study its solutions we present an algorithm implemented in Maple that allows to obtain analytic expressions for the effective potentials and for the transverse dipole moment matrix elements. The efficiency and accuracy of the derived algorithm and that of Kantorovich numerical scheme are confirmed by calculating eigenenergies and eigenfunctions, dipole moments and decay rates of low-excited Rydberg states at high |m| ∼ 200 of a hydrogen atom in the laboratory homogeneous magnetic field γ ∼ 2.35 × 10 −5 (B ∼ 6T ).
A New Formulation for the Herman-Wallis Coefficients for Infrared Transitions of A Diatomic Molecule
Internet Electronic Journal of Molecular Design, 2006
Motivation. The problem of the radial matrix elements in the infrared transition vJ v'J' of a diatomic molecule is considered. By using a new expansion in the perturbation theory of the eigenvalue and the eigenfunction of the two considered states in terms of the running number m we derived analytical expressions for the Herman-Wallis coefficients of the rotational factor in the rovibrational matrix elements. The numerical application to the ground states of the molecule HCl shows that the present formulation provides a simple and accurate method for the calculation of the Herman-Wallis coefficients, even for the high order coefficients, without any restriction on the potential function, the operator f(r) and the vibrational levels v and v'. Method. The most important methods used in this investigation are the Rayleigh-Schrödinger perturbation theory and the canonical functions approach. Results. The main results reported in the paper are the determination of the Herman-Wallis coefficients. Conclusions. The method used for the determination of the Herman-Wallis coefficients in this work allows the calculation of these coefficients for any type of potential function and to any order of correction in the perturbation theory. Keywords. Herman-Wallis coefficients for infrared transitions.
Computer Physics Communications, 2008
A FORTRAN 77 program is presented which calculates with the relative machine precision potential curves and matrix elements of the coupled adiabatic radial equations for a hydrogen-like atom in a homogeneous magnetic field. The potential curves are eigenvalues corresponding to the angular oblate spheroidal functions that compose adiabatic basis which depends on the radial variable as a parameter. The matrix elements of radial coupling are integrals in angular variables of the following two types: product of angular functions and the first derivative of angular functions in parameter, and product of the first derivatives of angular functions in parameter, respectively. The program calculates also the angular part of the dipole transition matrix elements (in the length form) expressed as integrals in angular variables involving product of a dipole operator and angular functions. Moreover, the program calculates asymptotic regular and irregular matrix solutions of the coupled adiabatic radial equations at the end of interval in radial variable needed for solving a multi-channel scattering problem by the generalized R-matrix method. Potential curves and radial matrix elements computed by the POTHMF program can be used for solving the bound state and multichannel scattering problems. As a test desk, the program is applied to the calculation of the energy values, a short-range reaction matrix and corresponding wave functions with the help of the KANTBP program. Benchmark calculations for the known photoionization cross-sections are presented.
Computer Physics Communications, 1999
Analytical and numerical procedures for the calculation of atomic transition matrix elements in the interaction with linearly and circularly polarized laser light are presented. The laser—atom interaction is treated beyond the dipole approximation. The procedures are derived explicitly for hydrogen but may be readily generalized to other one-electron systems with, e.g., quantum defect type radial wavefunctions. The numerical procedure is programmed in Matlab and tested for main quantum numbers up to n = 30. The results clearly indicate when the dipole approximation breaks down in connection with laser excitation of angular Rydberg wavepackets.
Consistent description of Klein-Gordon dipole matrix elements
Journal of Physics B: Atomic, Molecular and Optical Physics, 1998
Relativistic semiclassical radial dipole matrix elements are derived analytically in the length and velocity forms using Klein-Gordon wavefunctions. In contrast to our previous works in which KGWKB matrix elements were derived only in the length form, neglecting the angular displacement of the relativistic trajectory, we have obtained these matrix elements in both length and velocity forms without the above-mentioned approximation. In the latter form, an effective velocity operator has been used. The present approaches lead to two new intermediate states which are discussed. Detailed numerical results obtained through a systematic study of lithium, sodium and copper isoelectronic sequences, starting with the nonrelativistic WKB approaches, enable us to draw inferences as to the improvements and also the limitations of the new relativistic formulae. matrix element expression depends on the gauge in which the electromagnetic potentials are written. One particular gauge called the length gauge leads to the length-form oscillator strengths (f L -values) while another gauge, i.e. the velocity or Coulomb gauge leads to velocity-form oscillator strengths (f V -values). Note that the equivalence (or gauge invariance) of matrix element forms mentioned above also holds when the Hamiltonian used to study the problem is not exact (for detailed information see ). However, this equivalence is broken when approximate wavefunctions are used, since in this case these forms yield different results. The immediate question which arises is which of them is best to use for a particular approximate calculation. This question is of great importance since exact analytical wavefunctions are available for only a few problems in physics, so generally one has to use approximate wavefunctions.
Analytic matrix elements of the dipole moment and Herman-Wallis coefficients
Journal of Mathematical Chemistry, 1994
Perturbation theory proves to be a powerful approach to obtain in analytic form both vibration-rotational energies and matrix elements of the dipole moment of diatomic molecules in terms of the expansion parameter -y = 2Be~we, Bc and wc being, respectively, the equilibrium rotational and harmonic vibrational spectral parameters. A systematic and efficient algorithm has been developed to execute such calculations with sufficient accuracy for most physical applications when the potential-energy function is accurately represented in the Dunham form. The method also provides analytic expressions of the Herman-Wallis coefficients C~ and D~ for the vibration-rotational overtone bands J~-v for diatomic molecules in ly~ electronic states.