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Physical Review A

Derivation of the dipole approximation from the exact transition probabilities for hydrogen atoms

H. E. Moses

Published Jun 1, 1975• 2 citations
Publisher

Abstract

The usual dipole approximation for the transition probabilities for hydrogen atoms is derived from the exact transition probabilities by considering $j=1$ photons only, neglecting retardation, and using an identity derived from the wave equation for the radial wave function for the atom.

DOI
10.1103/physreva.11.2196

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Nepal Journal of Science and Technology•Jul 23, 2012•1 citation
The usual dipole approximation for the transition probabilities for hydrogen atoms is derived from the exact transition probabilities by considering j=1j=1j=1 photons only, neglecting retardation, and using an identity derived from the wave equation for the radial wave function for the atom.
The general theory of multiphoton ionization of hydrogen atoms by simultaneous absorption of several photons is discussed. In the nonrelativistic dipole approximation, the calculations lead to a recursion relationship concerning angular functions and involve radial matrix elements containing many summations over intermediate states. A general method for exactly evaluating several sums is described in detail and applied to the photo-ionization of atomic hydrogen in the ground state. Numerical results are presented as dispersion curves of the transition rate $\frac{{\ensuremath{\sigma}}_{n}}{{I}^{N\ensuremath{-}1}}$ for a range of wavelengths from 1000 up to 7200 \AA{}.
This chapter begins with the simple solution of the Schrödinger equation for hydrogen atom for which the energy value has been derived. The Bohr radius of H atom has also been computed. After this, this chapter provides the generalised solution of the Schrödinger equation for H atom/hydrogen-like species in which F equation, R equation, and T equations have been derived in a simple and lucid way. This chapter also deals with the Laguerre differential equation, Laguerre polynomials, Rodrigues formula for the Laguerre polynomials, Laguerre associated equation and its solution, associated Laguerre polynomials, Rodrigues formula for the associated Laguerre polynomials, and solution of the radial equation. It also provides normalisation of the radial wave function and complete wave function; for the hydrogen atom, hydrogenic atomic orbital; electron density of 1 s electron; mathematical background of shape of orbitals; the most probable distance of electron from nucleus of hydrogen atom, and average distance of electron from the nucleus of hydrogen atom. The end of this chapter describes the references and adequate computational problems (solved problems and question on concepts).
Within the framework of nonrelativistic quantum mechanics, the analysis of the two-photon absorption by atomic hydrogen is generalized to the case of two incident beams with arbitrary direction and polarization (including circular polarization). In the dipole approximation, the second-order matrix elements responsible for two-photon absorption are transformed into a finite sum consisting of the product of a radial part and an angular part. Exact calculation of the angular part predicts that, for the s–s transition only, transparency is obtained regardless of the frequencies of the two beams with perpendicular polarizations. In addition, circularly polarized light is found to be more efficient (up to a factor of 1.5) for the s–d two-photon transition. The radial parts of the matrix elements are accurately evaluated by using the Coulomb Green’s function technique and an implicit technique of Dalgarno and Lewis. Their calculation indicates that “zeros” exist in the two-photon absorption spectrum, thereby predicting that absorption is not possible at certain frequencies. It should be noted that in the calculated spectrum, near or at resonance points, the linewidths of the intermediate levels must be included in order to obtain accurate results.
In the framework of the quantum model for hydrogen atom, we use the wave functions provided by the solutions of the Schrödinger equation with Coulomb potential. Considering the radial and angular densities of the localization probability, we identify the representative quantum states that play a high-probability role in transitions for H-I spectra. Besides that, we obtain some other complementary results. So, we present the numerical tables for the radial distributions, the electron density functions and some observables like θ nodal angles, nodal surfaces, along with the ordered sequences of the average radii in the sub-shells associated with this n: ⟨r⟩4f = 18a0 < ⟨r⟩4d = 21a0 < ⟨r⟩4p = 23a0 < ⟨r⟩4s = 24a0 (a0 - the first Bohr radius). Further, we have added the significant scientific 3D-visualization of the orbitals shapes and describe them via the nodal values for θ angles.
Within the framework of second order time dependent perturbation theory and non relativistic dipole approximation, the exact expressions of the radial transition amplitude from 3d, 3p and 3s excited states, were obtained from the integral representation of the Green Coulomb function. Below threshold and above one photon ionization threshold, it can be expressed as an analytic combination of three Appell hypergeometric functions of first order, one of which is degenerated; while at threshold, it can be obtained in terms of an expansion of Gauss confluent hypergeometric functions. The angular coefficients are numerically evaluated and analysed for some available laser wavelengths.