Integral Representation of the Dispersion Formula for a Hydrogen Atom in the Ground State

M. I. Chibisov

Geoffrey L. Sewell

Rajalakshmi Radhakrishnan, R. B. Thayyullathil

We describe a unified approach for the determination of ac-Stark correction and Kramers-Heisenberg dispersion formula. In both cases the contribution from infinite intermediate summation appearing in the expression for the corresponding matrix elements are evaluated exactly in the dipole approximation for the ground state of hydrogen atom using a variation of the Dalgarno-Lewis method. The analytical expressions obtained can be efficiently used for the numerical evaluation of matrix element for all values of incident photon energy and comparison is made with results obtained by different methods.

Boris Podolsky, Vladimir Rojansky

It has been shown by Schr\"odinger and O. Klein that the Kramers-Heisenberg formulas for dispersion and incoherent scattering can both be obtained by an extension of Schr\"odinger's method for treating dispersion, the terms in the Hamiltonian involving squares of the potentials being neglected. The resulting formulas agree with those of Dirac. They are, however, inconvenient for calculations, as they contain summations with respect to all energy levels combining with the pair of levels under consideration, and thus imply complicated integrations when a continuous spectrum is present. This paper treats the problem of a hydrogen-like atom, acted on by light the wave-length of which is large compared to the size of the atom and the frequency of which is not too near a resonance frequency, and develops a method which obviates the necessity of integrating over the continuous range. It is an extension of the method used by one of us in treating the dispersion by atomic hydrogen. It is applied in detail to the first two levels of H. Formulas are derived for the intensities of the Smekal-Raman lines with a shift corresponding to the first absorption line.

Frank Steiner