D P Dewangan

First Born amplitude for transitions from a circular state to a state of large (l, m)

The use of cylindrical polar coordinates instead of the conventional spherical polar coordinates enables us to derive compact expressions of the first Born amplitude for some selected sets of transitions from an arbitrary initial circular state to a final state of large (lf, mf). The formulae for and transitions are expressed in terms of the Jacobi polynomials which serve as suitable starting points for constructing complete solutions over the bound energy levels of hydrogen-like atoms. The formulae for and transitions are in simple algebraic forms and are directly applicable to all possible values of ni and nf. It emerges that the method can be extended to evaluate the first Born amplitude for many other transitions involving states of large (l, m).

An accurate quantum expression for radiative transition between nearby Rydberg states

The paper gives three main results as follows. (1) An accurate quantum expression of the radial matrix element for radiative dipole transition between nearby Rydberg states. Its remarkable numerical accuracy is demonstrated over a very wide range of principal and orbital angular momentum quantum numbers covering low-lying states to very high Rydberg states. (2) A simple but accurate approximation to a class of terminating hypergeometric functions whose three arguments are large. This result essentially solves the problem of extracting analytic properties and performing numerical computations of such hypergeometric functions over a large range of values of the arguments which were earlier regarded to pose difficulties. (3) A derivation of the formula of the radial dipole matrix element of the correspondence principle method starting from the corresponding quantum expression, which, to the best of our knowledge, was not previously available in the literature.

An accurate quantum expression of the z-dipole matrix element between nearby Rydberg parabolic states and the correspondence principle

We give an exact quantum formula for the z-component of the dipole matrix element between parabolic states of a hydrogen atom in terms of the Jacobi polynomials. The formula extends the range of numerical computation to larger values of the parabolic quantum numbers for which computation from the standard textbook formula, which is in terms of the hypergeometric functions, is defined. We obtain an accurate quantum expression of the z-dipole matrix element in terms of the ordinary Bessel functions for transition between nearby Rydberg parabolic states. We derive for the first time the formula of the z-dipole matrix element of the correspondence principle method directly from the quantum expression, and in the process of derivation, clarify the nature of classical–quantum correspondence. The expressions obtained in this work solve the problem of computation of the z-dipole matrix element of hydrogen to a large extent.

An accurate quantum expression for radiative transitions between the Stark levels of nearby Rydberg states

The paper presents a new alternative exact quantum expression of the x-component of the dipole matrix element between the Stark states of a hydrogen atom in terms of the Jacobi polynomials by transforming the hypergeometric functions appearing in the standard quantum formula. The new quantum formula readily leads to analytic study and numerical computation for such large values of the parabolic quantum numbers for which difficulties had earlier been encountered. The paper goes on to derive an approximate but simple quantum formula of the dipole matrix element in terms of the ordinary Bessel functions and demonstrates its remarkable accuracy for transitions ranging from that between the Stark levels of the lowest lying states to that between the Stark levels of nearby Rydberg states. The formula enables accurate numerical computation to be performed over an extended range of large parabolic quantum numbers that had earlier defied evaluation. The expressions given in this paper in essence solve the problem of determination of analytic behaviour and numerical computation of the dipole matrix element for transitions between the Stark levels of nearby Rydberg states. The paper also presents, for the first time, a derivation of the formula of the correspondence principle method from the quantum expression without appealing to any classical or semiclassical argument, and clarifies the conditions of its applicability.

A complete solution of the first Born amplitude for the nis → nfs transition on the hydrogenic bound states

This letter reports a complete solution of the first Born amplitude (FBA) for the nis → nfs transition covering the entire set (ni, nf = [1,∞]) of the hydrogenic bound states. An exact compact expression of the nis–nfs FBA in terms of only one Jacobi polynomial is derived that serves as a basis for obtaining simple compact quantum expressions for a variety of cases, which include elastic scattering, transitions between adjacent Rydberg states, distant Rydberg states and transitions connecting any s-state to a very high Rydberg state. A general quantum formula for transitions between nearby Rydberg states is also derived, of which the expressions of the correspondence principle method previously reported in the literature are specific examples. All the formulae are derived using quantum mechanics without making any appeal to classical mechanics, even for states lying in the vicinity of the limiting point (ni, nf) →∞ of the hydrogenic bound energy levels.

Two Coulomb waves theory for direct excitation

A theoretical model of Dewangan, in which the total scattering wave function is approximated by a distorted wave containing two Coulomb wave functions, is discussed and its relation with the Brauner-Briggs-Klar model for ionization is examined. An important feature of the theory is that it includes a second Born amplitude naturally and in addition, contains, albeit approximately, both real and imaginary parts of all higher order Born terms. The theory is applied to study the 1s→2s excitation of hydrogen by electrons in the energy range 54.4 to 400eV. The differential and integral cross sections predicted by the theory are compared with the results of other theories and experimental data at 54.4eV and a good agreement is found.