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Journal of Physics B Atomic Molecular and Optical Physics

Stark mixing of degenerate states

Yukap Hahn, Predrag Krstić

Published Jun 28, 1993• 6 citations
Publisher

Abstract

The initial value problem and the field dressing associated with the time-dependent Stark effect are treated, which are of importance in applications. An exact solution of time-dependent field mixing of degenerate states as an initial value problem is obtained in the restricted subspace of a given principal quantum number of hydrogenic ions, with the field-dependent phases explicitly included. The field mixing in terms of the transformation from spherical to parabolic representation is clarified. The distinction between these two problems is stressed and previous work on the field-perturbed dielectronic recombination is re-examined.

DOI
10.1088/0953-4075/26/12/003

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The initial value problem and the field dressing associated with the time-dependent Stark effect are treated, which are of importance in applications. An exact solution of time-dependent field mixing of degenerate states as an initial value problem is obtained in the restricted subspace of a given principal quantum number of hydrogenic ions, with the field-dependent phases explicitly included. The field mixing in terms of the transformation from spherical to parabolic representation is clarified. The distinction between these two problems is stressed and previous work on the field-perturbed dielectronic recombination is re-examined.
Hydrogenic spectra A good starting point is photoexcitation from the ground state of H. The problem naturally divides itself into two regimes: below the energy of classical ionization limit, where the states are for all practical purposes stable against ionization, and above it where the spectrum is continuous. As an example, we consider first the excitation of the n = 15 Stark states from the ground state in a field too low to cause significant ionization of n = 15 states. From Chapter 6 we know the energies of the Stark states, and we now wish to calculate the relative intensities of the transitions to these levels. One approach is to calculate them in parabolic coordinates. This approach is an efficient way to proceed for the excitation of H; however, it is not easily generalized to other atoms. Another, which we adopt here, is to express the n = 15 nn 1 n 2 m Stark states in terms of their nℓm components using Eqs. (6.18) or (6.19) and express the transition dipole moments in terms of the more familiar spherical n ℓ m states. In the excitation of the Stark states of principal quantum number n from the ground state only p state components are accessible via dipole transitions, so the relative intensities for light polarized parallel and perpendicular to the static field, π and σ polarizations, are proportional to the squared transformation coefficients |〈 nn 1 n 2 m | n ℓ m 〉| 2 from the nn 1 n 2 m parabolic states to the n ℓ m states for ℓ = 1 and m = 0 and 1. In Fig. 8.1 we show the relative intensities by means of the squared transformation coefficients |〈15 n 1 n 2 m |15 p m 〉| 2 for m = 0 and 1.
The response of a hydrogen atom to a perturbative electric field is presented from a classical viewpoint. We find classical energy shifts in the hydrogen atom in situations that correspond to both the linear and quadratic Stark effects. We then compare these energy shifts to the exact quantum-mechanical values for the Stark effect. We find that the linear Stark effect can be accurately reproduced classically (given the correct initial conditions). In contrast, quantum-mechanical results for the quadratic Stark effect cannot be accurately reproduced classically, except in the correspondence principle limit of large principal quantum numbers n.
We present calculations of the photoionization spectrum of excited hydrogen using \ensuremath{\pi} polarization in the presence of a strong electric field as a function of the spherical (applicable to complex atoms) and parabolic quantum numbers of the excited state. We also present corresponding measurements of the photoionization yield from the individual parabolic states of n=2. Both the calculations and the measurements show an enhancement of the depth of the so-called ``Stark-induced modulation'' in the region E\ensuremath{\ge}0 when the initial excited state is a pure ml{m}_{l}ml​=0 blue state, and disappear almost completely when the initial state is a pure ml{m}_{l}ml​=0 red state. These results are understood using arguments based on the fact that the charge distribution of the Stark-induced states is tremendously extended up field. Because of the excellent signal to noise ratio of the enhanced modulations, we were able to measure the field dependence of the spacings with sufficient accuracy to confirm the (3/4) power law and rule out the recently suggested (2/3) power law.
General equations are presented for determining the field-dependent variation of the wavefunctions, matrix elements and intensities of dipole transitions between Stark levels of a hydrogen atom in a field range where the fine-structure effects may be neglected. The Sturm-series resolution for the reduced Coulomb-Green function is used to derive analytical expressions for the first-order corrections in terms of parabolic quantum numbers of initial and final states. Simple forms of these expressions as functions of the upper-state quantum numbers are presented explicitly for the Lyman and Balmer series. The selective action of the dc electric field on the intensity of Stark components of hydrogen lines is revealed in numerical calculations for both and types of emitted radiation. This selectivity may cause, in particular, enhancement of the high-frequency and damping of the low-frequency Stark -lines of the Lyman series.
This chapter presents approximation methods in quantum mechanics, focusing on both time-independent and time-dependent perturbation theory. The time-independent formulation is developed up to second-order corrections for both nondegenerate and degenerate systems. Applications include the analysis of the Zeeman and Stark effects, where energy level shifts due to magnetic and electric fields are calculated. The time-dependent perturbation theory is introduced, with harmonic perturbations used to derive the first-order transition probabilities and to establish Fermi's golden rule. As an application, the ionization probability of a hydrogen atom under a linearly polarized electromagnetic wave is computed in detail. The treatment is extended to include the case of an unpolarized electromagnetic field. Metadata/Keywords: perturbation theory, Zeeman effect, Stark effect, time-dependent perturbation, harmonic perturbation, Fermi's golden rule, hydrogen ionization
The profile of a hydrogen emission line is investigated on the assumption that the hydrogen atom is perturbed by passing ions. Provided that the ions do not approach to within ten times the average atomic radius the electric field of the ion may be assumed homogeneous and the matrix elements calculated for the usual Stark effect may be used. For a collision at low velocity or with close approach, the adiabatic approximation is valid, and the profile depends solely on the total phase shift produced by the collision. The Fourier analysis of the wave train may be carried through exactly for certain values of this phase shift. Nonadiabatic transitions are examined for the lowest lines, and an exact solution given for this case also. For higher lines a modified Born approximation is used. The Weisskopf collision broadening formula is shown to hold only in certain cases, and the dispersion curve smearing function for impact broadening is shown to lead in some instances to incorrect results.