Stark Effect In Hydrogen Research Articles

Glory oscillations in photoionization microscopy: Connection with electron dynamics and Stark spectral structures in the continuum

In classical scattering theory, the term “glory scattering” implies the divergence of the classical differential cross section that occurs as soon as the deflection function goes through zero for a nonzero value of the impact parameter. This critical effect also occurs in slow photoelectron imaging where near-threshold atomic photoionization is performed in the presence of an external static electric field. In this case, glory scattering manifests itself by the appearance of an intense peak at the center of the photoelectron momentum distribution. In the present work we examine the magnitude variation of this central peak as a function of electron energy. We experimentally study near-threshold two-photon ionization of ground state magnesium atoms, below as well as above the field-free ionization limit. It is found that, apart from its behavior of classical origin, the glory signal additionally exhibits strong oscillations and beating effects over the full spectral range of the recordings. Of particular interest are its oscillations above the zero-field limit, many aspects of which are expected to be independent of the atomic target. Our results are analyzed with the help of classical, semiclassical, and quantum mechanical calculations devoted to the hydrogenic Stark effect. It is theoretically found that these continuum glory oscillations are related to the resonantlike Stark structures appearing under certain conditions in the total photoionization cross section and implying energy quantization in the continuum. The striking outcome of the present study, however, is that both theory and experiment clearly support the connection between the energy- and static field-dependent periodicity of glory oscillations with the classical dynamics of electron motion. In particular, it is shown that the Fourier transform of the glory signal provides information on the differences between the origin-to-detector times of flight corresponding to specific pairs of classical electron trajectories.

Highly accurate calculation of the resonances in the Stark effect in hydrogen

We obtained accurate resonances for the Stark effect in hydrogen by means of three independent methods. Two of them are based on complex rotation of the coordinates and diagonalization of the Hamiltonian matrix (CRLM and CRCH). The other one is based on the Riccati equations for the logarithmic derivatives of factors of the wavefunction (RPM). The latter approach enabled us to obtain the most accurate results and extremely sharp resonances.

Nonperturbative Quantum Physics from Low-Order Perturbation Theory.

The Stark effect in hydrogen and the cubic anharmonic oscillator furnish examples of quantum systems where the perturbation results in a certain ionization probability by tunneling processes. Accordingly, the perturbed ground-state energy is shifted and broadened, thus acquiring an imaginary part which is considered to be a paradigm of nonperturbative behavior. Here we demonstrate how the low order coefficients of a divergent perturbation series can be used to obtain excellent approximations to both real and imaginary parts of the perturbed ground state eigenenergy. The key is to use analytic continuation functions with a built-in singularity structure within the complex plane of the coupling constant, which is tailored by means of Bender-Wu dispersion relations. In the examples discussed the analytic continuation functions are Gauss hypergeometric functions, which take as input fourth order perturbation theory and return excellent approximations to the complex perturbed eigenvalue. These functions are Borel consistent and dramatically outperform widely used Padé and Borel-Padé approaches, even for rather large values of the coupling constant.

Comment on “Stark effect in neutral hydrogen by direct integration of the Hamiltonian in parabolic coordinates”

We show that the resonances of the Stark effect in hydrogen reported by Fern\'andez-Menchero and Summers [Phys. Rev. A 88, 022509 (2013)] are considerably less accurate than the number of digits appear to suggest. In particular, the imaginary part of the lowest resonance is several orders of magnitude greater than it should be. We compare the results of that paper with those provided by the Riccati-Pad\'e method, perturbation theory, and an asymptotic expansion for the resonance width. The inaccuracy of those results can be traced back to the lack of precision in the calculation of the matrix elements of the secular equation. We carry out a more accurate calculation with the same method and show that the agreement with earlier results of other authors is greatly improved.

The trouble with orbits: The Stark effect in the old and the new quantum theory

The old quantum theory and Schrödinger׳s wave mechanics (and other forms of quantum mechanics) give the same results for the line splittings in the first-order Stark effect in hydrogen, the leading terms in the splitting of the spectral lines emitted by a hydrogen atom in an external electric field. We examine the account of the effect in the old quantum theory, which was hailed as a major success of that theory, from the point of view of wave mechanics. First, we show how the new quantum mechanics solves a fundamental problem that one runs into in the old quantum theory with the Stark effect. It turns out that, even without an external field, it depends on the coordinates in which the quantum conditions are imposed which electron orbits are allowed in a hydrogen atom. The allowed energy levels and hence the line splittings are independent of the coordinates used but the size and eccentricity of the orbits are not. In the new quantum theory, this worrisome non-uniqueness of orbits turns into the perfectly innocuous non-uniqueness of bases in Hilbert space. Second, we review how the so-called WKB (Wentzel–Kramers–Brillouin) approximation method for solving the Schrödinger equation reproduces the quantum conditions of the old quantum theory amended by some additional half-integer terms. These extra terms remove the need for some arbitrary extra restrictions on the allowed orbits that the old quantum theory required over and above the basic quantum conditions.

Method of complex coordinates: Application to the stark effect in hydrogen

The method of complex coordinates as applied to calculation of resonance parameters for dilitation analytic operators is discussed, and numerical evidence is presented that although the Stark potential Fz is not dilatation analytic, many of the results of the theory seem to hold. In particular, Stark widths and shifts for the H atom ground state in weak and strong fields are calculated in spherical coordinates using only L/sup 2/ basis functions, giving results in excellent agreement with those of methods applying detailed boundary conditions in parabolic coordinates.

The hydrogen atom in electric and magnetic fields: Pauli’s 1926 article

The results obtained by Pauli in his 1926 article on the hydrogen atom made essential use of the dynamical so(4) symmetry of the bound states. Pauli used this symmetry to calculate the perturbed energy levels of an hydrogen atom in a uniform electric field (Stark effect) and in uniform electric and magnetic fields. Although the Stark effect in hydrogen has been studied experimentally, Pauli’s results in mixed fields have been studied only for Rydberg states of rubidium atoms in crossed fields and for lithium atoms in parallel fields.

The complex absorbing potential method (CAP) to study the Stark effect inhydrogen andlithium

We present the complex absorbing potential (CAP) method to investigate theStarkeffect in hydrogen and lithium. The numerical technique as applied tocalculations of Stark shifts and widths yields an appealing, simple andpowerful tool for the computation of resonance parameters of thescatteringprocesses. The feasibility and accuracy of the present method aredemonstrated by the numerical evidence of the Stark shifts and widths ofHatom ground and excited states. The results agree satisfactorily with theavailable results in the literature. Moreover, the method providesunambiguous advantages over other existing methods forthe calculationofresonance parameters in a many-electron system and is illustrated for thelow-lying states of the Li atom under the influence of external DCelectric fields. Some new results are reported.

Slater Sum in Cylindrically Symmetric Inhomogeneous Electron Liquids: Towards A Differential Equation for the Stark Effect in Hydrogen Atom

Attention is focussed here on a variety of cylindrically symmetric inhomogeneous electron liquids. These include separable potentials in which a general variation along the (z) axis of cylindrical symmetry is combined with isotropic harmonic confinement in the (x, y) plane. in this case, an explicit differential equation is derived for the Slater sum along the z axis by projecting out of the (off-diagonal) canonical density matrix the states with zero angular momentum about the axis of symmetry. Some attention is then given to the calculation of the Slater sum for a hydrogen-like atom in a uniform electric field of arbitrary strength. the model of a separable potential with harmonic confinement, though no longer exact, is shown to lead directly to a (now approximate) equation for the Slater sum along the z axis for the Stark effect in hydrogen. Finally some further progress is shown to be possible in the extreme high field limit.

Use of the complex-scaling method in calculations of Stark resonances

Herbst and Simon [Phys. Rev. Lett. 41, 67 (1978)] have proven that the complex-scaling method (CSM) is applicable to the Stark effect in atoms. However, their proof is limited to complex scaling with an argument /3. Their method makes no statement as to the validity of CSM beyond this region. This article reports on a detailed numerical study of the use of CSM for the Stark effect in hydrogen. The results suggest a limitation on the range of energies of Stark resonances that can be calculated by CSM. We find that CSM does not produce resonances with energies $\mathrm{Im}E~\sqrt{3}\mathrm{Re}E,$ which corresponds precisely to the range of validity of the Herbst-Simon theory.

Stark effect in hydrogen: Reconstruction of the complex ground-state energy from the coefficients of an asymptotic perturbation expansion

Stark effect in hydrogen: Reconstruction of the complex ground-state energy from the coefficients of an asymptotic perturbation expansion

Classical examination of the Stark effect in hydrogen

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.

Direct calculation of Stark resonances in hydrogen.

We propose an alternative way of calculating the resonances of the Stark effect in hydrogen. The method is based on a rational approximation of the logarithmic derivative of the eigenfunction, and leads to a quantization condition for the complex energies of the metastable states. We present accurate results for the ground and some excited states for several field intensities. \textcopyright{} 1996 The American Physical Society.

Classical view of the Stark effect in hydrogen atoms

The unique properties of the hydrogen atom, especially those that result from the dynamic symmetry of the 1/r potential, are presented from both classical and quantal points of view. Using these properties as a starting point, the response of a classical hydrogen atom, an electron executing a Keplerian ellipse about a proton, to a weak electric field is described. Because the field is weak, the elliptical orbit may be treated as a dynamical element itself, variations in its shape and orientation depicting the classical Stark effect. The behavior of this orbit is then correlated with the quantum mechanical description of the first-order Stark effect in hydrogen, a subject that is included in most introductory courses in quantum mechanics.

Periodic orbit theory of broad resonances in two-dimensional hydrogenic Stark effect

The modified version of periodic orbit theory, developed by Robbins (1989) for systems which possess discrete symmetries is used to calculate parameters of broad resonances in the case of the two-dimensional hydrogenic Stark effect at energies above the zero field ionization threshold. It is shown that the contribution of single periodic orbits of the system give results that coincide with those of the comparison equation method also presented.

Periodic orbit theory of broad resonances in the hydrogenic Stark effect and the two-Coulomb-center problem

The periodic orbit theory is used to calculate the poles of the S -matrix which are related to broad resonances in two important realistic systems: the hydrogen atom in a uniform electric field and the two-Coulomb-center problem. It is shown that the contributions of certain complex periodic orbits in the periodic orbit sum give results that coincide with the comparison equation method calculations.

On broad resonances in the hydrogenic stark effect

In the hydrogenic Stark problem, the comparison equation is used to study poles of the S-matrix close to the positive real axis in the complex energy plane. simple WKB-type formulae which determine parameters of the resonances are derived. Comparisons with the full quantum mechanical calculations are presented.

Calculation of high-order electric polarizabilities via the perturbational-variational Rayleigh-Ritz formalism: Application to the hydrogenic Stark effect.

It is demonstrated that perturbation theory combined with the variational principle in the form of the perturbational-variational Rayleigh-Ritz (PV-RR) matrix formalism is a powerful method for applying large-order perturbation theory (LOPT) to the Stark effect for hydrogenic ions. Unlike other more specialized LOPT methods, which do not invoke the variational principle, application of the PV-RR formalism is not restricted to one-electron systems. A paradigm is derived and implemented which permits the precise construction of a Rayleigh-Ritz variational ansatz from which the real Stark ground-state Rayleigh-Schr\"odinger perturbation series for the eigenfunction, eigenvalue, and arbitrary expectation values can be generated via the PV-RR procedure to any prespecified order in the most economical way possible. A novel method of analysis, based on a combination of the Stark virial, Hellmann-Feynman, and remainder theorems, is developed for studying the physical origins of high-order atomic and molecular polarizabilities; the procedure is illustrated by application to the hydrogenic Stark effect. It is concluded on the basis of this work and of similar high-order PV-RR polarizability calculations for ${\mathrm{H}}_{2}$${\mathrm{}}^{+}$ that the prospects for extending the PV-RR formalism to accurate polarizability studies of more complex systems are promising.

The stark effect in hydrogen: Properties of threshold resonances in the photoionization spectrum of excited parabolic states

Abstract We have performed exact numerical calculations of the photoionization of highly excited parabolic states of hydrogen in the region of the zero field ionization limit (E ∼ 0) in an external electric field. We present results for the photoionization of blueshifted m=0 initial states (n2=0) by both π and σ polarized light. For highly excited initial states, the effects of the overlap of the initial and final state charge distributions become dominant in determining which final states will be excited, leading to approximate selection rules. The broad resonances in the photoionization spectrum near E=0 are shown to be dramatically enhanced and narrowed, resulting in the appearance of essentially isolated resonances in π polarization. In contrast to the situation for photoionization of low-lying states, large resonances are also predicted to be present in σ polarization.

Stark effect in a one-dimensional model atom

A very simple one-dimensional, quantum-mechanical analog of the second-order Stark effect in hydrogen is proposed. Its only resonance is found to be a complex root of an algebraic relation among the Airy functions and their derivatives. The Rayleigh–Schrödinger perturbation series is shown to arise from the real part of the resonance. The nonperturbative nature of the resonance width is stressed.