Energy Levels Of Hydrogen-like Atoms Research Articles

Toward XUV frequency comb spectroscopy of the 1 S–2 S transition in hbox {He}^+

The energy levels of hydrogen-like atoms and ions are accurately described by bound-state quantum electrodynamics (QED). hbox {He}^{+} ions have a doubly charged nucleus, which enhances the higher-order QED contributions and makes them interesting for precise tests of QED. Systematic effects that currently dominate the uncertainty in hydrogen spectroscopy, such as the second-order Doppler shift and time-of-flight broadening, are largely suppressed by performing spectroscopy on trapped and cooled hbox {He}^{+} ions. Measuring a transition in hbox {He}^{+} will extend the test of QED beyond the long-studied hydrogen. In this article, we describe our progress toward precision spectroscopy of the 1 S–2 S two-photon transition in hbox {He}^{+}. The transition can be excited by radiation at a wavelength of 60.8 nm generated by a high-power infrared frequency comb using high-order harmonic generation (HHG). The hbox {He}^{+} ions are trapped in a Paul trap and sympathetically cooled with laser-cooled hbox {Be}^{+} ions. Our recently developed signal detection scheme based on secular-scan spectrometry is capable of detecting hbox {He}^{+} excitation with single-event sensitivity.Graphic abstract

MAPLE program for modelling hydrogen-like atoms in quantum mechanics with non-negative distribution function

The program is proposed for a realization of the symbolic algorithm based on the quantum mechanics with non-negative probability distribution function (QDF) and for calculations of energy levels for hydrogen-like atoms. The program is written up in the language MAPLE. In the framework of the algorithm an original Maple package for calculations of necessary functions, such as hydrogen wave functions, Sturmian functions and their Fourier-transforms, Clebsch-Gordan coefficients, etc. is proposed. Operators of observables are calculated on the basis of the QDF quantization rule. According to the Ritz method, eigenvalues of Ritz matrices represent spectral values of the quantity under investigation, i.e. energy. As an example, energy levels of hydrogen-like atoms are calculated and compared with experimental data retrieved from the NIST Atomic Spectra Database Levels Data. It turns out that this theory seems to be equivalent to the traditional quantum mechanics in regard to predictions of experimental values. However, the existence of a phase-space probabilistic quantum theory may be an important advance towards the explanation and interpretation of quantum mechanics.

Relativistic Hydrogen-Like Atom on a Noncommutative Phase Space

The energy levels of hydrogen-like atom on a noncommutative phase space were studied in the framework of relativistic quantum mechanics. The leading order corrections to energy levels 2S 1/2, 2P 1/2 and 2P 3/2 were obtained by using the 𝜃 and the $\bar \theta $ modified Dirac Hamiltonian of hydrogen-like atom on a noncommutative phase space. The degeneracy of the energy levels 2P 1/2 and 2P 3/2 were removed completely by 𝜃-correction. And the $\bar \theta $ -correction shifts these energy levels.

Elliptical orbits in the phase-space quantization

The energy levels of hydrogen-like atoms are obtained from the phase-space quantization, one of the pillars of the old quantum theory, by three different methods - (i) direct integration, (ii) Sommerfeld's original method, and (iii) complex integration. The difficulties come from the imposition of elliptical orbits to the electron, resulting in a variable radial component of the linear momentum. Details of the calculation, which constitute a recurrent gap in textbooks that deal with phase-space quantization, are shown in depth in an accessible fashion for students of introductory quantum mechanics courses.

Retardation effect in the recoil corrections to the fine shift of S energy levels of hydrogen-like atoms

Contributions of recoil effects to the fine shift of S energy levels of hydrogen-like atoms have been considered. The detailed analysis of the influence of retardation effect on the value of the logarithmic in respect to the mass ratio correction from the two photon diagram of sequential exchange by Coulomb and transverse photons has been realized.

On the quantum Coulomb field

The quantum theory of the Coulomb field has been developed by Staruszkiewicz in the long series of papers. This theory explains the universality and quantization of the electric charge observed in Nature. Moreover, the efforts have been made to determine the value of the elementary charge from its mathematical structure. Nonetheless, no other immediate applications of this theory have been proposed. We make such an attempt by (i) considering the classical energy operator and defining its counterpart in the quantum theory of the Coulomb field; (ii) determining the eigenstates of the energy operator and assigning energy to the excitations of the theory; and (iii) proposing a simple theoretical scheme to estimate the effect of the quantum fluctuations of the Coulomb field on the energy levels of hydrogen-like atoms. We argue that the recent experimental advances in hydrogen and muonic-hydrogen spectroscopy may provide the unique window of opportunity for the verification of the Staruszkiewicz's theory.

Special features of the quasipotential approach to studying higher order corrections in the fine-structure constant to fine shifts of hydrogen energy levels

Important corrections to fine shifts of energy levels of hydrogen-like atoms can be obtained by studying in detail logarithmic contributions in the fine-structure constant. New results concerning the calculation of α 6µ3/(m 1 m 2) corrections are presented.

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).

Logarithmic contributions in the particle-mass ratio to the fine shift of S energy levels of hydrogen-like atoms in the fifth order in the fine-structure constant

A high-precision investigation of a logarithmic contribution in the particle-mass ratio to the fine shift of the S energy levels of hydrogen-like atoms from the exchange of a Coulomb photon is performed. It is shown that diagrams describing the exchange of one transverse photon and two Coulomb photons do not make such contributions.

Finite-temperature effects on the energy levels of hydrogenlike atoms.

It is shown that level-dependent finite-temperature effects on hydrogenlike atoms are of the order α3(Tm)2.Received 12 October 1984DOI:https://doi.org/10.1103/PhysRevD.32.1038©1985 American Physical Society

Dirac equation and energy levels of hydrogenlike atoms in Robertson-Walker metric

The Dirac equation of the hydrogenlike atom in the Robertson-Walker metric is studied in this paper by means of tetrad formalism. The decoupled Dirac equation is derived and energy levels of hydrogen atoms taking into account the cosmological radius are obtained by using the perturbation expansion.

Energy levels of hydrogen-like atoms predicted by quantum electrodynamics, 10≤ Z≤40

Energy-level separations predicted by quantum electrodynamics among the n=1 and n=2 states of hydrogen-like atoms with nuclear charge Z in the range 10≤Z≤40 are tabulated. For each level, the various Lamb-shift effects are listed separately.

Energy Levels of Hydrogenlike Atoms in a Magnetic Field

A new expansion of the wave function in Laguerre polynomials for hydrogenlike atoms in a magnetic field has been developed. This expansion reduces the Schr\odinger equation to an infinite set of algebraic equations for which a fast, accurate, and unambiguous calculation of the bound states of a hydrogen atom in a magnetic field is possible. Four quantum numbers ($K, C, M, N$) are used to label the energy levels of the system in cylindrical coordinates. When the magnetic field is zero, the new labeling of the states is equivalent to the standard ($n, l, m$) quantum numbers obtained for the Coulomb potential in spherical coordinates. Numerical results are presented for the first 14 energy levels with principal quantum number $n=1, 2, 3$. Several energy-level crossings occur in the region $0l\ensuremath{\gamma}l1$, where $\ensuremath{\gamma}=\frac{\ensuremath{\hbar}{\ensuremath{\omega}}_{c}}{2\mathcal{R}}$ and $\mathcal{R}$ is the effective Rydberg energy.

Theory of Critical Screening Radius of Energy Levels of Hydrogen-Like Atoms in Plasmas

The electronic states in the Debye-Huckel potential, -( Z e 2 / r )exp (- r / D ), are investigated and the critical screening radius D = D c ( n , l ) is given by the simple formulae

Correction to the Rydberg Constant for Finite Nuclear Mass

If the problem of the hydrogen-like atom is done with the assumption that the nucleus is at rest, the resulting energy levels are proportional to the Rydberg constant for infinite mass. It is traditional to obtain the Rydberg constant for finite mass of the nucleus by a simple substitution of the reduced mass of the system for the electron mass in the expression for the Rydberg constant for infinite nuclear mass. Realization that such a substitution (which is drawn from classical mechanics) does not give a correct transformation to the center-of-mass coordinate system in the case of charged particles, leads one to significant corrections for the Rydberg constants of finite nuclear masses. The corrections are drawn from classical, non-relativistic physics. Also, an interesting conjecture about the energy levels of hydrogen-like atoms, under the assumptions of Bohr theory and classical physics, is obtained.

The operational wave equation and the Zeeman effect

The object of this paper is to develop the theory of the Zeeman effect from the operational wave equation W ψ ={(p+ ec -1 G) A+ c -1 ( p 0 + e V) i A 4 + im 0 c ) ψ = 0, (1.1) V being the electrostatic potential of the nucleus and G the electromagnetic potential of the applied magnetic field. The same notation is employed as in a previous paper,* viz., the components of p are the momenum operators P 1 , P 2 , P 3 ; P 0 is the energy operator; A 1 , A 2 , A 3 , A 4 are wave operators, the first three being treated as the components of a vector A. It has been shown* that the energy levels of hydrogen-like atoms can be determined from the operational wave equation without making any restrictions on the wave operators other than Dirac’s conditions ½ (A m A n +A n A m )=0 if m ≠ n , 1 if m = n .