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Physical Review

On the Theory of the Stark Effect in Hydrogenic Atoms

Vladimir Rojansky

Published Jan 1, 1929• 43 citations
Publisher

Abstract

The Stark effect of the fine structure of hydrogenic atoms is discussed in the light of the quantum mechanics. Explicit formulas are given for the energy levels in either a weak or a strong electric field, including in each case both the relativity corrections and the Stark effect. (The conventional treatment neglects either one or the other.) The results are, except in form, similar to those recently given by Schlapp. In the new mechanics, in contradistinction to the old Bohr theory, there is a linear Stark effect even in weak fields, because of the identity of energy for the $\mathrm{ns}$ and $n{p}_{1}$ levels in the absence of external fields. This degeneracy is shown to account for the results of the recent experiments of Ornstein, Zernike, and Snoek, who found that the $2s$ level in $H$ is not metastable.The coefficients are found for the development of the "parabolic" eigenfunctions in terms of the "polar" ones, and a set of numerical values of these coefficients is tabulated.Methods are given for the calculation of the relative intensities in the Stark effect in terms of the already available theoretical fine structure intensities in the absence of fields.

DOI
10.1103/physrev.33.1

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Journal of Physics B Atomic Molecular and Optical Physics•Apr 28, 1991•1 citation
The Stark effect of the fine structure of hydrogenic atoms is discussed in the light of the quantum mechanics. Explicit formulas are given for the energy levels in either a weak or a strong electric field, including in each case both the relativity corrections and the Stark effect. (The conventional treatment neglects either one or the other.) The results are, except in form, similar to those recently given by Schlapp. In the new mechanics, in contradistinction to the old Bohr theory, there is a linear Stark effect even in weak fields, because of the identity of energy for the ns\mathrm{ns}ns and np1n{p}_{1}np1​ levels in the absence of external fields. This degeneracy is shown to account for the results of the recent experiments of Ornstein, Zernike, and Snoek, who found that the 2s2s2s level in HHH is not metastable.The coefficients are found for the development of the "parabolic" eigenfunctions in terms of the "polar" ones, and a set of numerical values of these coefficients is tabulated.Methods are given for the calculation of the relative intensities in the Stark effect in terms of the already available theoretical fine structure intensities in the absence of fields.
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.
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.
The explanation of the first-order Stark effect in hydrogen by Epstein and Schwarzschild in 1916 was seen as a great success for the old quantum theory. Yet, it also revealed some serious limitations of the theory. To recover the experimentally found line splittings, one had to make some arbitrary assumptions in addition to the basic quantum conditions to rule out certain orbits. The calculation of intensities of lines on the basis of Bohr's correspondence principle likewise required arbitrary additional assumptions. Finally, the actual orbits predicted by the old quantum theory depend on the coordinates chosen to impose the quantum conditions. Both Sommerfeld and Epstein recognized this problem but offered no solution for it. All these problems were solved in 1926 when Schrödinger and Epstein explained the Stark effect on the basis of the new wave mechanics. The calculations in the two theories are similar. In particular, both the Schrödinger equation in the new theory and the Hamilton-Jacobi equation in the old theory are separated in parabolic coordinates. The new quantum mechanics determines all allowed states and transitions without any additional assumptions. It also replaced the ambiguous guidelines based on the correspondence principle for calculating intensities by the straightforward prescription that intensities are given by the squares of the matrix elements of position, leading to results that agreed much better with the experimental data. Finally, the embarrassing non-uniqueness of orbits in the old quantum theory turned into the innocuous non-uniqueness of bases of eigenfunctions in wave mechanics. To this day, the Stark effect is remembered as one of the few qualified successes of the old quantum theory. We suspect that this is largely because after 1926 it became just one of the many unqualified successes of the new quantum theory.
Abstract Splitting the energy levels of a hydrogen-like atom by the electric field nonuniform at the atomic scale is studied. This situation is important for the multi-level treatment of the phenomenon of Rydberg blockade (Dumin Y V 2014 J. Phys. B: At. Mol. Opt. Phys. 47 175502 ). An explicit formula for the energy levels is derived. A typical value of the energy shift by the electric field gradient turns out to be proportional to the fourth power of the principal quantum number (i.e. the square of atomic size), as would be expected from a qualitative consideration. Finally, the fine spatial structure of the Rydberg blockade is analyzed when the electric-field-gradient term plays the dominant role, and the results are compared with the experimental data.
The Zeeman levels of hydrogenic atoms are determined by a new and simple method, neglecting terms in H2{\mathrm{H}}^{2}H2. The frequency of the Larmor precession is L=L0(1+Emc2)L={L}_{0}(1+\frac{E}{m{c}^{2}})L=L0​(1+mc2E​). L0{L}_{0}L0​ is the frequency of precession obtained by Newtonian mechanics and EEE the energy of the atom. This agrees with a formula of Pauli obtained by ordinary relativity mechanics. The formulae for the first order Stark effect and for the Paschen-Back effect can be obtained by similar methods.An extension of Ehrenfest's law of motion for a particle is proved. It is applicable to any conservative system. From it, a theorem of wave mechanics analogous to the virial theorem is obtained.