The eight-component relativistic wave equation for spin-
21 particles of Robson and Staudte (FV
21), a Feshbach--Villars linearization of the Feynman--Gell-Mann equation, reproduces the Dirac hydrogenic spectrum with a doubled solution space and an indefinite inner product. Its exact Stark shifts must equal the Dirac ones; whether its indefinite-metric perturbation theory delivers them is not guaranteed. We compute the first- and second-order Stark shifts of the
n=1--3 levels of hydrogenic ions from H to U
91+ in the Schrödinger, Dirac and FV
21 theories, evaluating all spectral sums, continuum included, by the Dalgarno--Lewis method. The FV
21 and Dirac shifts coincide level by level, to working precision at first order (closed-form roots) and to 45 significant figures at second order. Both ingredients are needed: without the explicit spin--field coupling the first-order roots become complex, and without the negative-norm states the
(Zα)2 coefficient is wrong. Norm sign and parity are locked in the doubled space: a potential lifting the
2s1/2--
2p1/2 degeneracy needs its own spin--gradient term, and shifts placed by hand on the
κ-labeled states give a spurious imaginary linear Stark shift. The relativistic linear Stark effect is smaller by finite factors, but a second-moment sum rule shows the non-relativistic linear Stark strength is conserved, the missing part reappearing in singular
(Zα)−2 second-order terms. For levels with a fine-structure partner, an exact intra-shell/regular split of the second-order shift gives the
(Zα)0 coefficients analytically and relates the
π2 terms of the
(Zα)2 coefficients to the singular ones. The Dirac weak/strong-field crossover for
n=2 of hydrogen lies near 2.8~kV cm
−1, and the perturbative window set by the fine structure grows as
Z5.
The energy level and wave function obtained by solving the Schrödinger equation are extremely important in quantum mechanics. In this study, the solution to the Schrödinger equation of the electron-proton system by separation of variables was briefly described and the visualizations of the various solution were generated by Mathematica. The perturbation theory was discussed with the exclusion of the correction terms coming from the degenerate subspace. Applying the perturbation theory to the wavefunctions of the electron-proton system to study the Stark effect, the perturbation Hamiltonian is a constant electrostatic field. For the principal quantum numbers n=1, n=2 and n=3, this paper obtains the energy corrections to the first and second degree, the corrections of the wavefunction to the first degree. The visualizations of the perturbed wavefunctions were obtained by setting the summation limits k=20 and other parameters to 1. The visualizations of various unperturbed-states and perturbed-states were generated by Mathematica version 13.2.
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.
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.