With the use of a simple similarity transformation which brings the radial wave equations of the Dirac-Coulomb problem into a form nearly identical to those of the Schr\"odinger and Klein-Gordon equations, we derive simplified solutions to the Dirac-Coulomb equation for both the bound and continuum states following the familiar standard procedure adopted in the derivation of the conventional solutions. We show that to obtain the desired form of the second-order radial equations we can still work with a first-order partial differential equation rather than with the second-order Dirac equation widely employed in the derivation of the simplified solutions, and thus we can avoid the task of reducing the solutions of the second-order equations to those of the original Dirac equation. The transformed Dirac-Coulomb radial equations are so simple that one can apply the WKB method to them in the same way that one applies the WKB approximation to the Schr\"odinger radial equation, without making the further approximations commonly invoked, and the Sommerfeld-Dirac discrete spectrum follows immediately. For small Z\ensuremath{\alpha}, we also present approximate expressions for both the transformed and the original Dirac-Coulomb wave functions, which are valid for all energies and certainly valid in the quasirelativistic approximation which takes account of relativistic effects. The structure of these approximate expressions also suggests one of the methods to obtain relativistic approximate wave functions from the Pauli-spinor wave function for more general central potentials arising in atoms, molecules, and solids.
A method for solving the two-center Dirac equation using a Gaussian-product basis set tailored to the axial symmetry of diatomic systems is presented. For the ground state of H
2+ at an internuclear distance
R=2.0 a0, we obtain a relativistic energy with relative precision at the
10−18 level. The properties of the basis set allow for an analytic Fourier transform of the wave function. As a first application we evaluate, for the first time, the zero-potential contribution to the one-loop self-energy for H
2+ in Feynman and Coulomb gauges. Although this contribution is not separately gauge invariant, its calculation provides a proof of principle for the use of the present framework in prospective QED calculations.
We present a method of simulating the Dirac equation in 3+1 dimensions for a free spin-1/2 particle in a single trapped ion. The Dirac bispinor is represented by four ionic internal states, and position and momentum of the Dirac particle are associated with the respective ionic variables. We show also how to simulate the simplified 1+1 case, requiring the manipulation of only two internal levels and one motional degree of freedom. Moreover, we study relevant quantum-relativistic effects, like the Zitterbewegung and Klein's paradox, the transition from massless to massive fermions, and the relativistic and nonrelativistic limits, via the tuning of controllable experimental parameters.
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.
Two-photon decay of hydrogenlike ions is studied within the framework of second-order perturbation theory, based on the relativistic Dirac's equation. Special attention is paid to the effects arising from the summation over the negative-energy (intermediate virtual) states that occur in such a framework. In order to investigate the role of these states, detailed calculations have been carried out for the $2{s}_{1/2}\ensuremath{\rightarrow}1{s}_{1/2}$ and $2{p}_{1/2}\ensuremath{\rightarrow}1{s}_{1/2}$ transitions in neutral hydrogen H as well as for hydrogenlike xenon
Xe53+ and uranium
U91+ ions. We found that for a correct evaluation of the total and energy-differential decay rates, summation over the negative-energy part of Dirac's spectrum should be properly taken into account both for high-
Z and low-
Z atomic systems.
No consensus has been reached on the appropriate operators for the spin and orbital angular momentum of free relativistic electrons. But there is a tendency to believe the existence of a spin-orbit coupling through the Berry connection. Here we show that this is not the case. Because the positive-energy solution of the Dirac equation is associated with the state of the electron, it is reasonable to postulate that only when projected onto the positive-energy subspace can observable operators on the Hilbert space of the Dirac equation represent physical quantities of the electron. On this basis, we find, after mapping the projected operators onto an irreducible representation, that the Berry connection describes the position of a quantum reference point with respect to the laboratory reference frame, relative to which the position of the electron is canonical. The orbital angular momentum of the quantum reference point about the origin of the laboratory reference frame is expressed in terms of the Berry connection. It combines with the canonical orbital angular momentum of the electron about the quantum reference point to give the orbital angular momentum of the electron about the origin of the laboratory reference frame.