The second-order hyperfine energy in the 1s and 2s states of the hydrogen atom is calculated analytically using the Dirac equation for an electron in a Coulomb potential perturbed by a point nuclear magnetic moment. The (divergent) result is compared with the calculation using the non-relativistic form of the hamiltonian. It is emphasized that in this comparison all the contributions to the non-relativistic second-order energy must be included. It is shown that, at least for the Fermi contact interaction, the results are the same to order mc 2α6, so that for this case the reduction of the Dirac equation to non-relativistic form is not suspect, contrary to previous assertions.
Hyperfine splittings play an important role in quantum information and spintronics applications. They allow for the readout of the spin qubits, while at the same time providing the dominant mechanism for the detrimental spin decoherence. Their exact knowledge is thus of prior relevance. In this work, we analytically investigate the relativistic effects on the hyperfine splittings of hydrogen-like atoms, including finite-size effects of the nucleis’ structure. We start from exact solutions of Dirac’s equation using different nuclear models, where the nucleus is approximated by (i) a point charge (Coulomb potential), (ii) a homogeneously charged full sphere, and (iii) a homogeneously charged spherical shell. Equivalent modelling has been done for the distribution of the nuclear magnetic moment. For the 1s ground state and 2s excited state of the one-electron systems H1, H2, H3, and He+3, the calculated finite-size related hyperfine shifts are quite similar for the different structure models and in excellent agreement with those estimated by comparing QED and experiment. This holds also in a simplified approach where relativistic wave functions from a Coulomb potential combined with spherical-shell distributed nuclear magnetic moments promises an improved treatment without the need for an explicit solution of Dirac’s equation within the nuclear core. Larger differences between different nuclear structure models are found in the case of the anisotropic 2p3/2 orbitals of hydrogen, rendering these excited states as promising reference systems for exploring the proton structure.
In this work alkali atoms hyperfine structure
D1 line all
π,
σ+ and
σ− transitions between magnetic sublevels are considered analytically. General
2×2 block matrices for ground and excited states were built in order to describe every transition. Eigenvalues and eigenkets for the mentioned above matrices are calculated and ``modified transfer coefficients as a function of nuclear angular momentum
I, magnetic quantum number
m and magnetic field magnitude
B are defined. Transition cancellations are observed only for
π transitions of each isotope. All the magnetic field values, which cancel transitions are calculated with an accuracy limited by the involved quantities uncertainties.
In a previous work the
π transitions of the
D1 line of alkali-metal atoms were shown to cancel at magnetic-field values given by a single closed-form expression. The
D2 line has until now resisted the same treatment, because the fixed-
m Hamiltonian blocks of the
2P3/2 manifold reach the dimension
4×4, and the corresponding formulas were considered to be too heavy to be useful. In this work we show that this difficulty is only apparent. Since the Zeeman interaction couples only levels with
ΔF=±1, every block, whatever its dimension, is tridiagonal; its characteristic polynomial therefore obeys a three-term recursion, and the components of its eigenvectors are polynomials in the eigenvalue. Using these two properties we obtain the eigenvalues in closed form by Ferrari's and Cardano's formulas, the eigenvectors without any further diagonalization, and finally a single relation which gives the magnetic-field value canceling a
D2 transition as an explicit function of the excited-state eigenvalue, the nuclear spin
I, the ground-state hyperfine splitting and the magnetic quantum number. A necessary condition on
m and on the polarization is derived, which replaces the selection rule known for the
D1 line. All the magnetic-field values canceling
π,
σ+ and
σ− transitions of
23Na,
39K,
40K,
41K,
85Rb,
87Rb and
133Cs are calculated up to 20~kG and given with their uncertainties; there are 234 of them. Contrary to the
D1 line, the
σ± transitions of the
D2 line do cancel, and they account for the majority of the values found. The accuracy of the calculated
B values is limited only by the uncertainties of the excited-state hyperfine constants.
We argue that the process of constructing the quantum mechanical current of the Pauli equation by copying the line of argument used in the spin-0 case, i.e., the Schrödinger equation, is ambiguous. We show that a nonrelativistic reduction of the relativistic Dirac four-vector current is, however, capable of fully resolving the problem. This analysis reveals that the nonrelativistic current of the Pauli equation should include an extra term of the form ∇×(ψ†σψ). We present an initial exploration of the potential consequences of this new “spin term” by solving the Pauli equation for crossed magnetic and electric fields and calculating the corresponding current.