The relativistic energy spectrum of hydrogen

Abstract

The exact eigenfunctions of the spin-orbit-coupling operators for a relativistic binary system are calculated. Concerning the eigenvalue problem and the radial part of the wavefunction of the bound state, we provide corrections for our previous calculations [1] that contained some sign errors. Eigenfunctions of the spin-orbit-coupling operatorsIn a previous publication [1] the energy levels of a relativistic two-particle Hamiltonian for a binary (hydrogen-like) atom bound by the static Coulomb force were calculated exactly. Some derivations in this paper were not presented in sufficient detail. Especially the eigenfunctions of the spin-orbit-coupling operators,K I = 1 +L ·Î andK S = 1 +L ·Ŝ, were not calculated explicity, a task that will be completed in this supplement. The spin in units of /2 is denoted asŜ for the electron andÎ for the proton, andL = L/ refers to the orbital angular momentum with respect to the relative position vector r.To calculate the eigenfunctions we use the natural basis constructed by tensor products of the spin eigenvectors in eq. (48) Operator algebra shows that the spin orbit-coupling operators and spin helicity operators mutually do not commute with each other:and therefore cannot have common eigenfunctions. However, as shown below, one can construct common eigenfunctions ofK S andK I . Their common eigenvalue is called k. *

The radial wavefunction of a relativistic binary of two fermions bound by the Coulomb force

“…The wavefunction (i.e. 16-component spinor) can be decomposed in products of eigenfunctions of the spin-orbit-coupling operators [2] and the operators acting in particle-antiparticle space. Therefore, we made in paper [2] the ansatz (16), involving four independent spatial functions for each particle species.…”

“…16-component spinor) can be decomposed in products of eigenfunctions of the spin-orbit-coupling operators [2] and the operators acting in particle-antiparticle space. Therefore, we made in paper [2] the ansatz (16), involving four independent spatial functions for each particle species. The resulting (4-component spinor) radial wavefunction suggests itself by the structure of the matrix (42) in the original paper [1] and because of parity reasons as explained there.…”

“…According to [1,2] the radial eigenvalue problem then can be written in the following matrix form: …”

“…Notice that we introduced in the previous paper [1] the signed differential operator: ∂ ± = −∂/∂x ± k/x, a definition that takes care of the sign change stemming from the anti-commutators (64) in [1] and (14,15) in [2]. Here the spin-orbit-coupling quantum number, k, may take the integer values, ±1, ±2, ±3, .…”

“…The energy levels of this relativistic two-particle Hamiltonian were calculated exactly. The eigenfunctions of the spin-orbit-coupling operators were not calculated explicitly, a task that was completed in an addendum and erratum [2] to that paper. However, in those papers the radial wavefunctions were neither calculated explicitly nor discussed in sufficient detail.…”