Dirac energy hydrogen-like atom

Fig. 1. The ground state energies of a Z = 90 hydrogen-like atom obtainedfrom the Dirac (D) andfrom the Levy-Leblond (L) equations as functions of the nonlinear parameters. In the upper-rowfgures (Dl and LI) the a (abscissa) and (3 (ordinate) dependence ofE is displayed when L and S are set equal to the values corresponding to the exact solutions. In the lower-row figures (D2 and L2 ) the L (abscissa) and S (ordinate) dependence of E is displayed when a and 5 are set equal to the exact value. The arrows show directions of the gradient their length is proportional to the value of the gradient The solid line crossing the saddle corresponds to the functions andLl) or S = Sj (L) D2 and L2. For the definitions of thesefunctions see text.

Fig. 5. The ground state energies of a Z 30 hydrogen-like atom in the Gaussian basis obtained from the Dirac (D) andfrom the L y-Leblond (L) equations as functions of a and. The saddle point coordinates equal to (282,168) in the Dirac case and (271,164) in the Levy-Leblondcase. The cross-sections of...

Figure 1. Variational ground state energy of a Z = 90 hydrogen-like atom obtained from the Dirac-Pauli equation as a function of a (abscissa) and (5 (ordinate) while a = b = s (left figure) and as a function of a (abscissa) and b while a = (5 = Z (right figure). The arrows are proportional to the gradient of . The saddle points correspond to the exact eigenvalues of the Dirac Hamiltonian.

The term Lamb shift of a single atomic level usually refers to the difference between the Dirac energy for point-like nuclei and its observable value shifted by nuclear and QED effects. Nuclear effects include energy shifts due to static nuclear properties such as the size and shape of the nuclear charge density distribution and due to nuclear dynamics, i.e. recoil correction and nuclear polarization. To a zeroth approximation, the energy levels of a hydrogen-like atom are determined by the Dirac equation. For point-like nuclei the eigenvalues of the Dirac equation can be found analytically. In the case of extended nuclei, this equation can be solved either numerically or by means of successive analytical approximation (see Rose 1961 Shabaev 1993). 

We therefore include the effect of the atomic nucleus by introducing the scalar potential energy operator of Eq. (6.3) into the Dirac equation for external fields in Eq. (5.121) with A = 0 to obtain the equation of motion for Dirac hydrogen-like atoms. 

The consideration of the nonrelativistic limit of the Dirac energy eigenvalue for the hydrogen-like atom with a Coulombic potential for the electron-nucleus attraction, Eq. (6.3), demonstrates the effect of subtracting the rest energy mgC and leads us to a discussion of the reference energy in the following section 6.7. 

Fig. 6. The application of the weak minimax principle to the ground state of a Z = 90 two-electron atom described by the simplified two-electron Dirac Hamiltonian using the hydrogen-like basis with L = S = 1. The thin solid lines represent the energy as a function of a when 3 = The broad solid lines give the energy...

Figure 6.6 Comparison of ground-state energies E[glZ scaled by I7 obtained tor hydrogen-iike atoms from Schrodinger quantum mechanics (horizontal line on top at -0.5 hartree), from Dirac theory with a Couiomb potential from a point-like nucleus (dashed line) and from Dirac theory with a finite nuclear charge distribution of Gaussian form (thin black line). The highest energy of the positronic continuum states, -2meC, appears as a thick black line, which is bent because of the l/Z scaling.

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