Hydrogen ground-state energy, Dirac

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 exact wavefunction corresponds to a = b = s and a = (3 = Z. The variational ground state energy of Z = 90 hydrogen-like ion in the Dirac-Pauli... 

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.

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

However, already in 1930s deviations were observed between the results of precision spectroscopy and the Dirac theory for simple atomic systems, primarily for the hydrogen atom. The existence of negative-energy states in the solutions of Dirac equation is the mathematical but not the physical grounds of the existence of particles and antiparticles (electrons and positrons). Besides, the velocity of light is finite. For an complete model we must turn to quantum field theory and quantum electrodynamics (QED) [4]. 

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