This would be very interesting. It might be easier to start with the non relativistic H atom equations and apply this method. There might be code in some packages such as MATLAB that can integrate these partial differential equations. However, the separability assumption psi = psi(r)psi(theta, phi) is fine. In order to introduce spin, a sigma basis can be introduced. Firstly I will write out these new equations in the non relativistic limit.
Sent: 10/03/2017 12:17:22 GMT Standard Time
Subj: Re: 372(6): The ECE2 Relativistic H atom
We can invetstigate the numerical solution of (7). The normal (non-relativistic) method is to vary E ans wee for which values psi converges. In the relativistic version there is no parameter E and a gamma instead. This changes the situation.
Your hint that this mehtod uses O(3) geometry is important. I remember calculational methods avoiding spinors in atomic and solid state physics which do the same (called “scalar-relativistic”). This even works for spin states to a certain degree. If you want to obtain spin-polarized states with spin-orbit coupling, howerver, you have to use the Dirac or Fermion equation.
Am 10.03.2017 um 11:59 schrieb EMyrone:
This note introduces an entirely new and original and ab initio theory of the relativistic H atom, using the lagrangian (1) of ECE2 relativity and three Euler Lagrange equations (14) to (16) which can be solved simultaneously using numerical methods such as the Runge Kutta methods of Maxima as developed by co author Horst Eckardt. The wavefunction is found by solving Eqs. (9) to (11) simultaneously, in general it is psi = psi(t, r, theta, phi). The energy levels are found from Eq. (7). This is much simpler than the traditional hamiltonain method (17) to (23). The usual development is to write the lagrangian for the Dirac equation using the Dirac matrices and Dirac wavefunction. In papers such as UFT172 to UFT177 (all classics by now) the Dirac equation is developed into the fermion equation. In the new method of this note the wavefunction is found in O(3) space, and not the SU(2) space of the fermion equation (chiral Dirac equation). The basic method described here for H can be extended to other atoms and molecules given the mainframes and supercomputers used routinely in computational quantum chemistry. Contemporary supercomputers are far more powerful than the IBM 3096 / LCAP system which I helped pioneer at IBM Kingston, Cornell, Zurich and ETH in the eighties. The desktop used by Horst Eckardt can crunch out the ab initio equations in this note. The results for the energy levels can be compared with the analytically known energy levels of the relativistic H atom given in note 372(4), Eq. (42).