Many thanks again. These are incisive remarks as usual. The separability assumption is a safe one so progress can be made. I think that the demonstraton of the precessing orbit from the lagrangian is a major step forward, and congratulations on that.
Sent: 10/03/2017 12:02:54 GMT Standard Time
Subj: Re: 372(5): Complete Set of equations for ECE2 Relativistic Quantum Mechanics: 1) 2D
The functions r(t), phi(t), r_dot(t), phi_dot(t) can be determined form the relativistic Lagrangian. However eqs. (14-20) are partial diff. equations. Therefore the simple Runge-Kutta method is not applicable. If we assume that the wave function is separable:
psi(r, phi) = psi_r(r) * psi_phi(phi)
then eqs. (19-20) would be solvable by Runge-Kutta, but r_dot and phi_dot are defined on a time grid and had to be interpolated to an r and phi grid. This is a bigger effort and the precision of this procedure would have to be checked.
Am 09.03.2017 um 14:31 schrieb EMyrone:
This note gives the complete set of equations available for 2D relativistic quantum mechanics using plane polar coordinates. It can be seen that there are enough equations to solve the problem completely, to find all the wavefunctions and energy levels. A computation scheme is suggested. The next note will extend this method to the relativistic H atom, using spherical polar coordinates in 3D. In both 2D and 3D there will be fine structure splitting. This does not occur on the non relativistic level as is well known. Having set the groundwork the method can be extended to the helium atom and eventually to the whole of computational quantum chemistry. It is a new ab initio method without use of perturbation theory or any approximation. It makes use of the ECE2 hamiltonian and lagrangian.