This will be very interesting again. I will have a look at helium with the Lagrangian method and see what can be done, then set up the relativistic lagrangian of ECE2 theory. That should give the precessing ellipse by quite simple integration. In dealing with any problem in quantum mechanics the Hamiltonain and Lagrangian methods should both be used. What we are developing now is an entirely new type of Lagrangian method, not the usual Lagrangian methods well known in quantum mechanics.
Sent: 06/03/2017 10:54:20 GMT Standard Time
Subj: Re: 372(2): Analytical Solutions
For a comparison with the analytical solutions, the Runge-Kutta solutions have to be normalized first and then compared. I can do that tonight or tomorrow. I remember that in Atkins or other literature not all radial wave functions were given correctly.
For other atoms than Hydrogen, the energy eigenvalues are not known. Normally one uses a numerical method to vary the energy until converging wave functions appear. If this can be simplified it would be a huge progress.
Am 06.03.2017 um 11:41 schrieb EMyrone:
The analytical solutions of the Lagrangian quantum mechanics of H, (Eq (1) of the attached) are related to the associated Laguerre polynomials as in Eq. (16). Maxima can probably work out all the analytical radial wavefunctions of atomic H for any valid n and l. This has been done many times by Horst Eckardt in previous UFT papers using the hydrogenic wavefunctions. This note checks that the Lagrangian and Hamiltonian quantum mechanics give the same result for H. The Lagrangian method may have computational advantages over the well known Hamiltonian method. That is the purpose of this work. To check the Runge Kutta integration of Eq. (1) with Maxima use the analytical results. The purpose of this is to check that the code works for H, then apply and develop the code for other atoms and molecules. Eq. (1) also holds for hydrogen like atoms, a term used in computational quantum chemistry (for example the alkali metals sodium, potassium and so on). I have used some interpretative material from Atkins, and I hope that he has not made an error. His calculations can be checked by computer algebra.