OK thanks, the easiest way forward is to simply use the lagrangian method for finding the classical r dot, phi dot and theta dot, and just to use these results as a supplement or addition to the well developed hamiltonian method of quantum mechanics for each atom or molecule considered in computational quantum chemistry. It is not worth the extra coding effort because these latest papers are designed to apply lagrangian theory to various problems using the Maxima method. So in summary, the Maxima method is good for classical orbits of all kinds, but the usual hamiltonian method is better for quantum mechanics. If you wish to use the scatter plot method out of curiosity then that is fine. So now I will proceed to writing up UFT372, the central result of which is the ability to produce a precessing orbit from the lagrangian alone, without any other input. I suggest that we compare the results with data on precession, for example that of the earth or mercury. This comparison was not easily possible with UFT328, but it is possible in UFT372. So that is clear progress.
Sent: 10/03/2017 19:16:39 GMT Standard Time
Subj: Re: 372(7): Non Relativistic Limit of Note 372(7)
The problem is that r_dot, phi_dot and theta_dot are primarily known on a grid of argument t. For solving the quantized equations (7-9) they have to be transformed to a grid of
This is possible by the so-called scatter plot method I used in paper 328 but then the routines of Maxima cannot be used directly. We have to use the hand-programmed Runge-Kutta solver I used for paper 328. This is connected with more effort since a sophisticated interpolation scheme has to be used.
Am 10.03.2017 um 14:57 schrieb EMyrone:
This looks promising, there is the time dependent Schroedinger equation (2) and the first order differential Eqs. (7) to (9) which can be solved simultaneously by Maxima. The numerical results can be checked with the analytical results. The radial wavefunctions psi(r) are the modified Laguerre polynomials and the angular psi (theta, phi) are the spherical harmonics. This looks like a new ab initio method which can be straightforwadly extended to helium without using perturbation theory. In the relativistic case it seems that the equivalents of Eqs. (7) to (9) are also soluble with Maxima for hydrogen. Then spin can be introduced with the Pauli matrices, and at a final stage the Dirac type wavefunctions introduced. The great advantage is the use of the Euler Lagrange equations to give r dot, theta dot and phi dot.