Many thanks again! Your very interesting computation can be used in UFT371 and crossed with Note 371(8), which shows that the new Lagrangian method gives the correct hydrogen atom wavefunctions. Your computation is for the radial wavefunction psi = R. The complete wavefunction as you know is psi = RY, where Y are the spherical harmonics. This completely new subject area can be called “Lagrangian quantum mechanics”.
Sent: 03/03/2017 11:16:21 GMT Standard Time
Subj: Re: 371(7): New Radial Wavefunctions of Atoms and Molecules
The Schrödinger-like eq.(12) has been solved numerically. The standard method in computational physics is to integrate the equation for a grid of energy values (here represented by L, alpha and a) and find non-diverging solutions for r –> inf. These are the eigen states. Here I used the direct integration with Maxima for given parameters. It can be seen that the solutions diverge in general. In FigD and FigE two solutions for L=1 and L=5 are shown. These (arbitrary) values correspond roughly to the number of extrema of psi, representing the angular momentum eigen state, as is the case for the physical, convergent solutions obtained with other methods.
Am 02.03.2017 um 15:40 schrieb EMyrone:
These are illustrated for the hydrogen atom H, and are found by solving Eq. (12) with Maxima, an equation obtained from the lagrangian. This is a simpler method than the usual one, based on the hamiltonian. Eq. (12) can be transformed into a differential equation in phi and theta using the conic section in beta, Eq. (2), and the definition of beta, Eq. (4). That gives new angular wave functions for atoms and molecules.