OK many thanks! I assume that these are Eqs. (17) to (22) of Note 371(4) on three dimensional orbits. By all means change the sign and proceed as you suggest. As usual the results will be full of interest. All the latest papers of the UFT series are already very popular because of the new method of solving differential equations and simultaneous differential equations of all kinds (early morning reports on the blog).
EMyrone@aol.com
Sent: 01/03/2017 11:38:02 GMT Standard Time
Subj: Re: 371(6): Wavefunctions from a Lagrange Method
I programmed eqs.(19-22). There is the problem that theta dot is always positive, i.e. there are no bound states (ellipses) possible. One would have to change the sign of the square root in dependence of theta. The results show an example for unbound states.
It is probably better to use the standard formulation of the problem in spherical polar coordinates and compute beta a posteriori.Horst
Am 28.02.2017 um 14:52 schrieb EMyrone:
This note begins a new numerical development of quantum mechanics and relativistic quantum mechanics starting from the classical lagrangian, then quantizing the results. This first note is a simple idealized atom modelled by an electron orbiting a proton in a plane. It could also be applied to quantize the usual conic section orbit of a mass m orbiting a mass M in a plane. After quantization, two simultaneous equations (17) and (18) are found. These can be solved for psi(r) and psi(phi). In the Born Oppenheimer approximation the complete wavefunction is psi = psi(phi)psi(r). It may be argued that these can be found analytically with well known methods, but the advantage of this numerical method is that it can be extended to three dimensions and to a new development of relativisic quantum mechanics. It gives a new method for computational quantum chemistry in general, given the supercomputer power. The Dirac equation, for example, can be solved in a new way. The Maxima code is very powerful and contemporary desktops are also very powerful. We begin with this baseline problem so that the numerical results can be checked against known analytical results, notably the non relativistic orbitals of the lodestone of quantum mechanics, the H atom. Numerical mthods such as these can also be applied ot the ECE wave equation inferred in 2003.