Many thanks again! These are excellent results which solve the problem completely. They elegantly complement the results obtained in UFT270 using the Maxima Runge Gutta numerical integrator. The existence of bound and unbound orbit states is very interesting. This is again a completely new discovery. The results for planar orbits are self consistent, there are ellipses and oblique ellipses in a plane. The typographical errors can be corrected as usual for the final paper. It will be very interesting to look for a precessing elliptical orbit and to find the conditions, if any, that give such a result if it appears. It will also be full of interest to graph the features of a three dimensional orbit in different ways. In general, all orbits are three dimensional. These numerical results have been found using r, theta and phi as the proper Lagrange variables, and solving the three Euler Lagrange equation simultaneously with numerical methods. I agree that with this powerful new numerical method, beta will not give new information, because beta is defined in terms of theta and phi. The beta parameter was introduced in UFT270 because an analytical solution was not possible at that stage if r, theta and phi were used as the proper Lagrange variables. However, the problem has been elegantly solved numerically now by Horst, so beta is no longer needed. In some problems however, the use of r and beta as the proper Lagrange variables is very useful, and leads to an entirely new quantum mechanics as in note 371(7). Eqs. (19) to (22) of Note 371(4) are the same as in UFT270, but this method is now obsolete. Also full of interest is that all these results can be quantized, giving an entirely new quantum mechanics.
Sent: 02/03/2017 23:27:58 GMT Standard Time
Subj: Computaton of Spherical Orbits
After a lot of tests the result is the following:
I used the equations of motion from the Lagrangian without pre-defined constants of motion. The equations for theta, phi, r are o8, o11, o14 in the protocol. These are of second order. I added the first-order equation for beta as an additional equation, see o15. There are typos in some notes concerning the missing square root.
The results are as follows: Bound and unbound states are obtained. For the initial conditions
theta dot = 0
theta = pi/2
a planar orbit in the XY plane comes out as expected. If the initial condition for theta dot is chosen different from zero, an oblique planar orbit appears, see Fig. 4. From Fig. 3 can be seen that for all planar orbits
beta = phi.
There is only a constant offset if the initial condition of beta is selected different from zero. This means that no additional information is obtained from beta. This may be different for non-planar orbits.
Fig. 5 shows the constants of motion L and L_phi computed from the resulting orbit. They are constant as expected, and for an orbit in the XY plane comes out
L = L_phi
as expected. So far we have arrived at a consistent state. I will check if the alternative equations (19-22) of note 371(4) will give the same results with adopted L and L_phi.
Am 02.03.2017 um 09:37 schrieb EMyrone:
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).
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.
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.