This is good, there are four equations in four unknowns, the generalized Lagrange variables of relevance. This discussion clarifies the meaning of degree of freedom in Lagrange theory and checks the basic concepts. Therefore the classical momentum can be found from the Euler Lagrange equations and the trajectories of r1, r2, beta1 and beta2 can be found. Also, their time derivatives can be found. Finally r sub 12 and its time derivative can be found. Quantization can then proceed and the energy levels found. The well known Hamiltonian approach to the helium atom introduces Coulomb and exchange integrals, Fermi theory and the Pauli exclusion principle as is well known to all first year students of chemistry. I was taught the subject in my first year by Mansel Davies, who had no grasp of quantum mechanics on the mathematical level. So it was a somewhat mysterious process, and I had to spend many hours in the library preparing notes. Mansel had no grasp of relativity at all. He seemed to approach things intuitively, a dangerous attitude in my opinion. One cannot get away with much without mathematics, especially in relativity and quantum mechanics, which are highly non intuitive. My third year undergraduate notes are on www.aias.us but my first and second year undergraduate notes seem to have been lost. The well known results of the Hamiltonian method should of course be used as a guide for the new Lagrangian approach, which does not use perturbation theory. It addresses the problem directly.
Sent: 06/03/2017 16:41:50 GMT Standard Time
Subj: Re: 372(3): Lagrangian Quantum Mechanics of the Helium Atom
I do not see why Lagrange multipliers should be used. The degree of freedom is 4 ( coordinates r1, beta1, r2, beta2), and this is independent from the space dimensions in Lagrange theory. I would not use the relative coordinate r12 here because it is derived from r1 and r2 and does not give an independent Lagrange equation.
Am 06.03.2017 um 15:56 schrieb EMyrone:
This sets up the problem in general. There are five Euler Lagrange equations in five unknowns. The space is only three dimensional, so Lagrange multipliers must be used as in molecular dynamics computer simulation of n interacting molecules. Having found the momentum classically it can be quantized to give the wavefunction of the helium atom without the use of perturbation theory. The Pauli exclusion principle can be used as usual. The next note will define the Lagrange multipliers.