Discussion of Note 372(1)

Fully agreed, by coincidence I sent over the analytical functions and checked that the result is right. This is very good computational work as usual.

To: EMyrone@aol.com
Sent: 06/03/2017 09:41:33 GMT Standard Time
Subj: Re: 372(1): The Fundamental Wave Equation of Lagrangian Quantum Mechanics

The integration has to start from a point r>0 since r=0 is a singularity as you mentioned rightly. I used r=0.001 . In Fig. 1-a (for the 1s state) with expanded r range about 0 you see that partial psi/partial r (blue curve) was not chosen appropriately for the first mesh point but it changes rapidly to the correct values. You have to guess the derivative for the first mesh point. Here it was -1, then the result of Fig. 1b appears, obviously right. For p states I could not find the correct derivative for the first mesh point but this is not relevant as could be seen from the graphics I sent over. One would have to use values from the analytical solutions in the H case. In professional software packages a logarithmic mesh is used and integration is sometimes done from outer to inner region so that this problem does not appear. I think we can say that this baseline test is successful.

Horst

Am 06.03.2017 um 09:12 schrieb EMyrone:

I think that these results are acceptable in the r to infinity limit, because the equations being used are exactly equivalent to the usual hamiltonian method as shown in UFT371. This note is meant to be a test of the new Lagrangian method, to check that it gives the right H wavefunctions.. I agree that the interesting development will be for more complicated atoms and molecules, starting with helium. It is already clear that all the highly developed methods of computational quantum chemistry can be adapted for the new Lagrangian method, there are many books of code and many code libraries as you know. The advantage is that the problem consists of only two proper Lagrange variables, r and beta. I agree that beta represents a tilted ellipse, again very interesting for tilted orbits in the solar system. This new lagrangian method should be regarded as being complementary to the hamiltonian method. For states of zero angular momentum quantum number, l = 0, the second term in Eq. (29) is zero. I will check, but Eq. (28) is the same as Eq. (4.3.4) of Atkins, second edition of “Molecular Quantum Mechanics”. This time Atkins manages to get it right, but as we know his section on quantum tunnelling is completely wrong (the computer algebra showed this). So the Lagrangian method produces all the radial wave functions correctly because for H it is equivalent to the hamiltonian method. This is the “base line” check. I cannot see rapid oscillations in the protocol, can you please indicate where they are? In the case of l = 0 the ellipse becomes r = 0 and an infinity develops in Eq. (28). The same problem exactly is present in the usual hamiltonian method but methods are available to deal with it.

To: EMyrone
Sent: 05/03/2017 16:41:38 GMT Standard Time
Subj: Re: 372(1): The Fundamental Wave Equation of Lagrangian Quantum Mechanics

I programmed the solution of eq.(28,29). The good message is: normalizable functions are obtained with psi –> 0 for r –> inf. The number of extrema is consistent with the known Hydrogen wave functions.
The bad message is: from eqs.(21,22) we obtain for s functions (l=0):

alpha = 0
epsilon = 1

These are parabolas (maybe alpha=0 is not allowed for a parabola). The numerical solution looks reasonable but there is no horizontal tangent for r –> 0 as has to be for s-like functions. Also for the other functions the initial behaviour near to r=0 is not regular but switches quickly to the desired behaviour. See strong oscillations of

partial psi/partial r

in the graphs near to r=0.

Horst

Am 05.03.2017 um 12:53 schrieb EMyrone:

This fundamental wave equation is made up of Eqs. (29) and (30), which give the radial wavefunctions in terms of the Bohr radius. The angular wavefunctions are the spherical harmonics. The three dimensional ellipse (8) of Lagrangian quantum mechanics is defined by the quantized half right latitude (21) and the quantized eccentricity (22). These equations agree with the results of previous work of the UFT series, notably UFT266 and UFT267. These results are for the H atom with Coulomb potential (7) describing the interaction of one electron with one proton. In general there are many electons and many protons (and also neutrons), and this method can be extended to describe the general atom or molecule, building up a new computational quantum chemistry. In the well known hamiltonian approach to quantum chemistry, only the H atom is analytical. All other atoms and molecules are developed with approximations or by computation. As Horst showed in UFT371, the Lagrangian method is much simpler than the Hamiltonian method for the H atom, and the same is true for other atoms and molecules.

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