Many thanks again, this is indeed interesting, another completely new result and excellent computer work! I am in favour of the modulus method because of Eq. (1) of Note 331(2). In that equation it is possible as a first step to use the classical interpretation of p0 squared and the quantum interpretation of orbital angular momentum, L0 psi = h bar m sub L psi. Finally use the expectation value of kinetic energy for the H atom energy levels. This gives the same result as your modulus method. So Note 331(3) is correct. Your excellent computer algebra shows that there is a sign change in the brackets of the right hand sides of Eqs. (8), (12) and (13) of Note 331(3). This will of course be fixed in the final paper as usual. The relativistic Zeeman splitting occurs in the megahertz region and is in fact a new kind of ESR and NMR. The main line is the 656.3 nm Balmer line. Non relativistic Zeeman theory splits this into three lines as is very well known, one line at the same frequency of 653.3 nm, one at higher and one at low frequency, equally spaced – th efamous Zeeman effect which won one of the first Nobel prizes. Relativistic Zeeman effect theory results in splitting of each of the three lines giving an entirely new spectroscopy. Even if we just use the classical p0 squared in Eq. (1) of Note 331(2), the new relativistic splitting exists. The classical result is the very well known p0 squared = 2m(H0 – V), where H0 is the classical hamiltonian and V the potential energy between the electron and proton in the H atom. The classical hamiltonian of the H atom is the expectation value of all teh textbooks, proportional to 1 / n squared, where n is the principal quantum number. In complex atoms and molecules a very rich and totally new spectrum emerges. The experimental challenge of course is how to measure it. Perhaps something like visible / radio frequency double resonance would be able to measure the new spectrum, or an adaptation of a Fourier transform ESR or NMR spectrometer. With the gear they have now this should be feasible. In computational quantum chemistry, spectra could be computed for any atom or molecule, and catalogued.

EMyrone

Sent: 29/10/2015 17:18:50 GMT Standard Time

Subj: Re: 331(4): Testing the Operator ApproximationI did some calculations. I computed

<Ekin>, <L_Z>, <Ekin*L_Z>, <L_Z*Ekin>,

see last page of protocol (this is not complete for any printing reasons).

There is an interesting result: The operator expectation values <Ekin*L_Z> and <L_Z*Ekin> give the same result, i.e. Ekin and L_Z are commutable. However there is a phase shift with the product of the single expectation values:

– i <Ekin> * <L_Z> = <Ekin*L_Z>

I am seeing two different interpretations. Either we say that <Ekin*L_Z> is imaginary and not physical, or we take the modulus and the original exp. value at the rhs can be computed by the lhs with omitting the phase factors and is exact. Then your “approximation” is exact.

Horst

Am 29.10.2015 um 15:48 schrieb EMyrone:

The rigorously correct expectation values can be computed from Eq. (10) adn compared with the approximation (2). In general there are terms given in Eq. (7), leading to a very rich spectrum. Only one of these terms has been used so far.