Many thanks again for these computer checks, with which I agree and which will be used in the final paper. The final result, Eq. (18), is the same, and was used in Note 331(3) in the discovery of relativistic Zeeman splitting, another of many effects on the list of new experiments, bearing in mind that this is the simplest possible theory (of atomic H). In Eq. (8), I used <L sub Z> = h bar m sub L, so the result in Eq. (8) follows, i. e. h bar m sub L can be taken outside the integral, i.e. integral psi* T L sub Zpsi d tau = L sub Z integral psi* T psi d tau. = L sub Z < T > . Finally L sub Z psi = h bar m sub L psi.
Sent: 29/10/2015 11:15:26 GMT Standard Time
Subj: notes 331(1,2)
Note 1 describes an interesting extension of spectra induced by relativistic magnetic effects.
Note 2: Eq.(9) is the Rydberg energy which is from Wikipedia:
This is different from eq.(9). There is no c^2 and m appears in the numerator. The correct result and all constants are listed in the attachment. The additional term in (12) is positive. The modulus of <H1> is enlarged.
How did you come to eq.(8)? It seems that you equated the expectation value of an operator product with the product of the single expectation values.
You can find all constants in the attachment. The numerical result in (18) is correct.