note 87(4)

Subject: Fwd: note 87(4)
Date: Thu, 28 Jun 2007 06:07:04 EDT

This is an interesting discussion, it brings me a clearer view of the background. I agree that the probability density has to be defined in such a way that interaction with r(vac) is described appropriately. This is different from the standard definition. Concerning my second suggested method you are right, I should use the replacement   r –> r+r(vac)  in all terms, not only on the right-hand side.

Horst

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Thanks for these comments, these definitions depend on what one is interested in. As you can see from Atkins' discussion of the H atom, we use R if we are interested in finding the most probable POINT at which an electron is found, and we use r squared R if we want to find the probability of finding the electron in A SPHERICAL SHELL. As you can see from Atkins, the radial distribution function is a maximum at the Bohr radius, the radius of the lowest energy orbit of atomic hydrogen. The electron is in this orbital, so the probability of finding it in this orbital is 4 pi r squared psi (squared). If we can replace psi by R then this becomes 4 pi r squared R squared, and this replacement of psi by R is possible because of the centrifugal term, so we don't have to worry about theta and phi.  The advantage of using r squared R is that the effect of r(vac) can be be incorporated directly in the charge density and driving force via

 

                                           r goes to r + r(vac)

 

If R squared is used there is no effect of r(vac) and no oscillatory driving force from the Lamb shift because R is still hydrogenic and unperturbed by the vacuum. For H this is accurate to 4 parts in ten million.The basic method is to assume that the radiative correction changes r of the orbital to r + r(vac), so I defined the charge density in terms of r + r(vac) using the radial distribution function. This charge density is the most probable charge density of the electron in a given orbital, it wobbles in its orbital because of the (cosinal) zitterbewegung from the radiative correction. So this is our cosinal driving term. If charge density is defined in terms of the radial part of the wave function, it becomes a maximum at the nucleus. However, the electron is never found at the nucleus, it is found in its orbitals. I agree that it is OK to use r squared R rather than r squared P, I think that is just a matter of choice, or definition. 

 

Eq. (23) of 87(4) should be regarded as the equation that defines the driving term initially. Then this driving term is used in eq. (22) of 87(4) to amplify phi. Finally this amplified phi is used in the densiy functional code to show that the electron is released by resonance amplification. No matter how small the initial driving term, resonance occurs. This is the very important point.  

 

It must be remembered that eq. (23) is an approximation, although a very good one for H, because R(r) is considered to be the hydrogenic radial wave function. Eq (22) is used to find the amplified phi, which is then inputted back in the Schrodinger equation as in our previous work. A more self-consistent scheme would need the simultaneous solution of eqs. (12), (22) and (23), but eqs. (12) and (23) are still approximations.

 

In your suggested method this morning, why is r changed to r + r(vac) only on the right hand side of the EB equation? I think that r should be changed to r + r(vac) throughout the EB equation. If this is done your method seems to be a good one. I will have a look at this method in more depth and report back. This is a very good discussion because improvements are being made all the time.

 

 

 

  

 

 

 

     

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note 87(4)

Datum:

Thu, 28 Jun 2007 2:12:13 AM Eastern Daylight Time

This looks indeed more consistent than in the earlier version. What I am concerned on is the definition of the probability density. You have defined it on base of the function P which is unusual. To my knowledge the density is always defined by the radial functions R even if the eqaution with the centrifugal term is solved.

Taking the radial distribution function r^2 P or r^2 R  instead of R for the charge density is also an unusual form. I wrote down a possible alternative in my second suggestion where a weight factor r^2 comes in in another way. In the first suggestion I describe a possible method to come to a self-consistency cycle, althouhg it is not clear if the alpha factor here is “strong” enough to initiate resonance. In the second solution there is no direct self-consistency cycle, but this should be sufficient to detect resonance enhancements as you described originally in the note.

 

Horst

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I sent over a suggested direct method, so both methods can be tried in paper 87. There are probably many resonances as in paper 63 combined with paper 85 and 86. The key advance would be to write this into density functional code.