The proposal that there is no magnetic field inside the orbitsphere appears to be an essential novel element in the above interpretation of GUTCP.
Such a hypothesis strengthens the case for calculating multi-electron atoms "from the inside out", since outer electrons then have neither electrical nor magnetic influence on inner electrons. This also naturally leads to the conclusion that electrons are not 'paired' to a single radius, supporting the Mod 1 theory.
However, how would this impact the force balance of muonium (μ+e-)?
(Muonium may seem a tad exotic, but it is actually a very pure, two-orbitsphere system to test GUTCP and alternatives for basic consistency: the rules that apply to electrons should equally apply to its cousins.)
In Mills' theory, the existence of magnetic (but not electric) fields within the electron orbitsphere provides the balancing force (GUTCP eq 2.234) for the muon orbitsphere and determines its size as 9pm, with the electron radius being 53pm. What would be the driver for the muon to even curve into an orbitsphere, if it weren't for the magnetic forces of a surrounding electron?
I think there must be magnetic fields inside the orbitsphere as Dr. Mills proposes. I believe each component of the great circle 'electron' sees the other electron as if it is at the center. I believe Dr. Mills gets that from Purcell's book Electricity and Magnetism chapter 11.
I agree we need a GUTCP and I applaud your efforts but still have some issues. As I understand it in SQM, the electron in a Hydrogen atom is most likely to be at the Bohr radius, not at zero because that is the peak of the radial probability density. Also, regarding the issue of whether the two electrons in Helium have the same energy, that's not the issue, it's that it takes 24.6ev to ionize one of them. I don't think SQM says both electrons are at the same energy or have to be because the wavefunction is for both together. What SQM says is that the total binding energy is ~ -79ev. And yes, one has to jump through hoops to get anywhere close with SQM.
Any version of GUTCP must also explain the ordering and filling of electron levels of multi-electron atoms and be consistent with the spectroscopic data. How do you align your model with S,P, D and F states? Do you predict the same order of filling higher states that Dr. Mills and SQM for that matter?
Finally, I was really hoping you would be able to help us understand how to compute the magnetic forces for multi-electron atoms as Mills' text is hard to follow why the different magnetic forces exist for different electron atoms. For example, why the magnetic forces for 17 electron atoms is different than 18 electron atoms and how to generate them in a consistent and logical way either for Mills' version or yours. Thanks.
As for your first question, the radial distribution function for s-electrons in SQM peaks at r=0. That is the maximum value of Psi^2 , where Psi is the SQM wave function, and Psi^2 is the time independent occupancy probability at a point for the ground state of hydrogen , is at r=0. Psi gets smaller and smaller as r increases.
The statement that it peaks at r=a0 is based on gross probability at a given radius. That is, the function 4*(pi)*(r^2)*(Psi^2), is maximum at r=a0. Note, 4*(pi)*(r^2)*(Psi^2) is not the same as (Psi(r))^2. The former is not a volumetric probability.
The added component 4*(pi)*r^2 accounts for the fact that the function is symmetric around the nucleus/origin, and the AREA that is 'encompassed' by 'r' increases according to the standard geometric relationship between radius and surface area of a sphere, area=4*pi*r^2....
A misinterpretation of this symmetry function is sometimes, and sometimes deliberately, misconstrued to argue that the most likely 'position' of the electron is at r=a0. In fact, the function yields the most likely radius.
A true probability comparison must be based on equivalent encompassed volumes. That is the comparison should be made on a probability/volume basis, so-called 'probability density'. Let's do a thought experiment: imagine a small spherical volume, r<<ao, at several radial values. It must be the SAME absolute spherical volume at each radius. The probability of an electron being in each of these volumes is a function of how far the test volume is from the nucleus/origin. It will decrease continuously as the radius increases.
Why? The relative probability in each volume will be proportional to AvgPsi^2, where AvgPsi is the average value of the wave function in that small volume. Squaring the avg value of Psi is a very good estimate of probability density in the volume if the radius of the little volume is <<a0.
As Psi gets smaller with increasing radius, Psi^2 also gets smaller...In sum, the statement in the text that the most likely position of the electron is in the vicinity of the nucleus is correct.
As for your other questions, they will be addressed in later chapters.
I'm having some difficulty with equation 4-8 due to the source text (ST) equation (1.261) where:
V = -Z e^3/(4 Pi e0 r1) = -Z^2 e^2/(4 Pi e0 a0)
why is Z squared in rightmost expression and not in the left?
These expressions are identical in all other respects, except for the substitution of r1->a0 which should be identical as well. I can't find a way of deriving one from the other.
aH is not the same as a0 because aH uses the reduced electron mass μe which is defined in terms of the electron mass, me, and proton mass, mp, in 1.255 and 1.258 yielding:
There is no reduced mass as typically defined in GUTCP since all masses are centered on the nucleus. That factor comes in for the Hydrogen atom because of the unique magnetic term which is different for Helium, as I understand it.
I'm not sure what your main point is. I don't expect 100% consistency over all Mills' tables or calculations. I view it more as an approach that could still use some fine tuning. My main interest is to understand the magnetic forces.
I believe I've discovered part of the discrepancy. There are two senses of the term "Ionization Energy": 1) The energy required to free the electron from the proton. 2) The energy required to elevate the orbital of electron to the next orbital. Only in sense #1 can "Ionization Energy" == "Binding Energy".
The sense of #2 might be called "activation energies". So, given this clarification it appears the calculation as given both in this article (4-8) and in the GUTCP book's relativistically corrected value (I-98) are outside the error tolerance of the empirical value. I don't know what other corrections might apply, such as fine structure+hyperfine structure+lamb shift+proton radius+further dirac corrections+higher order lamb shift, etc., but it would be nice to see enough orders of approximation that it brings the calculated value within the experimental value tolerance.
Excellent piece of writing and analysis. It's good that Dr Phillips sets out his own analysis and views and challenges as well as supports GUTCP. Theories need to be properly evaluated and challenged if they are to mature and grow.
The proposal that there is no magnetic field inside the orbitsphere appears to be an essential novel element in the above interpretation of GUTCP.
Such a hypothesis strengthens the case for calculating multi-electron atoms "from the inside out", since outer electrons then have neither electrical nor magnetic influence on inner electrons. This also naturally leads to the conclusion that electrons are not 'paired' to a single radius, supporting the Mod 1 theory.
However, how would this impact the force balance of muonium (μ+e-)?
(Muonium may seem a tad exotic, but it is actually a very pure, two-orbitsphere system to test GUTCP and alternatives for basic consistency: the rules that apply to electrons should equally apply to its cousins.)
In Mills' theory, the existence of magnetic (but not electric) fields within the electron orbitsphere provides the balancing force (GUTCP eq 2.234) for the muon orbitsphere and determines its size as 9pm, with the electron radius being 53pm. What would be the driver for the muon to even curve into an orbitsphere, if it weren't for the magnetic forces of a surrounding electron?
I think there must be magnetic fields inside the orbitsphere as Dr. Mills proposes. I believe each component of the great circle 'electron' sees the other electron as if it is at the center. I believe Dr. Mills gets that from Purcell's book Electricity and Magnetism chapter 11.
I agree we need a GUTCP and I applaud your efforts but still have some issues. As I understand it in SQM, the electron in a Hydrogen atom is most likely to be at the Bohr radius, not at zero because that is the peak of the radial probability density. Also, regarding the issue of whether the two electrons in Helium have the same energy, that's not the issue, it's that it takes 24.6ev to ionize one of them. I don't think SQM says both electrons are at the same energy or have to be because the wavefunction is for both together. What SQM says is that the total binding energy is ~ -79ev. And yes, one has to jump through hoops to get anywhere close with SQM.
Any version of GUTCP must also explain the ordering and filling of electron levels of multi-electron atoms and be consistent with the spectroscopic data. How do you align your model with S,P, D and F states? Do you predict the same order of filling higher states that Dr. Mills and SQM for that matter?
Finally, I was really hoping you would be able to help us understand how to compute the magnetic forces for multi-electron atoms as Mills' text is hard to follow why the different magnetic forces exist for different electron atoms. For example, why the magnetic forces for 17 electron atoms is different than 18 electron atoms and how to generate them in a consistent and logical way either for Mills' version or yours. Thanks.
Thanks for your comment Robert.
As for your first question, the radial distribution function for s-electrons in SQM peaks at r=0. That is the maximum value of Psi^2 , where Psi is the SQM wave function, and Psi^2 is the time independent occupancy probability at a point for the ground state of hydrogen , is at r=0. Psi gets smaller and smaller as r increases.
The statement that it peaks at r=a0 is based on gross probability at a given radius. That is, the function 4*(pi)*(r^2)*(Psi^2), is maximum at r=a0. Note, 4*(pi)*(r^2)*(Psi^2) is not the same as (Psi(r))^2. The former is not a volumetric probability.
The added component 4*(pi)*r^2 accounts for the fact that the function is symmetric around the nucleus/origin, and the AREA that is 'encompassed' by 'r' increases according to the standard geometric relationship between radius and surface area of a sphere, area=4*pi*r^2....
A misinterpretation of this symmetry function is sometimes, and sometimes deliberately, misconstrued to argue that the most likely 'position' of the electron is at r=a0. In fact, the function yields the most likely radius.
A true probability comparison must be based on equivalent encompassed volumes. That is the comparison should be made on a probability/volume basis, so-called 'probability density'. Let's do a thought experiment: imagine a small spherical volume, r<<ao, at several radial values. It must be the SAME absolute spherical volume at each radius. The probability of an electron being in each of these volumes is a function of how far the test volume is from the nucleus/origin. It will decrease continuously as the radius increases.
Why? The relative probability in each volume will be proportional to AvgPsi^2, where AvgPsi is the average value of the wave function in that small volume. Squaring the avg value of Psi is a very good estimate of probability density in the volume if the radius of the little volume is <<a0.
As Psi gets smaller with increasing radius, Psi^2 also gets smaller...In sum, the statement in the text that the most likely position of the electron is in the vicinity of the nucleus is correct.
As for your other questions, they will be addressed in later chapters.
I have some further thoughts but feel it best to continue this dialog in private if you allow or agree to that. If not, that's ok.
I've duplicated the calculations through (4-22) in this Mathematica notebook:
https://github.com/jabowery/GUTCP/blob/main/GUTCPHydrinoHypothesis.nb
I'm having some difficulty with equation 4-8 due to the source text (ST) equation (1.261) where:
V = -Z e^3/(4 Pi e0 r1) = -Z^2 e^2/(4 Pi e0 a0)
why is Z squared in rightmost expression and not in the left?
These expressions are identical in all other respects, except for the substitution of r1->a0 which should be identical as well. I can't find a way of deriving one from the other.
Please look at equation 1.257 which states r1=a0/Z.
Thanks.
That leaves another question open which is:
Why not 1.260 which is r1=aH/Z
aH is not the same as a0 because aH uses the reduced electron mass μe which is defined in terms of the electron mass, me, and proton mass, mp, in 1.255 and 1.258 yielding:
μe = (me mp)/(me+mp)
There is no reduced mass as typically defined in GUTCP since all masses are centered on the nucleus. That factor comes in for the Hydrogen atom because of the unique magnetic term which is different for Helium, as I understand it.
GUTCP Table 1.2 potential and kinetic energies are calculated using aH (from reduce mass):
-27.16 eV and 13.598 eV
GUTCP Table 1.3 potential and kinetic energies are calculated using a0 (from 1.261 and 1.262):
-27.21 eV and 13.61 eV
I'm not sure what your main point is. I don't expect 100% consistency over all Mills' tables or calculations. I view it more as an approach that could still use some fine tuning. My main interest is to understand the magnetic forces.
Where can I find the experimental uncertainty in hydrogen's ionization energy?
Using Mathematica's tables and evaluating your (4-8) the NON-relativistically corrected number is:
13.60569312(3±5)
Using the GUTCP book (I.98) the relativistically corrected number is:
13.60587425(8±6)
Asking GPT4 if there is a NIST-accepted uncertainty it said:
1312.0 ± 0.6 kJ/mol = 13.598 ± 0.006 eV
I believe I've discovered part of the discrepancy. There are two senses of the term "Ionization Energy": 1) The energy required to free the electron from the proton. 2) The energy required to elevate the orbital of electron to the next orbital. Only in sense #1 can "Ionization Energy" == "Binding Energy".
Here's a good web page on "ionization energy" in the sense of #1 (energy to free an electron from the proton): https://socratic.org/questions/594416b1b72cff161b4d924f
The sense of #2 might be called "activation energies". So, given this clarification it appears the calculation as given both in this article (4-8) and in the GUTCP book's relativistically corrected value (I-98) are outside the error tolerance of the empirical value. I don't know what other corrections might apply, such as fine structure+hyperfine structure+lamb shift+proton radius+further dirac corrections+higher order lamb shift, etc., but it would be nice to see enough orders of approximation that it brings the calculated value within the experimental value tolerance.
Keep up the good work 👌
Excellent piece of writing and analysis. It's good that Dr Phillips sets out his own analysis and views and challenges as well as supports GUTCP. Theories need to be properly evaluated and challenged if they are to mature and grow.