Archive for the 'Mathematics' Category

Nov 14 2021

Greatest Lower Bound

One would guess that the particle physicists and quantum field theorists may like a 1.3335 x 10-15 m diameter of a free electron, because it is closer to a point particle than many estimates of the diameter.  It is possible that 1.3335 x 10-15 m is also the limit inferior of the sequence S137 to Sn in an atomic orbital.

The maximum diameter, on the other hand, will depend on the element and on the orbital.  At a spin flip, electrons in all orbitals may reduce to 1.3335 x 10-15 m, before taking off on a new trajectory and increasing in diameter again.  We cannot speak of a limit superior of the sequence of diameters of the electron in an atomic orbital nevertheless.  That will depend on the direction of electron travel, and on whether the atom is at the surface of the earth, or at some other planet.  For the latter, it depends on the density of the gravitational field.

We may also ask whether 1.3335 x 10-15 m is the greatest lower bound at all locations in the universe.  This raises the question of whether the fine structure constant is a universal constant, or whether or not the Coulomb gauge is the same everywhere.

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Oct 24 2020

A Number

Published by under Mathematics

Fifteen years ago today I started an attempt to calculate gravity, after coming up with the concept on August 10th of the same year.  With a web search of “diameter of the electron”, reference number 3 of my April 2007 paper was found.  The link does not come in anymore, though there would be other places where Ernest Rutherford’s 1914 publication “The Structure of the Atom” can be found.

Starting with the diameter of the electron as the wavelength of a photon, I used hc/λ to produce an energy in Joules.  In the April 2007 paper it says: “Noticing that this number is on the order of the gravitational constant, it becomes worthwhile to proceed …”.

Sometime in 2007 or 2008 I sent a copy of the paper to Professor Converse Blanchard, with a note in the front thanking him for teaching me physics.  He sent the paper back to me with a cordial note in the front and a few markups throughout the paper.

One marking, at the spot noted above, was: “The grav constant is not an energy, and so this coincidence is without meaning.”  My answer to this is that a number is a number, – it has no units.  It is as Wilfred Kaplan states in Advanced Calculus on page 6: “We stress that det A is a number, …”  Otherwise, in the case of hc/λ we would speak of an “energy” and not a “number”.

It is worth noting that most of Professor Blanchard’s comments were constructive.  Later in the paper, relating to the quark coincidence, he wrote: “amusing!”.

My studies lately have been mostly in mathematics.  I have four books that are specifically on the topic of Advanced Calculus as well as several other math books.  Once in a while, I go back to a physics book and things come back quickly with my old markings and tabs sometimes leading the way.  Can do the same with engineering books, such as that Tds = du + Pdv.

Nineteen days after October 24, 2005, the calculation was made complete with
G = 4hf/3 m3 kg-1 s-2.  As can be seen in the diagram at the top of the blog, the units on the constant 4/3 are m/kg2.

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