Spherical harmonics are used to describe the wave function of the electron in a hydrogen atom, rpenner.
That
must be why I wrote:
Spherical harmonics are very important to the treatment of the analytical treatment of the quantum mechanics of a particle in a central potential which is the first degree of approximation used in textbook treatments of 1-electron atoms and ions like hydrogen, ...
Examples of hydrogen-like ions are $$\textrm{He}^{+}, \,\textrm{Li}^{++}, \, \textrm{Be}^{+++}, \, \dots$$ which validate the approximation of the 1-electron atoms as a particle in a spherical potential. Other details like spin-spin and spin-orbital interactions slightly alter this simple treatment to make the model correspond better with our understanding of electromagnetism and Dirac-modeled electrons. These improvements also lead to better agreement with the spectroscopic data of rarified atomic hydrogen gas and plasma of the other nearly completely ionized elements. Both electromagnetic effects and Pauli exclusion have removed the multi-electron atom from such simple analysis and so in general numerical methods and not expansion in terms of functions based on spherical harmonics is used in the fields of quantum and computational chemistry.
In short, unless you are comfortable in manipulating eigenfunctions and the approximations made in early quantum treatments of the atom, spherical harmonics do you no good and are not demonstrated as being applied by Farsight or Worsley. They are specifically used in the analytic treatment of the approximate hydrogen atom because when written in spherical coordinates this problem is seperable into a radial part and and an angular dependence, which is a topic in differential equations -- math that is not being used in Worsley's or Farsight's calculations.
See
this page which features a number of depictions.
Pointing to pictures does nothing to further your argument when I did not deny that spherical harmonics were used in some cases to study electron eigenfunctions.
And you have not "exposed the numerological trickery in earlier threads", you made an unsupported allegation in an attempt to discredit something you do not understand.
To review, from
post 180 of the thread "Lattices and Lorentz invariance" I demonstrated that there are physical expressions which are coincidentally close to a numerical value of 1 in SI units. This doesn't mean that they are 1 since they have units attached to them.
$$A = \frac{27 \pi^2 h }{4 m_{\tiny \textrm{electron}} c} \left( \frac{m_{\tiny \textrm{proton}}}{m_{\tiny \textrm{electron}}} \right)^3 \; \approx \; 1.000636 \, \textrm{m}$$
$$B = \left( \frac{m_{\tiny \textrm{electron}}}{m_{\tiny \textrm{proton}}} \right)^2 \frac{c}{9\pi^2} \; \approx \; 1.001062 \, \textrm{m} \textrm{s}^{\tiny -1}$$
But A not numerically close 1 in cgs or Imperial systems. A is about 100.06 cm or 39.40 inches or 0.00062 miles.
And B is not numerically close to 1 in other systems, being about 100.1062 cm/s or 2.239 mph.
And while $$4 \pi A B^{\tiny \frac{3}{2}} c^{\tiny -\frac{3}{2}}$$ is (by definition) the Compton wavelength of the atom $$\frac{h}{m_{\tiny \textrm{electron}} c}$$, the Worsley-derived expression $$4 \pi c^{\tiny -\frac{3}{2}}$$ is physically unrelated and has bizarre units of $$\textrm{m}^{\tiny -\frac{3}{2}} \, s^{\tiny \frac{3}{2}}$$.
Likewise $$\frac{1}{3 \pi} B^{\tiny -\frac{1}{2}} c^{\tiny \frac{1}{2}}$$ is the proton-electron mass ratio, but the Worsley-derived expression $$\frac{1}{3 \pi}c^{\tiny \frac{1}{2}}$$ is physically meaningless and has bizarre units of $$\textrm{m}^{\tiny \frac{1}{2}} \, s^{\tiny -\frac{1}{2}}$$. We don't have names for such collections of fractional units because nothing we observe has those combinations. But even if we did, that fact is $$\textrm{m}^{\tiny -\frac{3}{2}} \, s^{\tiny \frac{3}{2}}$$ is not a length and $$\textrm{m}^{\tiny \frac{1}{2}} \, s^{\tiny -\frac{1}{2}}$$ is not dimensionless.
From
Post 184 of that thread I demonstrated that two claims depend on $$D = \frac{4 \pi ( g_e - 1 )^8 c^3 e^2}{\varepsilon_0} \approx 0.99998 \, \textrm{kg} \, \textrm{m}^{\tiny 6} \, \textrm{s}^{\tiny - 5}$$ being conveniently close numerically 1 in SI units, but it is close to $$10^{15}$$ in cgs units.
I see that you remain stubbornly unaware that ...
Your misunderstanding of physics jargon in Wikipedia articles isn't at issue here. Your are operating below my tree-level understanding of quantum electrodynamics which covers all the ground exposed by those entries.
We have good experimental evidence that the electron is a standing-wave structure.
No -- we have good experimental evidence that the electron is well-modeled by a field of Dirac (massive) fermions coupled to a vector field of massless bosons. An electron in a well-defined energy state in a stationary potential is something that is well-modeled by a standing wave -- but that does not apply to
free electrons.
The metaphorical light of evidence more strongly favors the position that you choose to walk in darkness.
and in the light of how you deliberately ignore the dimensionality conversion factor n,
How have I ignored it, when I pointed out to you first that your expressions were garbage. Also, in this thread I directly addressed you new introduction of "n" when I wrote:
In the English Imperial system (where c is written as such-and-such miles per second, n is not 1, and so "Andrew"'s contention is that the SI units are physically special where there is no reason to suspect that they are.
Why we can even compute what n is from just my earlier posts. $$n=A^{\tiny -1} \, B^{\tiny -\frac{3}{2}} = \frac{4 m_{\tiny \textrm{electron}} c}{27 \pi^2 h } \left( \frac{m_{\tiny \textrm{electron}}}{m_{\tiny \textrm{proton}}} \right)^3 \; \times \; \left( \frac{m_{\tiny \textrm{proton}}}{m_{\tiny \textrm{electron}}} \right)^3 \frac{27\pi^3}{c^{\tiny \frac{3}{2}}} = \frac{4 \pi m_{\tiny \textrm{electron}} c}{h c^{\tiny \frac{3}{2}}} = \frac{4 \pi}{\lambda_{\textrm{Compton}} \, c^{\tiny \frac{3}{2}}$$ with units of $$\textrm{m}^{\tiny -\frac{5}{2}} \, \textrm{s}^{\tiny \frac{3}{2}}$$. It's close to 1 in SI units, but close to 0.0000093333 in cgs units or 449 $$\textrm{mile}^{\tiny -\frac{5}{2}} \, \textrm{hour}^{\tiny \frac{3}{2}}$$
I'm afraid your post comes over as dismissive carping from an envious naysayer. You sound like the sort of person who would reject a novel paper because it poses some kind of threat to your standing, then denigrate the author for settling for a low-impact journal. Exactly the sort of person, in fact, who is "unconnected with progress in physics". I would recommend that you desist, and instead apply yourself with sincerity to the quantum harmonics of particle structure.
Am I envious or am I protecting my standing? Can you have it both ways? I've read Worsley's self-published book and know just how shallow and sterile his ideas are. And I don't need to siphon Wikipedia or the source of your illustrations to make my arguments since I can type them up from my working understanding of physics.
Rpenner's response wasn't "There's no spherical harmonics in quantum mechanics!" it was "That work isn't an application of spherical harmonics". Didn't you read what he said?
It appears he jumped straight to assaination of my character rather than attempt to demonstrate the manner in which spherical harmonics were applied in the Worsley-derived expressions.
Furthermore your comments about numerology are completely wrong. The things you refer to are numerology, if you don't understand that then your level of understanding is even worse than basic quantum mechanics, it's stuff A Level students know about.
In the US, I cannot understand how one would expect to complete even an AP (high school course and exam which results in limited college credit) program in physics without mastering the need of understanding units. To understand 1 meter is distinct from 1 mile which is distinct from 1 second would seem to be material mastered by a six-year old. But Worsley, simply tells us to ignore that distinction because he is not interested in relating to the world of physical measurement and reality.
Huzzah! Mark this well!
rpenner would be getting stuck in telling us about toroidal harmonics
Actually, I got stuck. While I think you can describe toroidal motions with constant rates of angular motion, and this closes like in the diagram (whenever a/b is rational and 0 < S < R ), but the speed is not constant.
$$x(t) = \left( R + S \cos (a t + a_0) \right) \cos b t \, \; y(t) = \left( R + S \cos (a t + a_0) \right) \sin b t , \; z(t) = S \sin (a t + a_0)$$
$$\left| v(t) \right| = \sqrt{a^2 S^2 + \left( b R + b S cos(a t + a_0) \right)^2}$$
Similarly, for "diagonal" motion on a torus embedded in $$\mathbb{R}^3$$ with constant speed, I don't see that you are guaranteed to have the paths close as indicated in your diagram, unless you ignore the embedded speed and work in coordinates where the torus is flat. But in that case, you beggar the notion of movement.
), it's just human nature that we all share.