To understand this post in details, you need to read the following sub post and study them. I'll discuss this in the redit r/hydrino if you have some physical understanding you should be able to understand what Mills is doing with his approach to GUTCP and the novel compound he call's hydrinos. In principle what discussed here should with some more effort be able to produce answers that shows from from first principles an argument for what hydrinos are, answers why we have a quantization leading to the Rydberg series and why hydrinos cant excite by electromagnetic radiation and hence are in many way's black to us and invisible - e.g. a good candidate for dark matter just as Mills claims.
Consider the scaling \(r\to cr\), \(Z \to cZ\) Hence \(K\to K/c\) and \(w_0\) is consistent also in the Bohr Sommerfeldt model, hence the scaling is consistent and this also imply that \(r\to cr\). As \(v\) is constant (because \(mv\) is constant and m depends only on rest mass and v), frequency and \(k\) goes down as \(1/c\) hence a phase matched photon will decrease the energy as the scale \(1/c\) The condition for the charged field to not radiate are
And for this setup \(|k|r\) is invariant of the scaling hence the non radiation condition is still met.
Note that \(j_0(x)\) has zeros with the right properties at \(x = 2\pi n\), \(n >= 1\) and that essentially are standing waves with differnt number of nodes and hence represent fundamentally differnt forms of the EM field. For the same value they have a smooth path to transition for one radi to another. Anyhow one should expect that excited values of the EM fields would like to transition to lower states and \(n=1\) should be the norm and this transfer is mitigated by the electron shell as preferable they want to keep the same frequency as the underlying speed. But as energy condition will stop the hydrinos to move down the exited state of the standing wave and hence that exited state is locked.
For this setup of a standing EM wave we will also be non radiating using a boundary condition and it will interact with the photon and produce a central force that effectively reduces / increases the Z. This means that as Mills describes in GUTCP all condition will be met to lead to the Rydberg series.
As said this setup demands that the frequency changes and of if we on stead want other frequencies we could phase match them will integers multiples and for the the spherical symmetric setup, yielding \(j_0(n r |k|)\) that will indeed be zero for all integer \(n>0\) that we can scale.
So in a sense when en electron jump, it will first add n photons e.g. amplitude, and then they will go down in frequency loosing the phase and then radiate. In all we end up getting an understanding from where the quantization comes from. It has to do with non radiation and the properties of \(j_0(x)\).
Consider now a system in the normal ground state of Hydrogen. If we could get it to be temporarily in a exited state with higher frequency so e.g. we would have \($|k|r| = N2\pi\). By the logic of before this should mean that we temporarily can reduce the radius and hence get into a fractional hydrino state stuck with the value of \(|k||r|\) now in order for this electron state to capture a photon it has to capture one that does not follow \(E=hv\), but a scaling of it. But there is none to be found out there so it cannot excite and renounce back.
Philosophically yours Stefan