For those whose eyes glaze over in this mathematical analysis, a bit of history : physicists couldn't quite understand why the negatively charged electron didn't just spiral into the positively charged nucleus(unlike charges attract, like charges repel, as in the coulomb force of a fissioning U235 nucleus). Then came the 2 reasons : the electron travels at the c limit at the Bohr Radius(.5 angstrom)and the perimeter of the orbit is one matter wavelength. Thus it's difficult for delacoert and others to get past the Bohr Radius in their thinking. As to "infinity" : n/0=n why? Simple : division is repetitive subtraction. n/0 = n-0-0-0-...until the world wears flat and hell glaciates..-0-0-0-...=n. You see, n/0=n-0=n because infinity is NO THING, just as zero is NO THING. Only pharisees, the pied piper types...infer that infinity is some THING and their(PT Barnum's)suckers suck that nonsense right up....give them a radish and knife and ask that they carve it up into an infinite salad by slicing zero pieces off... Dr Mills is a true genius and his re-examination of the Rydberg Equation is where this all started; like all great scientists of the past, he took a critical look at accepted theory(something we all should do with "givens"). The proof of hydrinos(24 smaller orbits of the electron)is found in the solar spectrum : UV lines right where his theory predicts them(helium was discovered in much the same way). And if shrinking the hydrogen atom(80% of the universe is hydrogen)down to smaller sizes is a natural solar-burning process then the di-hydrino H2 atom explains the "dark matter" conundrum : stellar-burning "pollution", where is the EPA on that one? Dr Mills' hydrino development work has been fought tooth and nail by big oil and futurists. His hydrino compound patent(electric batteries with 500 times the energy density of your lead acid battery)was denied in federal court on these same specious Bohr Radius grounds with big oil behind the curtains of course. Futurists fight it because : if WATER(the hydrogen therein)has 37 times the heat-density of an equal weight of gasoline then : where does the WASTE HEAT go when everyone in the whole world(7 billion people)is cranking out GIGAWATTS of hydrino-heat? If you think global warming is bad now... Bottom line : the movie CHAIN REACTION.
In a graduate level treatment, with enough mathematics under the belt, one studies quantum mechanics not only independently of these early results, but also using a non-Hamiltonian approach. Quantum field theories (some via a Lagrangian formulation) can be derived independent of theoretical descriptions obtained during the historical development of quantum mechanics. We need not assume anything that Mills mistakenly thinks we need to assume.
Bettering an already obsolete incarnation of quantum mechanics while falling far short of current state-of-the-art descriptions is not a significant scientific achievement. It is an activity undertaken by all sorts, like Stephen Wolfram, who, like Mills, is brilliant, but went astray when his paranoid messiah complex took over (what is it with people developing "groundbreaking" theories that violate locality? They go for Einstein and QM in one go - is it the idea that if you can cannibalize the strongest warrior, you will gain his strength?).
That's really not what's going on -- both in general and specifically in my case.
Having been provoked by an accusation directed at me similar to the big oil conpiracy accusation you cited above, I looked more closely at the details of Mills' hydrino theory in which he claimed that states of the hydrogen atom below the ground state exist. When reading Randy Mills' book on Classical Quantum Mechanics, I found that a central postulate deals with the principle quantum number being less than one, i.e., integer fractions 1/2, 1/3, 1/4,... 1/137.
In particular, as I pointed out in a previous post, when discussing the principle quantum, n, for the solution to the Schrödinger equation, Mills states:
In the face of such a statement, I found that I had to contemplate what I thought I knew about the mathematics intrinsic to the solution of the Schrödinger equation in 3-D spherical coordinates for a hydrogen atom. I decided I knew that the solution arises out of the separation of variables technique. This technique poses a solution in a form containing the product of several independent functions, one of those being a function solely in terms of the independent variable for radial postion. The solution for this radial function is an infinite series involving Laguerre functions. Mathematically speaking, n is just the number of each term in the series. When terms are numbered, integers are used to number them. That's the way we count. Thus, for the simplest of reasons, n has to be an integer. The number, n, is subsequently interpreted to have a physical meaning, and is named the principle quantum number. This is an underlying "weakness" of all mathematical models; i.e., purely mathematical terms have to be interpreted to have a physical meaning.
When I examined the details of Mills' work, I found that just like the QM theory solution to the Schrödinger equation: 1) Mills' CQM theory solution to the classical wave equation is based on the mathematical technique known as separation of variables, 2) one of those functions depends solely of the independent variable for radial location and 3) the solution for this radial function is an infinite series forcing, n, the number of each term in the series, to be an integral.
This means that Mill's MATHEMATICAL MODEL for one-electron atoms suffers the same sort of underlying weaknesses of all mathematical models; i.e., purely mathematical terms (n in this case) have to be interpreted to have a physical meaning (interpretted to be the principle quantum number in this case). For the simplest of mathemaical reasons, n has to be an integer in Mills' CQM theory solution to the classical wave equation. Thus, it is impossible for Mills' principle quantum number, n, to take on fractional values. Therefore the hydrino states below the ground state are impossible (since they arise for fractional values of n).
I guess by saying that I'm stuck on the Bohr Radius, you are trying to ridicule the conclusion that it's impossible to achieve electron energy levels below the ground state. To me that appears to mean that you haven't grasped the math.
Do you understand separation of variables solutions to PDE's? Do you understand orthogonal functions, fourier series, and eigenvalues? If so, have you really thought about the mathematical meaning of the series index, n, and how that relates to the principle quatum number?
There are two parts to Mills' derivation for quantum states of hydrogen: 1) the part where the principle quantum number has integer values, i.e. n = 1, 2, 3,... 2) the part where the principle quantum number takes on fractional values, i.e. n = 1/2, 1/3, 1/4,..., 1/137.
The first part (i.e. n = 1, 2, 3,...) matches modern quantum theory. Thus part of what you say is true, Mills' results match the Rydberg states. So what? Mills' variation on the mathematics really isn't that cleaver, and if it didn't even match up with observed spectra we wouldn't be talking about it.
The second part is where the problem is. Fractional values for the principle quantum number are provably incorrect from a mathematical (theoretical) basis, and have never been observed in spectra from the sun or anywhere else.