[delacoert]

Is he being that stupid or is there some justifiable logic behind posing a fractional quantum number?

I have read enough of Mills' book on CQM to have found the answer to my own questions.

1. As to whether or not he is being stupid, it strikes me that someone clever enough to have raised $50M is anything but stupid.
2. I can definitely state that there is no justifiable reason behind posing a fractional quantum number, i.e., n = 1/integer.

What follows are my observations and an analysis leading to conclusion #2. It won't be easy to follow, but if you've studied quantum mechanics even just a little bit, and if you go and look in Mills' book (following the links I provide), the conclusion in #2 above is inescapable.

On page 24 of the current online version of the Introduction to Mills' book, when discussing the principle quantum, n, for the solution to the Schrödinger equation, Mills states:

Furthermore, it is a mistake to take the position that solutions of the POSTULATED Schrödinger equation which correspond to n = integer prohibit the existence of hydrogen atoms having a binding energy corresponding to n = 1/integer. Specifically, it is wrong to take the position that n = 1/integer values cannot exist according to conventional scientific theories since the Schrödinger equation may be solved for either case with equal validity.

Contrary to Mills' statement, the mathematics intrinsic to the solution of the Schrödinger equation in 3-D spherical coordinates for a hydrogen atom requires that n be an integer. This arises because the separation of variables technique used in arriving at the solution employs a product of independent functions, one of those being a radial function. The solution for the 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 strictly mathematical reasons, n has to be an integer.

The number, n, is subsequently interpreted to have a physical meaning. This is an underlying "weakness" of all mathematical models; i.e., mathematical terms are interpreted to have a physical meaning. In fact, Mill's spends quite a bit of time articulating the weaknesses of the QM mathematical model (i.e., the Schrödinger equation). Apparently, Mills has a blind spot here, i.e., the heart of Mills' CQM model for a hydrogen atom (or any one-elctron atom) is a mathematical model with its own underlying weaknesses.

On page 26 of the current online version of the Introduction, when discussing the principle quantum, n, for his CQM solution to the classic wave equation, Mills states:

The nonradiative state of atomic hydrogen, which is historically called the "ground state" forms the basis of the boundary condition of CQM to solve the wave equation. The solutions for electron states having principal energy levels corresponding to integers and corresponding to n = 1 reveal the mechanism of the corresponding transitions.

Here lies the heart of the problem! If he is correct that his theory allows for quantum numbers to be integer fractions, then his mathematical solution should not show otherwise. But guess what? His theory hinges on a MATHEMATICAL solution to the classical wave equation. His solution to this equation employs the separation of variables technique! He has a radial function that has a solution involving an infinite series! The terms in this series are numberedÂ… oops they're integers, not fractions!

Let me explain.

Mills' solution to the classic wave equation is found on pages 41-50 of Chapter 1 - The One Electron Atom On p. 43 he states:

The solutions of the classical wave equation are separable.
ρ(r,θ,φ,t)=ƒ(r)g(θ)h(φ)k(t) (1.9)

On page 42, regarding the radial function, ƒ(r), Mills states:
Then, the solution for the radial electron function, which satisfies the boundary condition is a delta function in spherical coordinates—a spherical shell [2]
ƒ(r)= 1/r²δ(r − r_n )                 (1.3)

where r_n is an allowed radius. This function defines the charge density on a spherical shell of a fixed radius, not yet determined, and Eq. (1.1) becomes the two-dimensional wave equation plus time with separable time and angular functions. Given time harmonic motion with angular velocity, ω_n, and a radial delta function, the relationship between an allowed radius and the electron wavelength is given by

2πr_n= λ_n                          (1.4)

where the integer subscript n here and in Eq. (1.3) is determined during photon absorption as given in the Excited States of the One-Electron Atom (Quantization) section. It is shown in this section that the force balance between the electric fields of the electron and proton plus any resonantly absorbed photons gives the result that r_n = nr₁ wherein n is an integer in an excited state.

Emphasis has been added to the phrases showing the stipulation that n must be an integer. This stipulation is mathematical and inescapable. Mills' own CQM model for one-electron atoms is a MATHEMATICAL MODEL that requires the principle quantum number, n, to be an integer! It can not be a fraction no matter how much that is wished!

The remaining details to Mills' solution can also be found in Chapter 1. It is more than a little interesting to note that on p. 44 Mills inserts a bolded section that is obviously part of an attempt to mislead the reader regarding possible values of n. That statement reads:

The subscript n is used hereafter; however, the quantization condition appears in the Excited States of the One-Electron Atom (Quantization) section. Quantization arises as "allowed" solutions of the wave equation corresponding to a resonance between the electron and a photon.

Following the trail on the mysterious fractional values for the principle quantum number, n, on page 136 in Chapter 2 - Excited States of the One-Electron Atom (Quantization), without any explanation, Mills' states:

Consider the hydrogen atom. The atom and the "trapped photon" caused by a transition to a resonant state other than the n = 1 state have neutral charge. As shown infra., the photon's electric field superposes that of the proton such that the radial electric field has a magnitude proportional to Z / n at the electron where n = 1,2,3,... for excited states and n = 1/2, 1/3, 1/4, …1/137 for lower energy states given in the Hydrino Theory—BlackLight Process section.

Chasing down the last lead on the possibility of a fractional value for the principle quantum number, n, we again follow Mills' direction and go to Chapter 5 - Hydrino Theory—BlackLight Process. On page 246, Mills states:

The n = 1 state is the "ground" state for "pure" photon transitions (the n = 1 state can absorb a photon and go to an excited electronic state, but it cannot release a photon and go to a lower energy electronic state). However, an electron transition from the ground state to a lower-energy state is possible by a nonradiative energy transfer such as multipole coupling or a resonant collision mechanism. These lower-energy states have fractional quantum numbers, n = 1/integer.

That is it! Just a couple statements out of nowhere to the effect that the principle quantum number, n, can take on the fraction values n = 1/2, 1/3, 1/4, Â…1/137. These statements are just plain untrue! These statements in no way overcome the requirement that n is an integer arising from the series solution to the radial function.

[delacoert]

I don't know what happened to the html display of Schrödinger in the above post, but it sure came out lame.

The only thing I can think of is I pulled the html coded post in an out of MS Word to spell/grammar check etc. Maybe MS Word scambled some of the html code.

Anyway, sorry about the formating problems.

[timer]

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.