A4. Expanding QM: Sub States of Hydrogen Atoms
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Cynthia Kolb Whitney
Figure A4.1. Identification of small-angle approximations in Hydrogen analysis.
The larger arrows on figure A4.1 point in the directions of these charge attractions. The
proton attraction to the electron has a radial component proportional to cos(α − β) and a
tangential component proportional to sin(α − β) . The electron attraction to the proton has a
radial component proportional to cos(β) and a tangential component proportional to sin(β) .
The small-angle approximations assumed for the Hydrogen analysis in A3 were:
cos(α − β) ≈ cos(β) ≈ 1 , sin(α − β) ≈ α , sin(β) ≈ β
(A4.1)
The small angle approximations allowed straightforward solution of the equation of
balance between energy loss rate due to radiation and energy gain rate solution due to
torquing. This was done algebraically by equating the energy gain rate due to torquing,
PT = (e4 / mp ) c(re + rp )3
(A4.2)
to the energy loss rate due to radiation,
PR = (25 e6 / me2 ) 3c3 (re + rp )4
(A4.3)
The relaxation of the small-angle approximations mandates the following complications:
1) The condition defining the angle α is:
α = (Ve / 2c) × [(re + rp ) / re ]
(A4.4)
Because rp << re , Eq. (A4.4) simplifies to α ≈ Ve / 2c .
2) The condition defining the angle β is
tan(β) = rp sin(α) ⎡ re + rp cos(α) ⎤
⎣
⎦
(A4.5)
Whatever angle α is, β will be near zero. But like α , angle β does depend upon Ve .
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3) The condition defining Ve is
meVe2 / re = e2 / (re + rp )2
Because rp << re , the approximation Ve ≈ e
(A4.6)
me (re + rp ) is acceptable.
4) Because the solution sought is a balance between torquing and radiation reaction, there
is no net acceleration of the charges in the angular direction. So the rate of energy loss due to
radiation PR depends only upon the radial components of forces. These components are
proportional to cos(α − β) and cos(β) . Because they are generally less than unity, the rate of
energy loss due to radiation is lessened. The scaling down is expressed by:
2
⎧1
⎫
PR → ⎨ ⎡⎣cos(α − β) + cos(β) ⎤⎦ ⎬ PR
⎩2
⎭
(A4.7)
5) The torque on the system has contributions arising from forces on both the electron
and the proton. Those contributions involve cross products between the forces and the radial
position vectors, and hence involve the sin(α − β) and sin(β) . The scaling down of the rate
of energy gain due to torquing is expressed by
⎧1
⎫
PT → ⎨ ⎡⎣sin(α − β) (α − β) + sin(β) β′ ⎤⎦ ⎬ PT
2
⎩
⎭
(A4.8)
where β′ is the factor within PT that sin(β) replaces. For small α , β ′ = β [from (A4.5)],
but for the full range of arbitrary α , β ′ = α / 1838 .
The changes developed above make the Hydrogen balance problem significantly more
complicated. Algebraic solution is no longer a practical approach. So graphical solution
becomes a more attractive approach. A simple EXCEL program is sufficient for a perfectly
reasonable graphical solution approach. To set the stage for graphical solution of the more
complicated problem with the small-angle approximations removed, figure A4.2 illustrates
the graphical solution approach as applied to the original problem, with the small-angle
approximations still embedded in it.
In figure A4.2, the independent variable is the system radius, re + rp . It is plotted on a
log scale, starting at re + rp = 10−11 cm and going through 6 decades to re + rp = 10−5 cm.
There are 301 data points for each dependent variable plotted. ‘Series 1’ refers to the steeper
line, representing energy loss rate due to radiation. ‘Series 2’ refers to the shallower line,
referring to energy gain rate due to torquing. The two lines on figure A4.2 are straight
because the energy gain and loss rates are power laws plotted on log scales. The graphical
solution comes approximately in the middle of the plots, approximately at point 138,
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Cynthia Kolb Whitney
approximately at re + rp = 5.5 × 10−9 cm, i.e. approximately the same as the algebraic solution
re + rp = 5.5 × 10−9 cm, and fairly close to the accepted value re + rp = 5.28 × 10−9 cm.
Figure A4.2 reveals something that was not obvious in the purely algebraic approach
used in [1]: the ground-state solution for Hydrogen is not stable. Any small perturbation right,
to higher energy, is encouraged by energy gain due to torquing, and any perturbation further
left, to lower energy, is encouraged by energy loss due to radiation. The revealed instability
does comport with observed facts: here on Earth, Hydrogen atoms tend to come together to
form molecules, and deep in space they tend to come apart to form a plasma. Nowhere do
they appear abundantly as individual atoms.
15
10
5
0
1
21 41 61
81 101 121 141 161 181 201 221 241 261 281 301
Series1
Series2
-5
-10
-15
Figure A4.2. Graphical solution for the Hydrogen ground state.
When the small-angle approximations are removed, the straight lines on figure A4.2
become the curved lines on figure A4.3. Again there are 301 data points, but now starting at
system radius re + rp = 10−15 cm and proceeding to re + rp = 10−5 cm.
The familiar first solution is located where the lines cross on the right side of figure A4.3,
between points 203 and 204, between re + rp = 5.41× 10−9 cm and re + rp = 5.84 × 10−9 cm.
The lower value comes a bit closer to the accepted value re + rp = 5.28 × 10−9 cm than did the
re + rp = 5.5 × 10−9 cm previously obtained with small-angle approximations. As one might
hope, including more trigonometric detail produces more numerical accuracy.
On the left side of figure A4.3, some dramatic features emerge. They are all associated
with α ≥ π . Note that in the vicinity of α = π , the electron speed is in the vicinity of
Ve = 2πc . This kind of speed is dramatically superluminal. Note that it does occur deep
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51
inside an atom – hardly the kind of ‘free-space’ situation for which the Einstein particle-speed
limit c was legislated.
30
25
20
15
10
Series1
Series2
5
0
1
21
41
61
81 101 121 141 161 181 201 221 241 261 281 301
-5
-10
-15
Figure A4.3. Extended graphical solution to the Hydrogen balance problem.
The first dramatic feature is a dip in the curve representing the rate of energy loss due to
radiation. This dip occurs where the cosine sum in Eq. (A4.7) touches zero. That happens as
angle α passes through π . The dip on figure A4.3 actually goes to zero, but since the plot is
on a log scale, the calculations have been set up to avoid having a data point occur exactly
where the zero occurs.
The second dramatic feature is a gap in the curve that represents the rate of energy gain
due to torquing. This gap occurs for the interval π < α < 2π . In this interval, the sin(α − β)
and the sin(β) in Eq. (A4.8) for PT are negative. That means the sign of PT is negative, and
PT cannot be plotted on a log scale. In this interval, the torquing effect on the system does
not oppose the radiation effect; it augments it. So system balance is not possible in this
interval.
But the third and most dramatic feature is that an additional solution emerges. It is
located just to the right of the radiation dip and torquing gap, between data points 26 and 27.
This second solution is clearly where the Hydrogen system would be driven to, should it be
perturbed from its first solution in the direction downward in radius, and hence downward in
energy. Throughout the radius regime from the first solution to the second solution, energy
loss due to radiation dominates energy gain due to torquing, and ever more energy is lost.
Looking further to the left on figure A4.3, it is clear that a family of viable superluminal
solutions is developing. Figure A4.4 begins at even smaller system radius, and shows even
more solutions. Superluminal solutions correlate with α = π + 4πn , n = 0,1, 2,3,... , These
solutions occur just to the right of the dips. Solution-prohibiting regions of negative torque
occur to the left of the dips. The dips themselves actually go to zero, but since the plots are on
a log scale, there are no data points placed exactly at the zeros. The depths of dips look
ragged, but that is just a computational artifact without physical meaning.
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Cynthia Kolb Whitney
The graphical analysis here gives an initial insight into the otherwise puzzling issue of
stability vs. instability of the Hydrogen ground-state solution to the problem of Hydrogen
torque vs. radiation balance. That solution is off the right side of figure A4.4, and it is
obviously unstable. The graphical analysis has then arrived at an infinite family of additional
solutions for Hydrogen torque vs. radiation balance. These solutions are on the left side of
figure A4.4, all below the ground state of Hydrogen in total energy, so they are ‘sub-ground’
states. They all have superluminal orbit speeds: with α ≈ Ve / 2c , we have
Ve = 2πc,10πc,18πc,... .
35
30
25
20
Series1
Series2
15
10
5
0
1
21
41
61
81 101 121 141 161 181 201 221 241 261 281 301
Figure A4.4. Multiple solutions to the Hydrogen balance problem.
The superluminal velocity results imply very high kinetic energies. Rewriting the results
as Ve = 2πc(1+ 4n) , n = 0,1, 2,3... , we see that kinetic energies, if proportional to Ve2 , must
advance with n2 . This behavior is like having a radial quantum number that is fractional. A
fractional radial quantum number is a phenomenon simply not within the current paradigm of
QM.
Though not the main topic of the present paper, these results are of further interest for the
future because there exists an extensive experimental literature on the spectroscopy of
Hydrogen, and some of it seems inexplicable in terms of traditional QM. In particular, the
work of Dr. Randall L. Mills and associates shows some extreme UV lines that are suggestive
of Hydrogen atom states with fractional radial quantum numbers. Dr. Mills tells that story
quite comprehensively in [28].
The refined Hydrogen analysis provides a stepping-stone to the analysis of systems in
which the participant charges are not of very dissimilar mass, as in Hydrogen, but rather of
equal mass, such as in the electron cloud in a trans-Hydrogenic atom.
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A5. An Example with Equal Masses: Positronium
The mass symmetry of Positronium makes for angle symmetry, illustrated by figure
A5.1.
Figure A5.1. Positronium.
From figure A5.1,
α −β ≡ β ≡ α / 2.
(A5.1)
The small-angle approximation for energy gain rate due to torquing becomes
PT = (e4 / me ) c(2re )3 .
(A5.2)
The small-angle approximation for energy loss rate due to radiation becomes
PR = (25 e6 / me2 ) 3c3 (2re )4 .
(A5.3)
The analysis then follows:
1) The condition defining the angle α becomes
α = (Ve / 2c) × (2re ) / re → Ve / c
(A5.4)
2) The formal condition for defining angle β becomes
tan(β) =
sin(α)
2sin(α / 2)cos(α / 2)
sin(α / 2)cos(α / 2)
=
=
= tan(α / 2) (A5.5)
1+ cos(α) 1+ cos2 (α / 2) − sin2 (α / 2)
cos2 (α / 2)
This relationship is indeed satisfied by β = α / 2 .
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Cynthia Kolb Whitney
3) The condition defining Ve becomes
meVe2 / re = e2 / (2re )2
The solution is then Ve ≈ e
(A5.6)
me (2re ) × (1 / 2) .
4) The factors for removing the small-angle approximations from PT and PR simplify.
The expression for energy loss rate due to radiation changes from
2
2
⎧1
⎫
PR → ⎨ ⎡⎣cos(α − β) + cos(β) ⎤⎦ ⎬ PR to PR → ⎡⎣cos(α / 2) ⎤⎦ PR
2
⎩
⎭
(A5.7)
5) The expression for the energy gain rate due to torquing changes from
⎧1
⎫
PT → ⎨ ⎡⎣sin(α − β) (α − β) + sin(β) β′ ⎤⎦ ⎬ PT to PT → ⎡⎣ sin(α / 2) (α / 2) ⎤⎦ PT (A5.8)
⎩2
⎭
35
30
25
20
Series1
Series2
15
10
5
0
1
24
47
70
93
116 139 162 185 208 231 254 277 300
Figure A5.2. Positronium solutions.
Being formally analogous to Hydrogen, Positronium also has a ‘ground-state’, low-speed
solution, plus a family of high-speed sub-ground solutions. Figure A5.2 illustrates the
solutions. The range of system radii starts at 10−16 cm, and goes to 10−11 cm via 301 points.
The ground-state solution occurs between points 269 and 270, at system radius of about
3 × 10−12 cm (quite a bit smaller than the 5.28 × 10−9 cm for the Hydrogen atom), with orbit