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A4. Expanding QM: Sub States of Hydrogen Atoms

# A4. Expanding QM: Sub States of Hydrogen Atoms

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48

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 .

Recent Progress in ‘Algebraic Chemistry’

49

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,

50

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

Recent Progress in ‘Algebraic Chemistry’

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.

Recent Progress in ‘Algebraic Chemistry’

53

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

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A4. Expanding QM: Sub States of Hydrogen Atoms

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