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European Journal of Applied Sciences – Vol. 12, No. 1

Publication Date: February 25, 2024

DOI:10.14738/aivp.121.16240

Bender, D. (2024). Quantum Gravity, Energy Wave Spheres, and the Proton Radius. European Journal of Applied Sciences, Vol -

12(1). 99-165.

Services for Science and Education – United Kingdom

Quantum Gravity, Energy Wave Spheres, and the Proton Radius

Darrell Bender

New Mexico Institute of Mining and Technology

ABSTRACT

We argue, from present considerations and a previous analysis of the

hydrogen atom as a miniature Michelson-Morley experiment in the

Material Point Universe Revisited, that the electron wave velocity in the

hydrogen atom is � and that the fine-structure constant � is the ratio of the

remaining mass of the electron to the initial electron mass, not the ratio of

velocities, as Sommerfield had it. We consider the proton energy wave

sphere, with mass �� and velocity �, so that the radius of the proton energy

wave sphere contained in an electron energy wave sphere is

�� = ħ

���

,

with �� the remaining mass of the proton equal to one fourth of the initial

proton mass so that

�� = �. ���������������� × ��"�� ������.

We argue that the hydrogen atom consists of nested energy wave spheres, including those for

the energies lost, so that energy is conserved. With a unit clock rate in the gravitational field

equal to

61 − �

�

;

&

'

,

not its inverse as Einstein had it, we show that energy wave sphere clocks have energy

=61 − �

�

;

&

'

h�(

(

so that the energy lost is

@1 − 61 − �

�

;

&

'

A=h�( = h� B

1

2

�

�

+

1

8 6

�

�

;

'

+

1

16 6

�

�

;

)

+

5

128 6

�

�

;

*

+ ⋯ I

(

,

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99

Bender, D. (2024). Quantum Gravity, Energy Wave Spheres, and the Proton Radius. European Journal of Applied Sciences, Vol - 12(1). 99-165.

URL: http://dx.doi.org/10.14738/aivp.121.16240

with the first term, after multiplication by h�, on the right being the Newtonion gravitational

energy of motion of the hydrogen atom. For a light ray moving perpendicular to a radius in a

gravitational field, we obtain

� = ��'

��*

M1 + tan' � = ��'

��*

sec �

With

��'

��*

= U1 − 1

�'

�

�

W

&

'

and

tan' � =

1 − 61 − 1

�'

�

�;

61 − 1

�'

�

�;

.

Abstract 1

The fine-structure constant �, also known as the Sommerfield constant, was introduced by

Sommerfield in 1916. In Sommerfeld, A. (1921). Atombau und Spektrallinien (in German)

pp. 241–242, Equation 8, Sommerfield considers � to be

v&

� ,

where �&is velocity of the electron in the first circular orbit of the Bohr model of the hydrogen

atom.

If we did not prove that the mass �+, not �, decreases by the factor � in this case, it is a

reflection that we did not need to prove it since we had already argued the result in the

consideration of the Lorentz contraction of the hydrogen atom and since we are prepared to

argue for it again via the principle that physical reality is comprehensible. Abandoning the

decreasing mass �+ requires abandoning the only rational theory of gravitation that we have.

Somewhere we have placed the rule for not abandoning the correct theory.

For the proton energy wave sphere, we calculated its radius as follows:

“What we see is a matter/energy wave in the form of a sphere, the wave having velocity � with

space-time coordinates along a radius in a quasi-static, spherically symmetric, gravitational

field about a point mass such that the measured velocity of the wave in any direction is always

�. For the proton energy wave, with the mass of the proton �, given by

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European Journal of Applied Sciences (EJAS) Vol. 12, Issue 1, February-2024

�, = 1.67272 × 10"'-��,

we have

�,�' = h�,

so that

�,� = h

�,

,

or

�, = h

�,�

,

where �, is the circumference of the proton energy wave sphere. We calculate the value of

this circumference as

�, = 6.62607015

(1.67272)(2.99792458)

10"&. ������

= 1.321332377387867 × 10"&. ������.

Dividing this last result by 2�, we obtain, for the proton energy wave sphere radius �,,

�, = 0.2102965793286309 × 10"&. ������.

This last value, which is the proton energy wave sphere radius without being inside of the

electron energy wave sphere, is one fourth the measured value of the proton radius with the

proton inside of the electron so that the proton inside of the electron has for energy

h�, = 1

4 h�,/

= 1

2' h�,/

and radius �,! so that

�,! = 4�, = 4

ħ

�,� = 0.8411863173145236 × 10"&. ������."

“When we are considering what we see,” which is the proton energy wave sphere or proton,

we know its mass-energy and we know that the velocity of the wave is � so that we know its

momentum, the denominator of the last expression for �,!. Thus, we have the wavelength and