The Hydrogen Atom Fractal Spectra, the Missing Dark Energy of the Cosmos and Their Hardy Quantum Entanglement

doi:10.4236/ijmnta.2013.23023

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International Journal of Modern Nonlinear Theory and Application, 2013, 2, 167-169

http://dx.doi.org/10.4236/ijmnta.2013.23023 Published Online September 2013 (http://www.scirp.org/journal/ijmnta)

The Hydrogen Atom Fractal Spectra, the Missing

Dark Energy of the Cosmos and Their Hardy

Quantum Entanglement

Mohamed S. El Naschie

Department of Physics, Faculty of Science, University of Alexandria, Alexandria, Egypt

Email: [email protected]

Received May 21, 2013; revised June 27, 2013; accepted July 5, 2013

cense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

ABSTRACT

In this letter, I outline the intimate connection between the fractal spectra of the exact solution of the hydrogen atom and

the issue of the missing dark energy of the cosmos. A proposal for a dark energy reactor harnessing the dark energy of

the Schrödinger wave via a quantum wave nondemolition measurement is also presented.

Keywords: Fractal Spectra; Dark Energy; Golden Mean; KAM Theorem; Quantum Entanglement; Special Relativity

The spectrum of the hydrogen atom was found in 2006

by V. Petruševski to harbor the golden mean for which

the discoverer could not give any deep rational explana-

tion [1,2]. On the other hand since the 1988 Physics Re-

view Article of C. DeVito and W. Little [3], it was

known that the spectral lines of the hydrogen atom [4]

are a kind of Cantor set fractal but they could not say

exactly what form of fractal set it is nor from where this

fractality stems [3,5].

The present work reports on an important discovery

which links the above mentioned two papers together and

gives a clear cut rational explanation for the fractality

and the golden mean of the hydrogen atom spectra and

moves from there to explaining dark energy [4,5-7]. It is

simply the involvement of random Cantor sets [8,9] in

the structure of the spectra on the very fundamental level

of Hardy type quantum entanglement in physics [5-10]

which leads to the golden mean and the random fractal

appearance [5-14]. A random Cantor set possesses with a

probability 1, a Hau sdorff dimension eq ual to the golden

mean



512 [11-14]. This is actually a theorem by

Mauldin and Williams [8-12]. Furthermore the Hardy

type probability of quantum entanglement of two parti-

cles is equal to the golden mean to the power of 5, i.e.

P(Hardy) = [5-7]. This particular result

was confirmed in various sophisticated experiments [5,10]

so that we could state confidently that it strongly sug-

gests that quantum spacetime is a Cantorian fractal as

59.01699%





advocated in various fractal-Cantorian spacetime theories

as well as the theory of non-commutative spaces [14] as

applied to Penrose fractal tiling [8-10,13,14]. The con-

certed action of all these facts and elements leads us to

recognize the golden mean random Cantor set nature of

the spectra of the hydrogen atom. For instance, in certain

cases (labeled 25 in Ref. [4], p. 192), which will be dis-

cussed in detail in a forthcoming paper, replacing the

Bohr radius in the corresponding probability density [4]

with the isomorphic length





2 of Penro se tiling

one obtains Hardy’s probability of quantum entangle-

ment. Consequently the present analysis adds more

weight to the assertion of the proponents of fractal-Can-

torian spacetime that quantum spacetime is indeed a Can-

torian [13,14] fractal with a topological dimension [6-9]

of exactly 4



and a likewise an exact Hausdorff

dimensi on gi ven by [6-9,13]

4.236067













(1)

where 0.6183398





is the golden mean [1-15].

Quantum entanglement is not only part and parcel of

an old basic problem of quantum theory such as the hy-

drogen atom but also at an unimaginably deeper level

covering the entire universe being the key to the missing

M. S. EL NASCHIE

168

dark energy of the cosmos [16]. In short, it turns out that

the famous relativity formula relating mass (m)

to energy (E) via the speed of light (c) does not distin-

guish between measurable real ordinary energy E(O) and

missing dark energy of the cosmos E(D) which cannot be

detected or measured directly using any of present day

technology [17,18]. The simple explanation for this un-

paralleled challenge to the foundations of modern theo-

retical physics and cosmology is again intimately con-

nected to Hardy’s quantum entanglement and conse-

quently to random Cantor sets and their golden mean

Hausdorff dimensions. Since at the quantum resolution

level, spacetime is made up of totally disjoint random

Cantor sets with an infinite number of graded dimensions,

the probability of quantum entanglement is governed by

the zero measure of this Cantorian space and is given

generally by

Emc



 



 where n is the num-

ber of particles [5,7]. For two particles, one finds 5





which is the theoretically and experimentally confirmed

Hardy quantum probability [5-7]. Intersecting the three

fundamental equations of physics, namely Newton’s ki-

netic energy 2

Emv where v is the velocity, Eins-

tein’s equation and

Emc5



 one finds a

quantum relativity energy-mass formula





255 22

122

Em vcmcmc









 2

This is exactly the energy density found via accurate

cosmic measurements COBE and WMAP which amount

to only 1224.5% of what E(Einstein) pred icted [10],

[16-19]. The rest namely







11 2295.5%is the

presumed missing dark energy of the Universe. It then

turned out that ordinary energy i.e. the measurable 4.5%

is simply the energy of the quantum particle face of the

particle wave duality of quantum mechanics [7,10] while

the missing 95.5% dark energy E(D) is the energy of the

other face of the duality, namely the quantum wave

[10,14,18]. Since the quantum particle is modeled by a

five dimensional zero set and the wave is modeled by a

five dimensional empty set we have here a zero set-

empty set duality [19].

Together



222Ec mc of the particle and









222

52 2122E Dmcmc



 of the wave add

up to a total exactly equa l to Einstein’s energy [16-19]:





 

522

total25 2E





 



mcmc. (2)

Two major conclusions follow after the above. First,

Einstein’s is correct but completely blind to

the distinction between dark energy and ordinary energy.

Second, since measurement collapses the quantum wave,

it is natural that our present day technology cannot detect

dark energy [10,16-18]. From the above we can see fu-

ture technological research going towards developing

quantum nondemolition measurement instruments and

nuclear dark energy reactors. None the less it is vital to

understand that we may have a wavy spacetime at the

quantum resolution simulating a quantum wave of a knot

in the fabric of spacetime. Consequently harvesting the

quantum wave may be harvesting the real vacuum mod-

eled by the empty set, similar to what KAM theorem

implies [8].

Emc

Finally we note that E(0) and E(D) could be readily

found from the ground energy state of the hydrogen atom

Emc



 where 1/137





[4,20] by running



as a function of energy as in quantum field theory and

replacing 2



by 23 5







so that we find

25 2

122.

Emcmc





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