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
Mohamed S. El Naschie
Department of Physics, Faculty of Science, University of Alexandria, Alexandria, Egypt
Received May 21, 2013; revised June 27, 2013; accepted July 5, 2013
Copyright © 2013 Mohamed S. El Naschie. This is an open access article distributed under the Creative Commons Attribution Li-
cense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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 , it was
known that the spectral lines of the hydrogen atom 
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
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
advocated in various fractal-Cantorian spacetime theories
as well as the theory of non-commutative spaces  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. , p. 192), which will be dis-
cussed in detail in a forthcoming paper, replacing the
Bohr radius in the corresponding probability density 
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]
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
opyright © 2013 SciRes. IJMNTA
M. S. EL NASCHIE
dark energy of the cosmos . 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
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
Emv where v is the velocity, Eins-
tein’s equation and
one finds a
quantum relativity energy-mass formula
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 ,
[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 .
222Ec mc of the particle and
52 2122E Dmcmc
of the wave add
up to a total exactly equa l to Einstein’s energy [16-19]:
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
Finally we note that E(0) and E(D) could be readily
found from the ground energy state of the hydrogen atom
[4,20] by running
as a function of energy as in quantum field theory and
by 23 5
so that we find
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