Journal of Modern Physics, 2013, 4, 29-32
doi:10.4236/jmp.2013.45B006 Published Online May 2013 (http://www.scirp.org/journal/jmp)
Spectra of Hydrogen Atom with GUP and Extra
School of Physical Electronics, University of Electronic Science and Technology of China, Chengdu, Sichuan, China
We argue that the Generalized Uncertainty Ptinciple (GUP) and the compact Extra dimensions will lead the hydrogen
atom has the non bounded state spectra which equal the free particle in extra dimensions. We use Generalized Uncer-
tainty Principle (GUP) to calculate the new energy spectra of hydrogen atom, and which lives in three dimensional
Euclidean spaces with one extra dimension. The result is not familiar with our known before that when n is
large enough. In our modified new spectra, we obtain that
,,0 (1 2)2
, which is lager than zero.
These new spectra give us new method to obtain the existenc e of extra dimensions. Finally, we find that the spectra are
same as part of extra dimensions in planck scale.
Keywords: Hydrogen Atom; GUP; Extra Dimensions
There exist many theories for solving the problem in
unifying the quantum theory and gravity, such as string
theory and loop quantum theory, which all of them pre-
dict that there exits the minimal observable length which
proportional to the Planck length cm. 
This minimal length was emerged by our observation
method under high energy scale, and which phenomenon
is not surprised for us. In usual quantum mechanics,
which does not consider the effect of gravity, has the
Heisenberg Uncertainty Principle which is described as
12xp . This uncertainty principle implies that the
large could make the position distance
trary small. How ever, when we cons ider the gravity, which
becomes more important in Planck scale, cannot lead us
to neglect the effect of gravity in many special cases.
This motivation gives us the advices that when we use
high energy to observe any arbitrary small scale, it will
causes the minimal blank hole, and hence its horizon
becomes the minimal observable length naturally. In this
analysis, the Heisenberg Uncertainty Principle must be
modifed for including the effects of quantum gravity and
the one of the most important models of this case is the
Generalized Uncertainty Principle (GUP). The modified
This new commutator relation gives us the po ssibilities
to contain the different conditions, which could describe
the usual uncertainty principle in our familiar quantum
and also is will includes the
nonzero position distance in high energy scale. In our
p is a positive function and which can
be expanded by Taylor series as
. Our previous work had been
introduced the running constant 0
p that would
varied with energy scales and be combined with another
quantum gravity quantity: extra dimensions . These
characteristics make us to construct the low energy effec-
tive theory which will contains some quantum gravity
phenomenon in intermediate energy scale. In GUP model,
we choose the first two terms as
and hence, the GUP model is
Generalizing the commutator to higher dimensions and
expressing it as tensor formalism, we obtain that
and the correspondent higher dimensional GUP model is
Copyright © 2013 SciRes. JMP
B. R. MU
where ii. This uncertainty principle easily
deduces the minimal observable length that
for every direction.
In this paper, we construct the toy model which the
extra dimension is only considered without any other
potentials, and use the GUP to redescribe the hydrogen
atom. This assumption is not unjustifiable in view of the
fact that the standard model charges cannot move in extra
dimensions in theory of brane world model. Now, extra
dimensions are widely accepted by theorists and there are
large works about it [7-13]. This model gives us the new
spectra, which looks same as our previous works ,
however, have different physical meaning. In usual spec-
tra of hydrogen atom, we know the spectra is given as
We know that the ground state energy is 113.6eVE
When , we obtain the upper boun d of the energy
is zero. In our following works, when consider the term
of one extra dimension
, the upper bound of
hydrogen atom will be larger than zero. On the other
hand, we obtain the energy spectra of hydrogen atom is
same as the spectra which describe the particle move in
the extra dimension freely in Planck scale. To sum up, it
gives us the new detect method to find the existence of
extra dimensions in intermediate energy scale.
In our case, we need that the wave function obeys the
, where x
represents the familiar Euclidean space and y yrepresents
the compact space. As we known, the function can be
described as the production of two parts that
, where represents the wave function in
compact space, and we have
. In this
way, the familiar wave function can be divided into two
parts. We hence can calculate the energy spectrum of
Euclidean and compact space respectively, and count up
these two parts for obtaining the total energy of the sys-
i and 00 j
, the usual
canonical operators. The unperturbed schrödinger equa-
with modifying the commutator relations, we can obtain
the new Hamiltonian of this quantum system that
the first order correction is then given as
3. Modified Spectra of Hydrogen Atom
The schrödinger eq uation of hydrogen ato m is
where potential ()Vr
comes from the Coulomb's law
and be given as
As we mentioned in introduction, we should consider
the hydrogen atom with an extra dimension at first.
When we consider the extra dimension, the term of
schrödinger equation should contain the momentum in
extra dimension and which can be rewritten as
where the correspondent wave function is
In this way, the equation be divided into two equations
which are unperturbed as follow
so, we have the unperturbed energy spectra of hydrogen
atom with one extra dimension, which can be given as
Now, let us modify the commutation relations by using
GUP, we obtain the perturbed term of Hamiltonian that
. So, we find the first order correction of
energy spectra is
Copyright © 2013 SciRes. JMP
B. R. MU 31
Finally, we obtain the modified energy spectra of hy-
drogen atom that
When n is large enough, in our introduction before,
will gives us . But, we obtain new value in here
when n is large, the spectra is
This result is too surprised. As we known, the usual
ground state of hydrogen atom is. With energy
level increase, the value of energy go up to, however,
specially, there exits the upper bound when is large
enough which equals zero. But, our result introduction
the new spectra of hydrogen atom when n is large. Above
the upper bound, we find new spectra! This result is not
end of journey, noteworthily, as our works before, we
have the relations
in Planck scale. It express
which is the energy spectra of extra dimension. Consid-
ering the physical meaning of the nonzero position dis-
tance in Planck scale, there exists the minimal black hole
and the horizon is that nonzero po int. This condition lead
the hydrogen atom has unbounded state in extra dimen-
sions and which is reason that we obtain the positive
spectra in final.
We use GUP and extra dimensions models to rewritten
the equation of hydrogen atom and obtain the modified
energy spectra which is far from the familiar solution that
has a upper bound. In this modified spectra, we find that
there also exits discrete spectra in field of >0
is same as the free particle that lives in extra dimensions.
This result is not familiar in usual hydrogen atom. As this
term is contributed by the wave function in extra dimen-
sion, it introduces the new method to find extra dimen-
sions in intermediate energy scale.
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