Journal of High Energy Physics, Gravitation and Cosmology, 2019, 5, 279-290
http://www.scirp.org/journal/jhepgc
ISSN Online: 2380-4335
ISSN Print: 2380-4327
DOI:
10.4236/jhepgc.2019.51015 Jan. 24, 2019 279 Journal of High Energy Physics, G
ravitation and Cosmology
Spectra of Harmonic Oscillators with GUP and
Extra Dimensions
Benrong Mu, Rong Yu, Dongmei Wang
*
Physics Teaching and Research Section, College of Medical Technology, Chengdu University of Traditional Chinese Medicine,
Chengdu, China
Abstract
In this paper, we address the spectra of simple harmonic oscillators based on
the generalized uncertainty principle (GUP) with a Kaluza-Klein compacti-
fied extra dimension. We show that in this scenario, to make the results
compatible with experiments, the minimal length scale equals to the radius of
compact extra dimension.
Keywords
Compactified Extra Dimension, GUP, Harmonic Oscillator
1. Introduction
All quantum gravity theories, such as string theory and loop gravity theory, pre-
dict that there exits a minimal observable length, which proportional to the
Planck length
33
10cm
p
−

[1]-[7]. This length scale can be easily justified by
combining black hole physics with quantum uncertainty principle. Quantum
mechanics is free of the theory of gravity. However, when the energy scale ap-
proaches Planck scale, the effect of quantum gravity cannot be neglected. It
means that quantum mechanics should be modified to accommodate the influ-
ences from gravity. Or, it implies that we need a gravitational quantum mechan-
ics. The simplest way is to modify the Heisenberg Uncertainty Principle. The
uncertainty principle
12xp∆∆≥
shows that the large
p∆
could make the po-
sition distance
x∆
arbitrarily small. When the gravity is included in the theory,
the
x∆
acquires a minimal length, caused by the emerged horizon of mini
black hole. To realize this minimal length scale, we introduce a simple model,
the so-called the Generalized Uncertainty Principle (GUP). Incorporating GUP
into black holes has been discussed in a lot of papers [8] [9] [10] [11] [12]. It is a
toy model of quantum gravity theory, which is expected to reveal some features
How to cite this paper:
Mu, B.R., Yu, R.
and
Wang, D.M. (2019) Spectra of Har-
monic Oscillators with GUP and Extra
Dimensions
. Journal of High Energy Phy
s-
ics
, Gravitation and Cosmology ,
5
, 279-290.
https:
//doi.org/10.4236/jhepgc.2019.51015
Received:
December 12, 2018
Accepted:
January 21, 2019
Published:
January 24, 2019
Copyright © 201
9 by author(s) and
Scientific
Research Publishing Inc.
This work is licensed under the Creative
Commons Attribution International
License (CC BY
4.0).
http://creativecommons.org/licenses/by/4.0/
Open Access

B. R. Mu et al.
DOI:
10.4236/jhepgc.2019.51015 280 Journal of High Energy Physics, G
ravitation and Cosmology
of the ultimate theory. Nearly all problems in quantum mechanics could be dis-
cussed with this modified commutator to model the system bathed in strong
gravitational background. The modified fundamental commutator (
1c==
) is
[13]-[25]
[]
()
,.xpifp=
(1)
This new commutator relation includes the different conditions, which could
describe the Heisenberg Uncertainty Principle in quantum mechanics by
(
)
1
fp=
and also includes the nonzero position distance in high energy scale.
In our requirement,
()
fp
is a positive function and which can be expanded in
Taylor series by
()
()
24
1fppp β =++
, and
22
00
pp
M βββ
==
. Our pre-
vious work had introduced the running constant
0
β
of
()
fp
, which will va-
ries with energy scales and be combined with extra dimensions [26]. It leads us
to construct the low energy effective theory which contains some quantum grav-
ity phenomenons in intermediate energy scale. In GUP model, we choose the
first two terms such that
[]
()
2
,1,xpip β =+
(2)
Note that this formalism is special case of generalized modification that
[]
()
22
,1xpixp
αβ =++
. Since the momentum has not minimal scale, it is re-
stricted by choosing
0
α =
. Therefore, this special expression gives the genera-
lized uncertainty principle (GUP)
()
2
1
1.
2
xpp
β

∆∆≥+∆

(3)
Generalizing the commutator to high dimensional space and expressing it as
tensor formalism, one obtains that
1
()
2
,2,
ijijijij
xpippp δβδβ

=++

(4)
and the correspondent higher dimensional GUP model is
()
() ()
2
2
2
2
1
12,
2
iiii
xppppp ββ

∆∆≥+∆++∆+


(5)
where
2
ii
ppp=
∑
. This uncertainty principle easily deduces the minimal ob-
servable length that
()
0
min
ip
x β ∆
for every direction.
In [26], the energy levels of a free particle including an extra dimension was
discussed. The results showed that the minimum observable length equals to the
compactification radius of the extra dimension. However, if the particle expe-
riences an non-vanishing potential, what is the result likely to be? In this paper,
we extend our discussion to the harmonic oscillator in the extra dimension and
consider the GUP’s modified energy levels. The result is agree with previous
work. The present works of extra dimensions can be refereed in [27]-[36].
This paper is organized as follows. In Section 2, the perturbation method
based on GUP is briefly reviewed. In Section 3, two dimensional harmonic os-
cillator including an compact extra dimension is discussed, and the energy spec-
1
The generic expression is
()
2
,
ijijijij
xpippp δβδβ
′

=++

. We choose
2 ββ
′
=
.

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ravitation and Cosmology
trum is obtained for the large and the small extra dimension. In Section 4, based
the perturbation method and GUP, the energy spectrum including quantum
gravitational effect is obtained. In Section 5, we summarize the conclusion ob-
tained in the Section 4.
2. The Perturbation Method Based on GUP
In our scenario, we require the wave function satisfies the periodicity
()()
,,2πxyxy ψψρ =+
, where x represents the usual Euclidean space and y
represents the compactified space. Obviously, the function can be separated into
two parts
()()()
,xx ψθφθ =Θ
, where
Θ
represents the wave function in
compactified space, and we have
()()
2π θθ Θ=Θ+
. Therefore, the energy
spectrum of Euclidean and compactified space can be obtained respectively. Af-
ter combining these two parts, we can find the total energy of the system. Fol-
lowing this step, we use the minimal length uncertainty relations, GUP specially,
to rewrite the wave function. The modified operators are given by
0
,
ii
xx
=
()
2
000
0
d
1,,
d
iii
i
ppppi
x
β
=+=−
(6)
where
2
000
ii
ppp
=
∑
and
00
,
ijij
xpi δ

=


are usual canonical operators. The
unperturbed Schrödinger equation reads
()
2
.
2
p
VE
m
ψψ

+=


x
(7)
Modifying the commutator relation, we can obtain the new Hamiltonian of
this quantum system
(
)
2
4
0
010
,
2
p
HHHVp
mm
β
=+=++
x
(8)
the first-order corrections to the energy eigenvalues are given as
()(
)
()
()
()
(
)
2
10
4
0
2
00
2
4,
42.
nn
nn
EpmEV
m
mEEVV
β
ψψβ
β
==−

=−+


(9)
3. Review of Harmonic Oscillator in Compactified Extra
Dimension
Considering the two dimensional harmonic oscillator which includes one of di-
mensional Euclidean space and another is compactified extra space, we need to
solve the Schrödinger equation
()
2
222
1
.
22
p
mxyE
m
ψωψψ ++=
(10)
The Schrödinger equation can be solved by the method of separation of va-
riables,
()()()
,xyxy ψφ =Θ
, with
()()
2πyy
ρ Θ+=Θ
. In this section, we do

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ravitation and Cosmology
not discuss the effects of GUP in Equation (10),
i
i
pi
x
∂
=−
∂
,
2
jj
ppp=
∑
. Only
the harmonic oscillator with periodic condition is reviewed in this section. In the
next section, we will discuss the GUP modification of Equation (10), and the
perturbation method will be applied since it is convenient enough for seeing
some features of quantum gravity. Now, we rewrite Equation (10) as
()
()
()
()
()
()
()
()
()()()
00000 00
,,.
xxyyxy
HxExHyEyEEE
φφ
=Θ=Θ=+
(11)
Since the harmonic oscillator lives in the compactified space, the physical
meaning of the equation
(
)
()
(
)
()
00
yy
HyEy
Θ=Θ
will be changed. The harmonic
oscillator in Euclidean space
x only satisfies the boundary condition by
()
0 φ ±∞=
. However, it should satisfy
()()
2π θθρ Θ=Θ+
and add an addi-
tional restriction for the potential by
()()
2πVyVy ρ =+
in the extra dimen-
sion
y , which identifies any two points that the length differ by
2πR
. However,
the potential of simple harmonic oscillator does not satisfy this condition, we
need to construct a new potential of harmonic oscillator with periodicity. Bezer-
ra and Rego-Monteiro had discussed the harmonic oscillator on a circle and
constructed a new wave function with periodic oscillator potential [37]. The
momentum operator
G in a compactified space is given by [37] [38]
d
,01.
d
G
i
αα
θ
→+≤<

(12)
Therefore, the Schrödinger equation in the extra space is constructed with the
unitary operator
e
i
W
θ
→
as
22
2
1
,
2
GmKWW
m
ω
ρ

Θ++Θ=Θ


†
(13)
where

is the eigenvalue of extra space. In Equation (13), the unitary operator
W describes the periodic potential in the extra space, the parameter K is used for
reducing the Equation (13) to the simple harmonic oscillator when
ρ →∞
.
In order to determine the value of
K , we set
0 α =
and simplify Equation (13)
as
()
22
2
22
d
2cos0.
2d
mK
m
ωθ
ρθ
Θ
+−Θ=


(14)
Let
2 ηθ =
, Equation (14) becomes
()
22
2
22
d
48cos20.
2d
mK
m
ωη
ρη
Θ
+−Θ=


(15)
with
()
()
π ηη ΘΘ+
. Moreover, we define
1
2π,0π,yy
θηρ
ρ
==+≤≤
(16)
and set
2
2EmK ω =+
, Equation (15) can be rewritten as
2
22
22
22
d1sin2
0.
22
d
y
EmKy
my
y
ρ
ω
ρ
ρ


Θ

+−Θ=





(17)

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ravitation and Cosmology
When
ρ →∞
, Equation (17) must reduce to the simple harmonic oscillator.
This means the coefficient of
2
y
equals 1. Then we have
2
1
.
2
K ρ
=
Substituting the value of
K back into Equation (15), and set
22
8am ρ =
,
2242
4qm ωρ =
, we obtain the Mathieu equation
()
2
2
d
2cos20.
d
aq η
η
Θ
+−Θ=


(18)
The solutions of Mathieu’s equation cannot be expressed in terms of hyper-
geometric functions. We address two extremal situations in the following dis-
cussions.
1)
1q
case. This situation corresponds to the large extra dimension. The
characteristic values
j
a
of Mathieu function, which represent energy spectrum
of harmonic oscillator in the compactified space, are given by [39] [40]
()
()
()
2342
312
7
135349
22
22
2
jjjjj
jj
vvvvv
aqvq
q
q
++++
≈−+−−−−
(19)
where
21
j
vj=+
and
2242
4qm ωρ =

. Define [39]
2
2
,q
m
ω
ρ
′
=

(20)
the spectrum in (15) is given by
(
)()
(
)
2342
242132
82
135349
1
4
2
22
2
jjjjj
vvvvv
q
j
mmmq
mq
ω
ρρρ
ρ
++++

′
−++−−−−



(21)
It includes the usual energy spectrum of simple harmonic oscillator and the
energy spectrum which comes from the quantization of extra dimension.
2)
1q
case. This situation corresponds to the small extra dimension.
When
1q

, the solution to Equation (18) has two branches, one is even, the
other is odd [41]. In our model,
η
has the periodicity
π
, so we only adopt the
even solutions in [41]. We rearrange the notation of the even solutions as follows
(
)
()
()
(
)
[
]
2
222
0
2
,,cos2,
π
m
lmmr
r
qceqAqr
ηηη
∞
=
=
Θ≡≡
∑
(22)
()
()
()
()()
22
212 222
0
2
,,sin22,
π
m
lmmr
r
qseqBqr ηηη
∞
+
=+++
=
Θ≡≡+


∑
(23)
where
0,1,m=
, and [40]
()()
()
()
()
22
2222
21!
,0,220,
!21!
mm
r
mrmr
mr
ABtrmr
rm
−−
−−
>−≥
−

(24)
()()
()
()
()
()
22
2222
2!
1,0,20,
!2!
r
mm
r
mrmr
m
ABtrm
rrm
++
−>>
+

(25)
with
4tq=
.
The normalization of wave function
()
,
q η Θ
requires the functions
2m
ce

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and
2m
se
normalized as follows
(
)
()
()()
π
π
2222
00
π
,,d,,d.
2
mnmnmn
ceqceqseqseq
ηηηηηηδ ==
∫∫
(26)
From Equation (22) and Equation (23), the Mathieu Equation (18) can fall
into into two categories
()
2
2
22
2
d
2cos20,
d
m
mm
ce
aqce η
η
+−=


(27)
()
2
22
2222
2
d
2cos20.
d
m
mm
se
bqse η
η
+
++
+−=


(28)
Corresponding to the wave function (22) and (23), we write the energy spec-
trum as
222122
,,0,1,
lmmlmm
aaabm
==++
≡≡=

(29)
More specifically, for
3m≤
, we have the energy spectrum that [42]
()
246810
0
172968687
,
2128230418874368
aqqqqq=−+−++
(30)
()
246810
1
1528921391
4,
121382479626240458647142400
aqqqqq=−+−++
(31)
()
246810
2
576310024011669068401
4,
121382479626240458647142400
aqqqqq=+− +−+
(32)
()
2468
3
131710049
16.
308640002721600000
aqqqq=+−−+
(33)
()
2468
4
14335701
16,
308640002721600000
aqqqq=++−+
(34)
()
2468
5
11875861633
36,
704390400092935987200000
aqqqq=++−+
(35)
()
2468
6
11876743617
36.
704390400092935987200000
aqqqq=++++
(36)
Notice that when
l→∞
,
()
1ll
ll
abql
−
−=
. In fact, for
3m>
and
2lm=
, we have
()
()
(
)()
(
)
()()
()
24426
2
2
35
2
22222
5795829
21
321464149
ll
lqllq
q
abl
l
lllll
+++
≈≈++++
−
−−−−−

(37)
Combining the spectrums in the part of extra dimension and the part of Euc-
lidean space, one can easily find the total energy spectrum of simple harmonic
oscillator:
()
()
()()
2
0
00
or
2
,1,
1
,1.
2
2
j
xy
sjl
l
aq
EEEs
aq
m
ω
ρ


=+=++








(38)
4. The Energy Spectrum with GUP
Now, we begin to modify the commutator relations. The Schrödinger equation is

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given by Equation (7), and p is determined by Equation (6). The first-order cor-
rections of the energy eigenvalues are given by Equation (9). We only consider
the small extra dimension, that is,
1q

. Then we rewrite Equation (9) as
(
)(
)
(
)
(
)
(
)
()
22
1000
2
442.
nlnlnl
EmEVmEEVV
ββ

=−=−+


(39)
From Equation (10) and Equation (18), the potential is
222 2
1
4cos2.
2
Vmxm ωωρη
=+
(40)
Then using Equations (22)~(25), we have
222 2
22
1
4cos2
2
11
4,
22
l
Vmxm
nm
ωωρη
ωωρ
=+

=++Λ



(41)
(
)
22442422244
2
22
232244
1
4cos2162
cos
4
1
663416,
162
ll
Vmxmxm
nnmnm
ωωρηωρη
ω
ωρωρ
=++

=+++Λ++Ξ




(42)
where
(
)(
)
()
(
)
2222
202222
0
,
mmmm
lmrr
r
AAAA
∞
=+
=
Λ=+
∑
(43)
()
()
2222
212224
0
,
mm
lmrr
r
BB
∞
++
=+++
=
Λ=
∑
(44)
()()
(
)
()
2222
204224
0
111
,
222
mmmm
lmrr
r
AAAA
∞
=+
=
Ξ=++
∑
(45)
()
()
2222
212226
0
11
,
22
mm
lmrr
r
BB
∞
++
=+++
=
Ξ=+
∑
(46)
Therefore, we obtain the total energy spectrum
(
)()
()
()
()
2
000
22
22
232244
1
48
2
1
663416,
162
nlnlnlnll
ll
EEmEEnm
nnmnm
βωωρ
ω
ωρωρ



=+−++Λ








++++Λ++Ξ







(47)
with
()
2
0
2
1
.
2
2
l
nl
a
En
m
ω
ρ

=++




(48)
Now, for analyzing the result, we should calculate the value of
l
Λ
and
l
Ξ
.
We know that when
0q→
,
()()
1
mm
mm
AB==
. From Equation (24) and Equation
(25), we calculate four states’ values of
l
Λ
and
l
Ξ
()
()
21
0
2
0
1
,
!1!
r
l
r
tt
rr
∞
+
=
=
Λ =−+−
+


∑
(49)

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ravitation and Cosmology
()()()
()
21
1
0
4
,
!1!2!3!
r
l
r
t
rrrr
∞
+
=
=
Λ=−
+++
∑
(50)
()()
()
21
2
2
0
4
2,
!1!2!
r
l
r
tt
rrr
∞
+
=
=
Λ=+−
++


∑
(51)
()()
()
21
3
2
0
1576
,
3
!1!4!
r
l
r
tt
rrr
∞
+
=
=
Λ=+−
++


∑
(52)
and
()
222
0
2
0
1111
,
822
!2!
r
l
r
tt
rr
∞
+
=
=
Ξ=++
+


∑
(53)
()
()
22
1
2
0
141
,
22
!2!4!
r
l
r
t
rrr
∞
+
=
=
Ξ=+
++


∑
(54)
(
)()
222
2
2
0
1141
,
322
!2!4!
r
l
r
tt
rrr
∞
+
=
=
Ξ=−++
++


∑
(55)
()
()()
222
3
0
115761
.
102!2!4!6!2
r
l
r
tt
rrrr
∞
+
=
=
Ξ=−++
+++
∑
(56)
Now, considering Equation (47), and using
(
)
()
()
2222
2
0
22
33
66663,
816216
n
Ennnn
ωω

=++++




(57)
we obtain the asymptotic formalism of total energy spectrum
()()()
()
000
222
22
0
222
22
344
3
116
2
14232
2
64.
nlnnl
ll
nl
l
EEmEm
aa
mEm
m
m
βωβρ
ββωβρ
ρρ
ωβρ

+−Λ



+++−Λ


+Ξ


(58)
In Equation (58), the terms
2
βρ
are small, and which can be neglected.
Then Equation (58) should be rewritten as
()()()
22
000
22
3
1142.
2
2
ll
nlnnn
aa
EEmEmE
m
βββ
ρρ


++++






(59)
This expression has same expression with our previous work [26]. According
to the analysis in [26], for the small extra space, the second term is larger than
the first term since it has the term
2
1 ρ
. In the high energy case, the leading
order of spectrum, which includes the first excited state of extra dimension effect,
can be written as
22
11
11
22
12.
2
aa
E
m
β
ρρ

+




(60)
In an intermediate energy scale where the gravity becomes important, we have
0
.
IP
MM β 
(61)

B. R. Mu et al.
DOI:
10.4236/jhepgc.2019.51015 287 Journal of High Energy Physics, G
ravitation and Cosmology
It is easy to find
22
11
11
222
12.
2
I
II
aa
ME
MM
ρρ

+




(62)
With simple calculation, we obtain
2
22
11
2222
1,
2
II
aa
MM
ρρ

+




(63)
this gives us the surprise relation
min0
1and.
Ip
Mx ρββρ ∆==
(64)
5. Conclusion
In summary, we have discussed the GUP effect of harmonic oscillator in the
compactified extra dimension. We use the modified momentum operator and
modified harmonic oscillator potential to build the periodic Hamiltonian. The
correspondent wave equation is so-called Matheu function. Using the operators
of GUP to modify this periodic Hamiltonian, we find the modified Hamiltonian.
For obtaining the modified energy levels, we use the perturbation method for
Matheu function. Keeping the leading order of the energy levels, we obtain the
same energy eigenvalue formation with our previous work [26]. This result
shows the minimum observable length equals the compactification radius of the
extra dimension.
Acknowledgements
We are grateful to Houwen Wu and Zheng Sun for useful discussions. This work
is supported in part by NSFC (Grant Nos. 11747171) and Natural Science Founda-
tion of Chengdu University of TCM (Grants Nos. ZRYY1729 and ZRQN1656). Dis-
cipline Talent Promotion Program of “Xinglin Scholars” (Grant No. QNXZ2018050)
and the key fund project for Education Department of Sichuan (Grant No.
18ZA0173).
Conflicts of Interest
The authors declare no conflicts of interest regarding the publication of this pa-
per.
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