Acta Polytechnica


Acta Polytechnica 53(3):268–270, 2013 © Czech Technical University in Prague, 2013
available online at http://ctn.cvut.cz/ap/

THE TWO-DIMENSIONAL HARMONIC OSCILLATOR ON A
NONCOMMUTATIVE SPACE WITH MINIMAL UNCERTAINTIES

Sanjib Dey∗, Andreas Fring

Department of Mathematical Science, City University London, Northampton Square, London EC1V 0HB, UK
∗ corresponding author: sanjib.dey.1@city.ac.uk

Abstract. The two dimensional set of canonical relations giving rise to minimal uncertainties
previously constructed from a q-deformed oscillator algebra is further investigated. We provide a
representation for this algebra in terms of a flat noncommutative space and employ it to study the
eigenvalue spectrum for the harmonic oscillator on this space. The perturbative expression for the
eigenenergy indicates that the model might possess an exceptional point at which the spectrum becomes
complex and its PT-symmetry is spontaneously broken.

Keywords: noncommutative space, non-Hermitian operators, 2D-systems.

In [1] we demonstrated how canonical relations im-
plying minimal uncertainties can be derived from a
q-deformed oscillator algebra for the creation and an-
nihilation operators A†i, Ai

AiA
†
j −q

2δij A
†
jAi = δij,

[A†i,A
†
j] = 0,

[Ai,Aj] = 0,


 for i,j = 1, 2, 3; q ∈ R,(1)

as investigated for instance in [2–6]. Starting from
the general Ansatz

X = κ̂1(A
†
1 + A1) + κ̂2(A

†
2 + A2)

+ κ̂3(A
†
3 + A3), (2a)

Y = iκ̂4(A
†
1 −A1) + iκ̂5(A

†
2 −A2)

+ iκ̂6(A
†
3 −A3), (2b)

Z = κ̂7(A
†
1 + A1) + κ̂8(A

†
2 + A2)

+ κ̂9(A
†
3 + A3), (2c)

Px = iκ̌10(A
†
1 −A1) + iκ̌11(A

†
2 −A2)

+ iκ̌12(A
†
3 −A3), (2d)

Py = κ̌13(A
†
1 + A1) + κ̌14(A

†
2 + A2)

+ κ̌15(A
†
3 + A3), (2e)

Pz = iκ̌16(A
†
1 −A1) + iκ̌17(A

†
2 −A2)

+ iκ̌18(A
†
3 −A3), (2f)

with κ̂i = κi
√
~/(mω) for i = 1, . . . , 9 and κ̌i =

κi
√
mω~ for i = 10, . . . , 18 we constructed some par-

ticular solutions and investigated the harmonic oscil-
lator on these spaces. Here we provide an additional
two dimensional solution previously reported in [6].
Setting κ3 = κ6 = κ7 = κ12 = κ15 = κ16 = κ17 =
κ18 = 0 in equations (2a)–(2f), employing the con-
straints reported in [6] together with the subsequent

nontrivial limit q → 1, the deformed oscillator algebra

[X,Y ] = iθ
(
1 + τ̂Y 2

)
, [Px,Py] = iτ̂

~2

θ
Y 2,

[X,Px] = i~
(
1 + τ̂Y 2

)
, [Y,Py] = i~

(
1 + τ̂Y 2

)
,

[X,Py] = 0, [Y,Px] = 0, (3)

was obtained, with τ̂ = τmω/~ having the dimension
of an inverse squared length. By the same reasoning as
provided in [1, 5–9], we find the minimal uncertainties

∆Xmin = |θ|
√
τ̂ + τ̂2〈Y 〉2ρ,

∆Ymin = 0,
∆(Px)min = 0,

∆(Py)min = ~
√
τ̂ + τ̂2〈Y 〉2ρ, (4)

where 〈.〉ρ denotes the inner product on a Hilbert
space with metric ρ in which the operators X,Y,Px
and Py are Hermitian. So far no representation for
the two dimensional algebra (3) has been provided.
Here we find that it can be represented by

X = x0 + τ̂y20x0,
Y = y0,
Px = px0,

Py = py0 − τ̂
~
θ
y20x0, (5)

where x0,y0,px0,py0 satisfy the common commutation
relations for the flat noncommutative space

[x0,y0] = iθ, [x0,px0 ] = i~, [x0,py0 ] = 0,
[px0,py0 ] = 0, [y0,py0 ] = i~, [y0,px0 ] = 0,

for θ ∈ R. (6)

Clearly there exist many more solutions that one
may construct in this systematic manner from the

268

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vol. 53 no. 3/2013 Two-dimensional Harmonic Oscillator on a Noncommutative Space

Ansatz (2a)–(2f), which will not be our concern here.
Instead we will study a concrete model, i.e. the two-
dimensional harmonic oscillator on the noncommuta-
tive space described by algebra (3). Using representa-
tion (5), the corresponding Hamiltonian reads

H2Dncho =
1

2m
(P 2x + P

2
y ) +

mω2

2
(X2 + Y 2)

= H2Dfncho +
τ̂

2

[
mω2{y20x0,x0}−

~
mθ
{y20x0,py0}

]
+
τ̂2

2

[
mω2 +

~2

mθ2

]
y20x0y

2
0x0, (7)

where we used the standard notation for the anti-
commutator {A,B} := AB + BA. Evidently this
Hamiltonian is non-Hermitian with regard to the stan-
dard inner product, but respects an antilinear symme-
try PT± : x0 → ±x0,y0 → ∓y0,px0 → ∓px0,py0 →
±py0, i →−i. This suggests that its eigenvalue spec-
trum might be real, or at least real in parts [10–12].
Let us now investigate the spectrum perturbatively
around the solution of the standard harmonic os-
cillator. In order to perform such a computation
we need to convert flat noncommutative space into
the standard canonical variable xs, ys, pxs and pys .
This is achieved by means of a so-called Bopp-shift
x0 → xs − θ~pys , y0 → ys, px0 → pxs and py0 → pys .
The Hamiltonian in (7) then acquires the form

H2Dncho = H
2D
ho +

mθ2ω2

2~2
p2ys −

mθω2

2~
{xs,pys}

+
τ̂

2

[
mω2{y2sxs,xs}−

~
mθ
{y2sxs,pys}

]
+
τ̂

2

[( 1
m

+
mθ2ω2

~2
)
{y2spys,pys}

−
mθω2

~

(
{y2spys,xs} + {y

2
sxs,pys}

)]
−
τ̂2

2

[
mθω2

~
+

~
mθ

](
y2spysy

2
sxs + y

2
sxsy

2
spys

)
+
τ̂2

2

[
1
m

+
mθ2ω2

~2

]
y2spysy

2
spys

+
τ̂2

2

[
mω2 +

~2

mθ2

]
y2sxsy

2
sxs

= H2Dho (xs,ys,pxs,pys ) + H
2D
nc (xs,ys,pxs,pys ). (8)

In this formulation we may now proceed to expand
perturbatively around the standard two dimensional
Fock space harmonic oscillator solution with normal-
ized eigenstates

|n1n2〉 =
(a†1)

n1 (a†2)
n2

√
n1! n2!

|00〉,

a
†
i|n1n2〉 =

√
ni + 1

∣∣(n1 + δi1)(n2 + δi2)〉,
ai|00〉 = 0,

ai|n1n2〉 =
√
ni
∣∣(n1 − δi1)(n2 −δi2)〉, (9)

for i = 1, 2, such that H2Dho |nl〉 = E
(0)
nl |nl〉. The energy

eigenvalues for the Hamiltonian H2Dncho then result to

E
(p)
nl = E

(0)
nl + E

(1)
nl + E

(2)
nl + O(τ

2)

= E(0)nl + 〈nl|H
2D
nc |nl〉

+
∑

p,q 6=n+l=p+q

〈nl|H2Dnc |pq〉〈pq|H2Dnc |nl〉
E

(0)
nl −E

(0)
pq

+ O(τ2)

= ω~(n + l + 1) +
1
16

~ωΩ
[
2n− (2l + 1)Ω + 10l + 6

]
+

1
8
~τω

[
Ω
(
8nl + 4n + 6l2 + 10l + 5

)
+ 10nl + 5n + 5l2 + 10l + 5

]
+ O(τ2), (10)

where Ω = m2θ2ω2/~2. We note the minus sign in
one of the terms, which might be an indication for
the existence of an exceptional point [13, 14] in the
spectrum. Naturally it would be very interesting to
obtain a more precise expression for the eigenenergies,
but nonetheless as has turned out to be very useful
in the one dimensional setting [15] the first order
approximations is very useful for the computation of
coherent states [16].

Acknowledgements
S.D. is supported by a City University Research Fellowship.

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	Acta Polytechnica 53(3):268–270, 2013
	Acknowledgements
	References