Coherent States for Tremblay-Turbiner-Winternitz Potential
In this study, we construct the coherent states for a particle in the Tremblay-Turbiner-Winternitz potential by finding the conserved charge coherent states of the four harmonic oscillators in the polar coordinates. We also derive the energy eigenstates of the potential and show that the center of t...
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irk-123456789-1486792019-02-19T01:25:05Z Coherent States for Tremblay-Turbiner-Winternitz Potential Sucu, Y. Unal, N. In this study, we construct the coherent states for a particle in the Tremblay-Turbiner-Winternitz potential by finding the conserved charge coherent states of the four harmonic oscillators in the polar coordinates. We also derive the energy eigenstates of the potential and show that the center of the coherent states follow the classical orbits of the particle. 2012 Article Coherent States for Tremblay-Turbiner-Winternitz Potential / Y. Sucu, N. Unal // Symmetry, Integrability and Geometry: Methods and Applications. — 2012. — Т. 8. — Бібліогр.: 25 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 81R30; 81Q05; 81Q80; 81S99 DOI: http://dx.doi.org/10.3842/SIGMA.2012.093 http://dspace.nbuv.gov.ua/handle/123456789/148679 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України |
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In this study, we construct the coherent states for a particle in the Tremblay-Turbiner-Winternitz potential by finding the conserved charge coherent states of the four harmonic oscillators in the polar coordinates. We also derive the energy eigenstates of the potential and show that the center of the coherent states follow the classical orbits of the particle. |
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Coherent States for Tremblay-Turbiner-Winternitz Potential |
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Coherent States for Tremblay-Turbiner-Winternitz Potential |
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Coherent States for Tremblay-Turbiner-Winternitz Potential |
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Coherent States for Tremblay-Turbiner-Winternitz Potential / Y. Sucu, N. Unal // Symmetry, Integrability and Geometry: Methods and Applications. — 2012. — Т. 8. — Бібліогр.: 25 назв. — англ. |
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Symmetry, Integrability and Geometry: Methods and Applications |
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 8 (2012), 093, 9 pages
Coherent States for Tremblay–Turbiner–Winternitz
Potential?
Yusuf SUCU and Nuri UNAL
Department of Physics, Faculty of Science, Akdeniz University, 07058 Antalya, Turkey
E-mail: ysucu@akdeniz.edu.tr, nuriunal@akdeniz.edu.tr
Received July 31, 2012, in final form November 28, 2012; Published online December 01, 2012
http://dx.doi.org/10.3842/SIGMA.2012.093
Abstract. In this study, we construct the coherent states for a particle in the Tremblay–
Turbiner–Winternitz potential by finding the conserved charge coherent states of the four
harmonic oscillators in the polar coordinates. We also derive the energy eigenstates of the
potential and show that the center of the coherent states follow the classical orbits of the
particle.
Key words: Tremblay–Turbiner–Winternitz potential; generalized harmonic oscillator; non-
central potential; coherent state
2010 Mathematics Subject Classification: 81R30; 81Q05; 81Q80; 81S99
1 Introduction
The coherent states were derived for the one-dimensional linear harmonic oscillator by Schrö-
dinger [11] and used in the quantum theory of electrodynamics in 1963 and were recognized as
the Glauber states [4, 5]. The coherent states are particle like localized, non-dispersive solutions
of the linear Schrödinger equation. In these states, the probability density is a time dependent
Gaussian wave packet and the center of the packet follows the classical trajectory of the particle.
Recently, the coherent states were constructed for the Kepler problem by transforming it into
four harmonic oscillators evolving in a parametric-time [17]. This technique has been applied
to derive the coherent states for a particle in the Morse potential [18], 5-dimensional Coulomb
potential [21] and the non-central generalized MIC-Kepler potential [20]. For the coherent states,
the expectation values of the position and momenta give the classical trajectories. Furthermore,
the stationary quantum eigenstates and corresponding eigenvalues may be derived from the
coherent states. Therefore, they are very important to discuss the relation between classical and
quantum mechanics of a system.
Two important properties of a physical system in classical and quantum mechanics are the
exact solvability and superintegrability. Some time ago, Tempesta, Turbiner and Winternitz
conjectured that for two dimensional systems, all maximally superintegrable systems are exactly
solvable [13]. Recently, in two dimensional plane, the following Hamiltonian has been proposed
by Tremblay, Turbiner and Winternitz
H = p2
r +
p2
θ
r2
+ ω2r2 +
1
r2
(
αk2
sin2 kθ
+
βk2
cos2 kθ
)
, 0 6 θ 6
π
2k
and 0 6 r <∞, (1)
where ω is the angular frequency of the oscillators, α and β are nonnegative constants and k is
constant. The Schrödinger equation has been exactly solved for this Hamiltonian [12]. It includes
all known examples of the two dimensional superintegrable systems for k = 1, 2, 3 [2, 3, 9, 15, 25]
?This paper is a contribution to the Special Issue “Superintegrability, Exact Solvability, and Special Functions”.
The full collection is available at http://www.emis.de/journals/SIGMA/SESSF2012.html
mailto:ysucu@akdeniz.edu.tr
mailto:nuriunal@akdeniz.edu.tr
http://dx.doi.org/10.3842/SIGMA.2012.093
http://www.emis.de/journals/SIGMA/SESSF2012.html
2 Y. Sucu and N. Unal
or α = 0 [1]. The quantum superintegrability of the system has been also discussed for odd k
values [10]. The conjecture about the superintegrability of the system has been supported by
showing that for the classical version of the system all bounded trajectories are closed for all
integer and rational values of the constant k [14]. It has been proven that such systems in
classical mechanics are superintegrable and supported the conjecture that all the orbits are
closed for all rational values of k [8]. Also, in order to present a constructive proof for the
quantum superintegrable systems for all rational values of k, the canonical operator method has
been applied [7]. In this connection, the superintegrability of a system is related into, in classical
sense, the presence of closed and periodic orbits and, in quantum mechanics sense, the presence
of a non-Abelian algebra for its integrals of motion and degenerate energy levels.
The k = 1 case of equation (1) corresponds one of the four maximally superintegrable
Smorodinsky–Winternitz potentials for which coherent states have been constructed [16, 19, 23].
The symmetry algebra of the system in k = 1 case is quadratic, but, for other integer and rational
values of k, these systems are separable in only one coordinate frame and the symmetry algebra
is not quadratic. In classical and quantum sense, the integrability properties of the system have
been investigated, in detail, for the other values of k, but the coherent states hasn’t been dis-
cussed so far. Since the coherent states are the macrostates as a superposition of microstates
of a system, they give both the classical and quantum behavior of the system. Therefore, it is
interesting to generalize the coherent states and to find the expectation values of a system in
equation (1) for the all integer and rational values of k.
The aim of this study is to construct the coherent states for a particle in a potential given
by equation (1). In Section 2 we first discuss the mapping of the two dimensional system
into four dimensional system in the harmonic oscillator potential, and the second, construct
the coherent states for the four harmonic oscillators and the third, find the coherent states for
the Hamiltonian H. We also derive the energy eigenstates of the system in polar coordinates.
Section 3 we find the expectation values of radial and angular coordinates of the particle in
the Tremblay–Turbiner–Winternitz (TTW) potential between the coherent states of the system.
Section 4 is our conclusions.
2 Coherent states for Tremblay–Turbiner–Winternitz
potential
To derive the coherent states for the TTW potential, we start by writing the action A in the
following way
A =
∫
Ldt =
∫ (
pr
dr
dt
+ pθ
dθ
dt
−H
)
dt,
where pr and pθ are the radial and angular momenta of a particle in the two dimensional space
with the following line element
ds2 = (dr)2 + r2(dθ)2.
The Hamiltonian H is similar to the radial Hamiltonian of the four harmonic oscillators. We
change the angular coordinate θ as Θ = kθ, with 0 6 Θ 6 π
2 . Then the Hamiltonian becomes
H =
(
p2
r +
k2L2
r2
)
,
where the constant of motion L2 is given as
L2 =
[
p2
Θ +
(
α
sin2 Θ
+
β
cos2 Θ
)]
.
Coherent States for Tremblay–Turbiner–Winternitz Potential 3
Therefore, to construct the coherent states of the system, we first reduce the system into the
k = 1 case and take the coherent states of the Smorodinsky–Winternitz potential. We finally
replace the quantum numbers corresponding to L2 by k2L2.
In order to represent these potentials as the kinetic energy contribution of the angular mo-
menta of the particle on S3, we first introduce the cartesian coordinates of the particle u and v
as
(u, v) =
√
ωr(cos Θ, sin Θ),
with the condition of u, v ≥ 0 which corresponds to represent the system as two isotropic
polar harmonic oscillators with centrifugal barrier
(
α
u2
+ β
v2
)
. In polar coordinates
(
∂
∂u ,
∂
∂v
)
Ψ is
replaced by
(
∂
∂u + 1
2u ,
∂
∂v + 1
2v
)
Ψ√
uv
and centrifugal barrier
(
α
u2
+ β
v2
)
become
(
α+1/4
u2
+ β+1/4
v2
)
.
By using this prescription we get the following Hamiltonian in dimensionless units
H̃ = p2
u + p2
v + u2 + v2 +
α+ 1
4
u2
+
β + 1
4
v2
.
Second, we introduce two dummy angles φ and ψ by the Lagrange multipliers without changing
the dynamics of the particle. We also change pr and pΘ as the radial and angular momenta of
a particle in four dimensional space with the the line element
ds2 = (du)2 + (dv)2 + u2(dφ)2 + v2(dψ)2.
Thus the Lagrangian L becomes
L̃ = pu
du
dωt
+ pv
dv
dωt
+
dφ
dωt
(
pφ −
√
α+
1
4
)
+
dψ
dωt
(
pψ −
√
β +
1
4
)
− H̃,
where the Hamiltonian H̃ is rewritten as
H̃ = p2
u + p2
v + u2 + v2 +
p2
φ
u2
+
p2
ψ
v2
.
The coherent states are given in configuration space [22] ui, vi with i = 1, 2 as
Ψ̃λi(t)(ui) = N
4∏
i=1
e−4iωte−
1
2
u2+κi(t)uie−
1
2
v2+λi(t)vi .
Here we use the natural units ~ = 1 and the complex eigenvalues of the harmonic oscillators
lowering operators κi(t) and λi(t) are given as
κi(t) = κi(0)e−2iωt, λi(t) = λi(0)e−2iωt.
In polar coordinates (u, φ) and (v, ψ) the coherent states are given as
Ψ̃λi(t)(ui) = Ne−4iωte−
1
2
u2e
u
(
κ1(t)−iκ2(t)
2
eiφ+
κ1(t)+iκ2(t)
2
e−iφ
)
e
v
(
λ3(t)−iλ4(t)
2
eiψ+
λ1(t)+iλ2(t)
2
e−iψ
)
.
If we expand the exponential expressions into the power series we get
Ψ̃λi(t)(ui) = Ne−4iωte−
1
2(u2+v2)
+∞∑
µ1=−∞
[K(0)]µ1Jµ1
(√
u2(−iκ(t))2
)
exp iµ1φ
×
+∞∑
µ2=−∞
[Λ(0)]µ2Jµ2
(
2
√
v2((−iλ(t)))2
)
exp iµ2ψ, (2)
4 Y. Sucu and N. Unal
where K(0), Λ(0) and κ(t), λ(t) are defined as
κ(t) =
√
[λ2(t) + iλ1(t)][λ2(t)− iλ1(t)], λ(t) =
√
[λ4(t) + iλ3(t)][λ4(t)− iλ3(t)],
K(0) =
√
[λ2(t) + iλ1(t)]/[λ2(t)− iλ1(t)], Λ(0) =
√
[λ4(t) + iλ3(t)]/[λ4(t)− iλ3(t)].
We notice that in equation (2) the eigenvalues K(0) and Λ(0) are time independent and they
correspond to the conserved quantities (charges) L12 and L34 or pφ and pψ. Therefore, the
eigenvalues corresponding to the polar parts of the four oscillators are time dependent.
To derive the coherent states of the physical particle Ψ̃phys we consider elimination of the
dummy coordinates φ and ψ. There are two methods of elimination: in wave function formalism
we consider physical eigenvalues of corresponding conjugate momenta p̂φ and p̂ψ by taking care
of that these dummy coordinates are not cyclic and the conjugate momenta have continuous
eigenvalues
p̂φΨ̃phys =
√
α+
1
4
Ψ̃phys, p̂ψΨ̃phys =
√
β +
1
4
Ψ̃phys.
In the path integration formalism, we integrate over all the possible final values of these variables.
Then Ψ̃phys becomes
Ψ̃phys
λi(t)
(ui) = Ne−4iωte−
1
2(u2+v2)[K(0)eiφ
]pφ[Λ(0)eiψ
]pψ
× Jpφ
(√
u2(−iκ(t))2
)
Jpψ
(√
v2(−iλ(t))2
)
. (3)
We parameterize the product of two Bessel functions as
Jpφ
(√
u2(−iκ(t))2
)
Jpψ
(√
v2(−iλ(t))2
)
= Jpφ
(√
u2 + v2 cos Θ(−i)
√
κ2 + λ2 cos Φ
)
Jpψ
(√
u2 + v2 sin Θ(−i)
√
κ2 + λ2 sin Φ
)
,
where
(u, v) =
√
u2 + v2(sin Θ, cos Θ), (κ, λ) =
√
κ2 + λ2(sin Φ, cos Φ).
We write the product of two Bessel functions Jpφ and Jpψ in terms of one Bessel function [24]
Jpφ
(
−i
√
u2 + v2 sin Θ
√
κ2 + λ2 sin Φ
)
Jpψ
(
−i
√
u2 + v2 cos Θ
√
κ2 + λ2 cos Φ
)
=
∞∑
l1=0
N
(pφ,pψ)
l1
J2l1+pφ+pψ+1
(
−i
√
u2 + v2
√
κ2 + λ2
)
−2i
√
u2 + v2
√
κ2 + λ2
dl1pφ,pψ(cos 2Φ)dl1pφ,pψ(cos 2Θ),
where the constant N
(pφ,pψ)
l1
is
N
(pφ,pψ)
l1
=
i(−1)l1 l1!Γ(pφ + pψ + l1 + 1)
Γ(pφ + l1 + 1)Γ(pφ + l1 + 1)
,
and the angular wave functions dl1pφ,pψ(cos 2Θ) are defined in terms of Jacobi polynomials
P l1pφ,pψ(cos 2Θ) as
dl1pφ,pψ(cos 2Θ) = (sin Θ)pφ(cos Θ)pψP l1pφ,pψ(cos 2Θ).
In previous equation the generalized conserved charge coherent states are parameterized by the
time dependent eigenvalues√
κ2 + λ2 =
√
κ2(0) + λ2(0)e−2iωt,
Coherent States for Tremblay–Turbiner–Winternitz Potential 5
and the time independent eigenvalues√
κ2/λ2 =
√
κ2(0)/λ2(0).
Here
√
κ2/λ2 corresponds to the conserved charge related to L2. In equation (3) the time
independent phase factors are omitted.
Then except some constant phase factors Ψ̃phys becomes
Ψ̃phys√
κ2+λ2
(r,Θ) =
N
4
e−4iωte−
1
2
ωr2
∞∑
l1=0
N
(pφ,pψ)
l1
dl1pφ,pψ(cos 2Φ)
×
J2l1+pφ+pψ+1
(
2
√
ωr2
√
(−i)2(κ2+λ2)
2
)
√
ωr2
√
(−i)2(κ2+λ2)
2
dl1pφ,pψ(cos 2Θ).
Here we expand the Bessel functions in terms of Laguerre functions [6]. The conserved charge
coherent states of the four oscillators are given as
Ψ̃phys√
κ2+λ2
(r,Θ) =
N
4
∞∑
l1=0
N
(pφ,pψ)
l1
dl1pφ,pψ(cos 2Φ) (sin kθ)pφ (cos kθ)pψ P l1pφ,pψ
(
2 sin2 kθ − 1
)
×
∞∑
nr=0
e
(|κ|2+|λ|2)
4 e−4iωt
(
−(κ2+λ2)
4
)(2l1+pφ+pψ+1)
2
+nr
Γ (nr + 2l1 + pφ + pψ + 2)
× e−
1
2
ωr2
(
ωr2
)(2l1+pφ+pψ+1)−1
2 L
(2l1+pφ+pψ+1)
nr
(
ωr2
)
.
In order to find the coherent states of a system described by TTW Hamiltonian H for the
terms under the nr summation, we first replace (2l1 + pφ + pψ + 1) by (2l1 + pφ + pψ + 1) k and
multiply the wave function Ψ̃phys√
κ2+λ2
(r,Θ) by
√
ωr2 sin Θ cos Θ.
Then, we find the coherent states of the TTW system as
Ψphys√
κ2+λ2
(r, θ) =
N
4
∞∑
l1=0
N
(pφ,pψ)
l1
dl1pφ,pψ(cos 2Φ)(sin kθ)pφ+ 1
2 (cos kθ)pψ+ 1
2
× P l1pφ,pψ
(
2 sin2 kθ − 1
) ∞∑
nr=0
e
(|κ|2+|λ|2)
4 e−4iωt
(
−(κ2+λ2)
4
) (2l1+pφ+pψ+1)k
4
+nr
2
Γ[nr + (2l1 + pφ + pψ + 1)k + 1]
× e−
1
2
ωr2
(
ωr2
) (2l1+pφ+pψ+1)k
2 L
(2l1+pφ+pψ+1)k
nr
(
ωr2
)
.
Here time independent energy eigenstates are given as
Ψphys
nr,l1
(r, θ) = (sin kθ)pφ+ 1
2 (cos kθ)pψ+ 1
2P l1pφ,pψ
(
2 sin2 kθ − 1
)
× e−
1
2
ωr2
(
ωr2
) (2l1+pφ+pψ+1)k
2 L
(2l1+pφ+pψ+1)k
nr
(
ωr2
)
,
and the energy eigenvalues are
E = 2(2l1 + pφ + pψ + 1)k + 2nr. (4)
The spectrum is degenerate for integer and rational values of k. So, the classical orbits are
closed for these k cases. The energy eigenstates and eigenvalues are the same with the result
given in [12].
6 Y. Sucu and N. Unal
3 Expectation values
For the coherent states of the isotropic four harmonic oscillators the expectation values in the
configuration space 〈ui〉, 〈vi〉 are given as
〈ui〉t =
κi(t) + κ∗i (t)
2
= |κi(0)| cos(2ωt− ϕi),
and
〈vi〉t =
λi(t) + λ∗i (t)
2
= |λi(0)| cos(2ωt− χi),
where ϕi and χi are the phases of κi(0) and λi(0). Then for the square of radial coordinate 〈ui〉2t
the expectation value is given as
〈u〉2t = 〈u1〉2 + 〈u2〉2 =
|κ1(0)|2 + |κ2(0)|2
2
+
|κ(0)|2
2
cos(4ωt− 2ϕ).
Here we have neglected the uncertainties and the constants, κ(0) and 2ϕ are given as
|κ(0)|2 =
[
|κ1(0)|4 + |κ2(0)|4 + 2|κ1(0)κ2(0)|2 cos 2ϕ1 − ϕ2)
] 1
2 ,
and
tan 2ϕ =
|κ1(0)|2 sin 2ϕ1 + |κ1(0)|2 sin 2ϕ2
|κ1(0)|2 cos 2ϕ1 + |κ1(0)|2 cos 2ϕ2
.
In order to derive the expectation values between the conserved charge coherent states, we
evaluate the angular momentum L12 as
L12 = −(κ1(t)κ∗2(t)− κ2(t)κ∗1(t))
2i
= − |κ1(0)κ2(0)| sin (ϕ1 − ϕ2) .
We fix the value of L12 as
√
α+ 1
4 . Then
cos 2(ϕ1 − ϕ2)
2
= 1−
2
(
α+ 1
4
)
|κ1(0)κ2(0)|2
,
and
|κ(0)|2 =
[(
|κ1(0)|2 + |κ2(0)|2
)2 − 4
(
α+
1
4
)] 1
2
.
As a result of this, for k = 1, 〈u〉2t is given as
〈u〉2t = 〈u1〉2 + 〈u2〉2
=
|κ1(0)|2 + |κ2(0)|2
2
+
[(
|κ1(0)|2 + |κ2(0)|2
)2
4
−
(
α+
1
4
)] 1
2
cos(4ωt− 2ϕ). (5)
In the similar way 〈v〉2t is given as
〈v〉2t = 〈v1〉2 + 〈v2〉2
=
|λ1(0)|2 + |λ2(0)|2
2
+
[(
|λ1(0)|2 + |λ2(0)|2
)2
4
−
(
β +
1
4
)] 1
2
cos(4ωt− 2χ).
Coherent States for Tremblay–Turbiner–Winternitz Potential 7
These are the expectation values of 〈u〉2t and 〈v〉2t between the conserved charge coherent states
for the radial oscillators given by the Hamiltonian H̃. In two-dimensional oscillator with polar
coordinates (u, φ) the expectation value of unit vector
〈
φ̂
〉
also oscillates in time. For the
reduction of dimension we fix the oscillations in 〈φ̂〉 and take it as a constant unit vector.
To find the expectation value of radial coordinates we arrange the following product[
κ2
1(t) + κ2
2(t) + λ2
1(t) + λ2
2(t)
] [
κ∗21 (t) + κ∗22 (t) + λ∗21 (t) + λ∗22 (t)
]
=
[
|κ1(0)|2 + |κ2(0)|2 + |λ1(0)|2 + |λ2(0)|2
]2
+ [κ1(0)κ∗2(0)− κ2(0)κ∗1(0)]2 + [λ1(0)λ∗2(0)− λ2(0)λ∗1(0)]2
+ [κ1(0)λ∗1(0)− λ1(0)κ∗1(0)]2 + [κ1(0)λ∗2(0)− λ2(0)κ∗1(0)]2
+ [κ2(0)λ∗1(0)− λ1(0)κ∗2(0)]2 + [κ2(0)λ∗2(0)− λ2(0)κ∗2(0)]2
= 4
(
E
ω
)2
− 4L2.
Then, for k = 1 case, 〈r〉2t becomes
ω〈r〉2t =
(
E
2ω
)
+
[(
E
2ω
)2
− L2
] 1
2
sin 2(2ωt− δ). (6)
Here the constants of motion are defined as
E
ω
=
|κ1(0)|2 + |κ2(0)|2 + |λ1(0)|2 + |λ2(0)|2
2
,
and
cot 2δ =
|κ(0)|2 sin 2ϕ+ |λ(0)|2 sin 2χ
|κ(0)|2 cos 2ϕ+ |λ(0)|2 cos 2χ
.
In order to derive the expectation values for TTW Hamiltonian we replace L2, α+ 1
2 , and β+ 1
2
by k2L2, k
(
α+ 1
2
)
and k
(
β + 1
2
)
, respectively. Then, ω〈r〉2t becomes
〈r〉2t =
E
2ω2
+
[(
E
2ω2
)2
− A
ω2
] 1
2
sin 4ω(t− t0),
where A is k2L2. The 〈r〉t shows the center of the coherent states following the classical tra-
jectories of the particle in this TTW potential. It should be emphasized that the result on the
radial coordinate of expectation value derived from the coherent states agrees with the result
of [14] in which bounded trajectories are investigated for the integer and rational values of k.
The expectation value of sin Θ may be derived from 〈u〉2t and ω〈r〉2t in equations (5) and (6) as
〈
sin Θ
〉2
=
〈u〉2t
ω〈r〉2t
∣∣∣∣
L2
12=k(α+ 1
2)
.
4 Conclusion
In this study, we have constructed the coherent states for the Tremblay–Turbiner–Winternitz
potential. One of the constants of motion is k2L2. Therefore, for the case k 6= 1, we have
integrated the equations of motion in polar coordinates by using the solutions for k = 1 case,
8 Y. Sucu and N. Unal
since the separability in this coordinates do not depend on the value of k. For k = 1 case, the
Schrödinger equation is also separable in cartesian coordinates (u, v).
For the Tremblay–Turbiner–Winternitz potential, the coherent states correspond to the con-
served charge coherent states of the four oscillators with the eigenvalues of k
√
α+ 1
4 , k
√
β + 1
4
and k2 [2l1(2l1 + pφ + pψ + 1)] of the operators kL12, kL34, and k2
[
p2
Θ +
(
α
sin2 Θ
+ β
cos2 Θ
)]
, re-
spectively. The energy eigenvalues are given as by equation (4), and as it was shown previously
in [12], the spectrum becomes degenerate for the integer and rational values of the parameter k.
This result corresponds to the periodicity of the Hamiltonian: if the Hamiltonian has angular
periodicity of πm
n for any of two integers m and n, then k must be an integer or rational num-
ber. Furthermore, we have evaluated the expectation value of the radial coordinate 〈r〉t, and
shown that the center of the coherent states follow the classical trajectories of the particle in
the Tremblay–Turbiner–Winternitz potential.
Acknowledgements
This work was supported by Akdeniz University, Scientific Research Projects Unit.
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1 Introduction
2 Coherent states for Tremblay-Turbiner-Winternitz potential
3 Expectation values
4 Conclusion
References
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