Uncertainty relations for modified isotropic harmonic oscillator
and Coulomb potentials
S.H. Patil a , K.D. Sen b,∗
a Department of Physics, Indian Institute of Technology, Mumbai 400 076, India
b School of Chemistry, University of Hyderabad, Hyderabad 500 046, India
Communicated by V.M. Agranovich
Abstract
The dimensional analyses of the position and momentum variances which define the Heisenberg uncertainty product are carried out for two
non-relativistic model central potentials generated by adding a/r 2 term to (i) the isotropic harmonic oscillator, and (ii) the Coulombic hydrogenlike potentials. The uncertainty products are shown to be independent of the scaling of the part (i) and (ii) but are dependent on the strength a of
the additional term. The scaling properties are found to be reflected in the entropic uncertainty measure of the Shannon information entropy sum
and the Fisher information product. Numerical results are presented in support of the analytic results derived.
Keywords: Isotropic harmonic oscillator; H atom; Davidson potential; Kratzer potential; Central potentials in D dimensions; Heisenberg uncertainty relation;
Shannon entropy; Fisher information measure
1. Introduction
Uncertainty relations are the basic properties of quantum
mechanics. In particular, Heisenberg uncertainty principle [1]
for the product of the uncertainties in position and momentum,
in terms of Planck’s constant,
1
σx σp h̄,
2
2 σx2 = x − x ,
2 σp2 = px − px ,
(1)
is an important element of quantum properties. In this case the
Gaussian wave functions have the minimum uncertainty product of h̄/2. For extensive numerical tests of Eq. (1) for the
central potentials we refer the reader to the published literature [2]. Here it may be observed that the uncertainty product
for the bound states in homogeneous power-law potentials, is
independent of the strength of the potential. This follows from
the dimensionality argument that for this case h̄ is the only
quantity which has the dimension of xp. Recently, an extension of this effect for the eigendensities of the homogeneous
potentials has be proposed [3]. An interesting case would be the
uncertainty relation for the bound states for a superposition of
two power-law potentials, and its dependence on the strengths
of the two terms.
We consider two special cases of superpositions of potentials, the isotropic harmonic oscillator (h.o.) potential with an
additional a/r 2 term,
1
a
V1 (r) = kr 2 + 2
2
r
(2)
and the Coulomb potential with an additional a/r 2 term,
Z
a
+ 2.
(3)
r
r
We note here that V1 (r) represents the modified oscillator
potential proposed by Davidson [4] which has been found useful in analyzing [5] the roto-vibrational states of diatomic molecules. A five-dimensional Davidson model potential has been
employed to study the nuclear rotations [6] and vibrations. FurV2 (r) = −
110
ther, a generalization of the free isotropic harmonic oscillator
potential given by Br 2 + A/r 2 , with A and B constants, expressed as the Gol’dman and Krivchenkov Hamiltonian [7] has
been recently used to represent the unperturbed part of a class
of anharmonic singular potentials [8]. Similarly, the potential
V2 (r) can be identified with the Kratzer potential [9,10] which
has been successfully applied in the analysis of diatomic molecular spectra [11]. Recently there has been renewed interest [12]
in finding accurate solutions of Kratzer potential for all angular
momentum states.
We begin with a dimensional analysis for σr and σp where
2 σr2 = r − r ,
2 ,
σp2 = p − p
(4)
which defines the (square of ) Heisenberg uncertainty product,
HP, as σr2 σp2 . The purpose of this Letter is to show that the
HP corresponding to the potentials V1 (r) and V2 (r) is independent of the parameters k and Z, respectively. Equivalently,
HP depends only on the parameter a of the potentials given by
Eqs. (2)–(3). We deduce explicit expressions for these uncertainties using Feynman–Hellmann theorem and other relations.
These expressions lead to the uncertainty relations which confirm the prediction of the dimensional analysis. A representative
set of numerical calculations are presented to verify these results. Further, the arguments based on the inter-dimensional
degeneracy of the Schrödinger equation has been used to discuss the HP in two-dimensional forms of the potentials as well.
We also consider the well-known entropic formulation of the
uncertainty relationship in terms of sum of the Shannon information entropy [13] of the probability density in the position and momentum spaces given by the lower bound due to
Bialynicki-Birula and Mycielski [14], and test its validity and
scaling behavior in case of V1 (r) and V2 (r) for the arbitrary values of the parameters Z, k and a. Finally, the application of our
results in obtaining the Fisher information measure [15] corresponding to a single particle in a central potential is presented.
2. Some general properties
Here some general properties of the uncertainties will be
considered by analyzing the dimensional properties.
For the modified isotropic h.o. potential in Eq. (2), the
Schrödinger equation is
−
h̄2 2
1
a
∇ ψ + kr 2 ψ + 2 ψ = Eψ.
2M
2
r
(5)
Here, the basic parameters are h̄, M, k and a. Of these Ma/h̄2
is the only dimensionless quantity, so that
1 σr2 = f1 Ma/h̄2 ,
α
2
σp = α h̄2 f2 Ma/h̄2 ,
1/2
α = Mk/h̄2
,
(6)
where 1/α has the dimension of (length)2 . This implies that the
uncertainty product
σr2 σp2 = h̄2 f1 Ma/h̄2 f2 Ma/h̄2
(7)
depends on a but is independent of k. It is also interesting to
note that the bound state energies are of the form
E = h̄(k/M)1/2 f3 Ma/h̄2 .
(8)
For the modified Coulomb potential in Eq. (3), the Schrödinger equation is
−
h̄2 2
Z
a
∇ ψ − ψ + 2 ψ = Eψ.
2M
r
r
(9)
Here also Ma/h̄2 is the only dimensionless quantity, so that
2 σr2 = h̄2 /MZ g1 Ma/h̄2 ,
2 σp2 = h̄2 MZ/h̄2 g2 Ma/h̄2 ,
(10)
where h̄2 /MZ is the Bohr radius. This implies that the uncertainty product
σr2 σp2 = h̄2 g1 Ma/h̄2 g2 Ma/h̄2
(11)
depends on a but is independent of Z. The bound state energies
are of the form
E = MZ 2 /h̄2 g3 Ma/h̄2 .
(12)
These forms are implied by just the simple dimensional properties of the parameters. Clearly, the product of the uncertainties
is also of the same form as in Eqs. (7) and (11) for other power
potentials with an additional a/r 2 term.
3. Uncertainty relation for the modified isotropic h.o.
potential
The radial equation for the modified isotropic h.o. potential
in Eq. (2) can be written as
h̄2 d 2
l(l + 1)
1
a
−
−
ηl + kr 2 ηl + 2 ηl = Eηl ,
2M dr 2
2
r2
r
ηl = rRl (r).
(13)
Now the a/r 2 term can be combined with the angular momentum term leading to
l (l + 1)
1
h̄2 d 2
−
ηl + kr 2 ηl = Eηl ,
−
2M dr 2
2
r2
2Ma
⇒
l (l + 1) = l(l + 1) + 2
h̄
1
1 2 2Ma 1/2
l =− + l+
(14)
+ 2
.
2
2
h̄
This equation is of the same form as the usual s.h.o. equation
except that l is replaced by l . The solutions therefore are of the
same form,
E = (k/M)1/2 h̄(2n + l + 3/2),
1/2
,
l + 1/2 = (l + 1/2)2 + 2Ma/h̄2
1
2
Rl = Ar l F −n, l + 3/2, αr 2 e− 2 αr ,
1/2
,
α = Mk/h̄2
(15)
111
where n is the radial quantum number, and F is the confluent
hypergeometric function. Now, since the solutions have welldefined parity, expectation values r and p
are zero, so that
σp2 = p 2 .
σr2 = r 2 ,
(16)
These expectation values can be obtained in a very simple form
by using the Feynman–Hellmann theorem,
∂H /∂λ = ∂E/∂λ,
(17)
of Eq. (20) in that the HP is independent of the parameter k but
depends on a. It is interesting to note that the uncertainty product increases with n and a. In particular, for a → ∞, carrying
out an expansion in inverse powers of a leads to
1/4
,
σr σp → h̄(2n + 1)1/2 2Ma/h̄2
for a → ∞.
(21)
4. Uncertainty relation for the modified Coulomb potential
where λ is a parameter in the Hamiltonian. Taking k and M as
the parameters and using Eqs. (5) and (15), we get
2
r = h̄(Mk)−1/2 (2n + l + 3/2),
(18)
2
2
2Ma/h̄
.
p = h̄(Mk)1/2 (2n + l + 3/2) − (19)
l + 1/2
The radial equation for the modified Coulomb potential in
Eq. (3) reduces to
This leads to
Combining the a/r 2 term with the angular momentum term,
one gets
σr σp = h̄(2n + l + 3/2) 1 −
1/2
2Ma/h̄2
(l + 1/2)(2n + l + 3/2)
1/2
l + 1/2 = (l + 1/2)2 + 2Ma/h̄2
.
,
(20)
This is of the form in Eq. (7) implied by the dimensional properties, independent of parameter k but dependent on a. In Table 1
we have presented the results of our numerical calculations for
the HP using the solution of the Schrödinger equation with the
potential V1 (r) and the method described in our earlier work
[16]. A representative set of our calculations for the three lowest states with l = 0, 1 shown in Table 1 confirm the predictions
Table 1
Numerical results of energy (E), expectation values r 2 , p2 and the product
r 2 p 2 corresponding to the potential V1 (r) = kr 2 /2 + a/r 2 with different
values of k and a. The first three levels with the angular quantum number 0 and
1 are given. At a fixed value of a, the product is independent of k. The entries
in italics display the dependence on the value of a. All values are in a.u.
n+1
k/2
a
E
r 2 p 2 1
1
1
1
2
2
2
2
3
3
3
3
1
1
1
1
2
2
2
2
3
3
3
3
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1
1
1
0.5
0.7
1
0.5
0.5
0.7
1
0.5
0.5
0.7
1
0.5
0.5
0.7
1
0.5
0.5
0.7
1
0.5
0.5
0.7
1
0.5
0.5
0.5
0.5
1.5
0.5
0.5
0.5
1.5
0.5
0.5
0.5
1.5
0.5
0.5
0.5
1.5
0.5
0.5
0.5
1.5
0.5
0.5
0.5
1.5
2.11803
2.50609
2.99535
2.80278
4.11803
4.87252
5.82378
4.80278
6.11803
7.23896
8.65221
6.80278
2.80278
3.31629
3.96372
3.29129
4.80278
5.68272
6.79215
5.29129
6.80278
8.04915
9.62058
7.29129
2.11803
1.79007
1.49768
2.80278
4.11803
3.48037
2.91189
4.80278
6.11803
5.17068
4.3261
6.80278
2.80278
2.36878
1.98186
3.29129
4.80278
4.05909
3.39608
5.29129
6.80278
5.74939
4.81029
7.29129
1.22361
1.44779
1.73044
1.13868
3.22361
3.81422
4.55887
3.13868
5.22361
6.18065
7.3873
5.13868
2.24808
2.65996
3.17926
1.98198
4.24808
5.02639
6.00769
3.98198
6.24808
7.39282
8.83611
5.98198
HP
2.59164
2.59164
2.59164
3.19145
13.27492
13.27492
13.27492
15.07435
31.9582
31.9582
31.9582
34.95725
6.30085
6.30085
6.30085
6.52327
20.40255
20.40255
20.40255
21.06981
42.50426
42.50426
42.50426
43.61634
l(l + 1)
Z
a
h̄2 d 2
−
ηl − ηl + 2 ηl = Eηl ,
−
2M dr 2
r
r2
r
ηl = rRl (r).
h̄2 d 2
l (l + 1)
Z
−
ηl − ηl = Eηl ,
2M dr 2
r
r2
2
1/2
1
2Ma
1
+ 2
.
l = − + l +
2
2
h̄
(22)
−
(23)
This equation is of the same form as the usual equation for the
Coulombic potential except that l is replaced by l . The solutions for this case are
MZ 2
1
E=−
,
2
+ 1)2
(n
+
l
2h̄
1/2
,
l + 1/2 = (l + 1/2)2 + 2Ma/h̄2
Rl = Ar l F (−n, 2l + 2, 2αr)e−αr ,
1/2
,
α = −2ME/h̄2
(24)
with n being the radial quantum number. In these cases, mean
squared deviations are
σr2 = r 2 ,
σp2 = p 2 .
(25)
Now the expectation value of r 2 can be obtained by using
integrals involving the confluent hypergeometric functions [17],
or by using Pasternak–Kramers recursion relations [18], leading
to
2 2
1 2
2
h̄
2
N 5N − 3l (l + 1) + 1 ,
r =
MZ 2
N = n + l + 1.
(26)
Note that l and N are generally not integers. The expectation
value p2 can be obtained by using the Feynman–Hellmann
theorem in Eq. (17). For the Hamiltonian in Eq. (9) and the
energy in Eq. (24), with M as the parameter one obtains
p2 =
MZ
h̄2
2
h̄2
1
2Ma/h̄2
.
1
−
(l + 1/2)N
N2
(27)
112
Table 2
Numerical results of energy (E), expectation values r 2 , p2 and the product
r 2 p 2 corresponding to the potential V2 (r) = −Z/r + a/r 2 with different
values of Z and a. The first three levels with the angular quantum number 0 and
1 are given. At a fixed value of a, the product is independent of Z. The entries
in italics display the dependence on the value of a. All values are in a.u.
−Z
a
E
r 2 p 2 HP
1
1
1
2
2
2
3
3
3
1
1
1
2
2
2
3
3
3
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
−1
−2
−1
−1
−2
−1
−1
−2
−1
−1
−2
−2
−1
−2
−2
−1
−2
−2
0.5
0.5
1.5
0.5
0.5
1.5
0.5
0.5
1.5
0.5
0.5
1.5
0.5
0.5
1.5
0.5
0.5
1.5
−0.19098
−0.76393
−0.09429
−0.07295
−0.2918
−0.04584
−0.0382
−0.15279
−0.02701
−0.09429
−0.37716
−0.2567
−0.04584
−0.18335
−0.13914
−0.02701
−0.10803
−0.08712
14.51722
3.62931
49.08747
110.5927
27.64817
253.8457
415.2912
103.8228
782.8537
49.08747
12.27187
24.30535
253.8457
63.46142
103.9757
782.8537
195.7134
289.2001
0.17082
0.68328
0.0523
0.09605
0.38421
0.04548
0.05751
0.23003
0.03312
0.14315
0.57262
0.27258
0.07628
0.30511
0.18218
0.04705
0.1882
0.12663
2.47984
2.47984
2.56741
10.62279
10.62279
11.54584
23.88247
23.88247
25.93107
7.0271
7.0271
6.62507
19.36249
19.36249
18.94218
36.83349
36.83349
36.62081
These expressions together lead to
1 2
2
2Ma/h̄2
= h̄ 5N − 3l (l + 1) + 1 1 − ,
2
(l + 1/2)N
1/2
,
l + 1/2 = (l + 1/2)2 + 2Ma/h̄2
σr2 σp2
(28)
This again is independent of the strength Z of the Coulombic
potential, but depends on Ma/h̄2 , consistent with the result in
Eq. (11) implied by the dimensional properties. In Table 2 we
have listed the results of our numerical calculations for the HP
using the potential V2 (r). Our results for the three lowest states
with l = 0, 1 given in Table 2 present the numerical evidence in
support of the predictions of Eq. (28), i.e. the HP is independent
of the parameter Z but depends on a. In particular, for a → ∞,
one can carry out an expansion in inverse powers of a to get
1/4
, for a → ∞.
σr σp → h̄(n + 1/2)1/2 2Ma/h̄2
(29)
It is interesting to observe that this is similar to the corresponding expression for the modified isotropic h.o. potential,
in Eq. (21) except for the factor of 2.
5. Extension to the potentials in 2D
The results we have obtained can be extended to the potentials in 2D by using inter-dimensional degeneracy [19]. The
radial equation for the Schrödinger equation in 2D can be written in the form
m2 − 1/4 1/2 (2)
h̄2 d 2
(2)
−
r Rm + V (r)r 1/2 Rm
−
2M dr 2
r2
(30)
where m is the angular momentum quantum number in 2D.
Comparing it with the corresponding equation in 3D,
(31)
one obtains the relations
(3)
(2)
= Er 1/2 Rm
,
(3)
= ErRl ,
(2)
= r 1/2 Rm−1/2 ,
Rm
n+1
N = n + l + 1.
(l + 1/2)2 − 1/4
h̄2 d 2
−
rRl(3) + V (r)rRl(3)
−
2M dr 2
r2
(3)
(2)
Em
= Em−1/2
(32)
for the wavefunctions and the energies in 2D and 3D. Therefore
all the energies and expectation values in 2D can be obtained
from the 3D results by replacing l with m − 1/2. In particular,
for the modified 2D isotropic h.o. potential in Eq. (2) one gets
(2)
= (k/M)1/2 h̄(2n + m + 1),
Em
1/2
m = m2 + 2Ma/h̄2
,
1/2
2Ma/h̄2
,
σr σp = h̄(2n + m + 1) 1 − m (2n + m + 1)
(33)
where n is the radial quantum number. For the modified 2D
Coulomb potential in Eq. (3), one obtains
MZ 2
1
(2)
,
Em = −
2
(n
+
m
+ 1/2)2
2h̄
1/2
,
m = m2 + 2Ma/h̄2
1
2Ma/h̄2
σr2 σp2 = h̄2 5N 2 − 3 m 2 − 1/4 + 1 1 −
,
2
m N
N = n + m + 1/2.
(34)
6. Heisenberg product, Shannon entropy,
Fisher information
In this section we discuss the relevance of the results obtained in Eqs. (18)–(20) and (26)–(28) to the other popular
information theoretical measures, namely, the Shannon information entropy [13] and the Fisher information [15].
The Shannon information entropy of the electron density
ρ(r) in coordinate space is defined as
Sr = − ρ(r) ln ρ(r) dr
(35)
and the corresponding momentum space entropy Sp is given by
Sp = − ρ(p) ln ρ(p) dp
(36)
where ρ(p) denotes the momentum density. The densities ρ(r)
and ρ(p) are each normalized to unity and all quantities are
given in atomic units. The Shannon entropy sum ST = Sr + Sp
contains the net information and obeys the well-known lower
bound derived by Bialynicki-Birula and Mycielski BBM [14].
This lower bound provides a stronger version of the uncertainty
measure than the Heisenberg uncertainty product. According
to it, the entropy sum in D dimensions satisfies the inequality
[14,20–22]
ST = Sr + Sp D(1 + ln π).
(37)
The lower bound in Eq. (37) is saturated by a Gaussian distribution. We have generated the momentum space wave function
113
through Fourier transformation of the position space wave function [16] and the corresponding densities were used to compute
the Shannon entropies.
The relevance of Eqs. (18)–(20) and (26)–(28) on the above
information measures will now be considered with the example
of the potential V1 (r) in an equivalent formulation as follows.
This can be similarly extended to V2 (r).
We begin by defining = [h̄2 /(Mk)]1/4 . Since has dimensions of length, a dimensionless coordinate x can be defined via
r = x. As already noted, β = Ma/h̄2 is a pure (i.e., dimensionless) number.
Expressing the Laplacian in terms of x as ∇ 2 = ∇˜ 2 /2 and
after dividing by h̄ω, where ω = [k/M]1/2 , the Schrödinger
equation becomes
1
β
1
− ∇˜ 2 φ + x 2 φ + 2 φ = φ.
2
2
x
(38)
Here, = E/(h̄ω) and φ(x) depends only on the pure number β which in turn depends on a. Therefore r 2 = 2 x 2 and,
similarly, p 2 is proportional to 1/2 . It follows that the uncertainty product depends on β but not on k. Since the position
space wavefunction φ(x) depends on β but not on k, all the uncertainty measures that depend of the position and momentum
space density will be independent of k but depend on β. In addition, all products of the expectation values such as r n r −n and p n p −n will exhibit similar dependence on the parameters of V1 (r). Analogously, for V2 (r) the information measures
considered above can be shown to be independent of Z but depend on a. We shall now present the numerical data to support
these observations.
In Table 3 we have collected numerical results corresponding to Sr , Sp and ST derived from the potentials V1 (r) and
V2 (r) for a set of parameters Z, k and a. The variation of ST
under scaling of the parameters Z, k and a in the potentials
is found to be the same as in the case of HP. The results also
confirm that the BBM bound is valid for these sums of power
potentials for the arbitrary values of the parameters.
Very recently Dehesa and coworkers [23,24] have reported
novel results connecting the HP with the Fisher information
measure [15,25]. The Fisher information measure or intrinsic
Table 3
Numerical results of the Shannon information entropy in position (= Sr ) and
momentum space (= Sp ) and their sum (ST ) for the ground state corresponding
to the Davidson potential, V1 (r) = kr 2 /2 + a/r 2 , and the Kratzer potential,
V2 (r) = −Z/r +a/r 2 , with different values of (k, a) and (Z, a). The parameter
C1 represents k/2 and −Z in V1 (r) and V2 (r). The entries in italics display the
dependence on the value of a. All values are in a.u.
Potential
n+1
C1
a
Sr
Sp
ST
Davidson
1
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0
0.5
0.7
1
0.5
1
1.5
2
1
0.5
0.5
0.5
0.6
0.5
0.5
0.5
0.6
3.70352
3.45116
3.18366
3.75631
6.59449
5.37809
4.51505
6.86299
2.82717
3.07952
3.34703
2.7941
−0.1191
1.09729
1.96033
−0.3911
6.53068
6.53068
6.53068
6.55041
6.47538
6.47538
6.47538
6.47189
Kratzer
accuracy in position space is defined as
|∇ρ(r)|2
dr
Ir =
ρ(r)
(39)
and the corresponding momentum space measure is given by
|∇ρ(p)|2
dp.
Ip =
(40)
ρ(p)
The individual Fisher measures are bounded through the
Cramer–Rao inequality [26] according to Ir 12 and Ip 12 .
σr
σp
The equality corresponds to a Gaussian distribution. A sharp
and strongly localized probability density gives rise to a larger
value of Fisher information in the position space. For a variety
of applications of the Fisher information measure we refer to
the recent book [25] and for applications to the electronic structure of atoms, to the pioneering work of Dehesa et al. [23,24].
These authors have shown that for a single particle in a central potential in the quantum state defined by the set of quantum
numbers (n, , m) is given by
Ir = 4 p 2 − 2(2l + 1)|m| r −2 ,
(41)
−2 2
Ip = 4 r − 2(2l + 1)|m| p .
(42)
Using the dimensional analysis arguments following
Eq. (38), it is clear that all the four products of the radial and
momentum expectation values of the form r n p n defining
the product Ir Ip via Eqs. (41) and (42), follow similar dependence on k, Z, and a for the potentials given by Eqs. (2)–(3).
This leads to an interesting result that for any central potential
including those given by Eqs. (2)–(3), the Fisher information
product is independent of the scaling parameters k and Z but
depends only upon a. The parameters can vary arbitrarily. Following Dehesa et al. [23], our results on r 2 p 2 , ST and Ir Ip
can be extended to the D-dimensional hyperspherical systems.
It is to be noted that a universal lower bound on the Fisher
product for an arbitrary potential is not as yet available. Very
recently [24] the lower bounds for the single particle in any
central potential has been obtained as
2 2 r p (l + D/2)2 .
(43)
We have numerically tested the above bound for a very large
number of states corresponding to the potentials given by
Eqs. (2)–(3) with several (l) values and found Eq. (43) to be
valid for the arbitrary values of the scaling parameter k, Z,
and a.
7. Summary
We have analyzed the uncertainties in the vectorial position r and momentum p for the isotropic h.o. potential and
the Coulomb potential, with an additional a/r 2 term which are
identified with the empirical potentials generally known as the
Davidson and Kratzer potentials, respectively. The dimensional
analysis indicates that the product of the associated uncertainties is independent of the strength of isotropic h.o. and Coulomb
potentials, but depends on the dimensionless parameter Ma/h̄2 .
114
Explicit analytical expressions have been obtained for the uncertainties and their products, which confirm the predictions of
the dimensional analysis. The results obtained on the variances
based Heisenberg uncertainty have been shown to hold good
for (a) the sum of the Shannon entropy, and (b) the product
of the Fisher information measure in the position and momentum spaces. The results obtained here can be generalized to
D dimensions (D 2). We are currently carrying out numerical tests of the recently proposed entropic formulation [27]
based on the Rényi entropies for the potentials considered in
this work.
Acknowledgements
S.H.P. acknowledges support from AICTE, New Delhi for
emeritus fellowship. K.D.S. thanks Professor Jacob Katriel for
a comment leading to Eq. (38), and gratefully acknowledges the
constant encouragement received from Professor H.E. Montgomery Jr.
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