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Local coordinate, wave vector, Fisher and Shannon information
in momentum representation
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Á. Nagy
a,∗
b
, E. Romera , S.B. Liu
c
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a
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Department of Theoretical Physics, University of Debrecen, H–4010 Debrecen, Hungary
b
Departamento de Física Atómica, Molecular y Nuclear and Instituto Carlos I de Física Teórica y Computacional, Universidad de Granada,
Fuentenueva s/n, 18071 Granada, Spain
c
Research Computing Center, University of North Carolina, Chapel Hill, NC 27599-3420, USA
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a r t i c l e
i n f o
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a b s t r a c t
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Article history:
Received 9 August 2012
Accepted 5 November 2012
Available online xxxx
Communicated by V.M. Agranovich
The formalism of Luo [Int. J. Theor. Phys. 41 (2002) 1713] on local values of quantum observables is
generalized for N-electron systems both in coordinate and momentum spaces. It is shown that the
imaginary part of the total local coordinate (momentum) is the half of the local wave vector (or the
half of the gradient of the local momentum(coordinate)-space Shannon information per particle).
© 2012 Elsevier B.V. All rights reserved.
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1. Introduction
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2. Local functions of quantum observables in one-electron
systems
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Nowadays, there is a growing interest in applying information
theoretical concepts in several fields of physics and chemistry. Local information quantities attracted less attention. Here we show
that local Shannon and Fisher information can be related to local
quantum observables.
Local values of quantum observables were derived by several
authors [1–3]. Here the formalism of Luo [1] is applied and generalized for N-electron systems. Luo showed that the real part of the
local value of a quantum observable is the expectation value of the
quantum observable, while the imaginary part comprises the fluctuation closely related to the Fisher information [4]. In this Letter
a relation to the local Shannon information [5] and the local wave
vector is explored.
In the following section the formalism of Luo [1] is summarized. The method is generalized for N-electron systems in Section 3. We show that the imaginary part of the total local momentum is the half of the local wave vector or the half of the gradient
of the local Shannon information per particle. Local functions of
quantum observables in N-electron systems in momentum space
are revealed in Section 4. We point out that the imaginary part of
the total local coordinate is the half of the local momentum-space
wave vector or the half of the gradient of the local momentumspace Shannon information per particle. Local Shannon and Fisher
momentum densities in neutral atoms are presented in Section 5.
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*
Corresponding author. Fax: +36 52415102.
E-mail addresses: [email protected] (Á. Nagy), [email protected] (E. Romera),
[email protected] (S.B. Liu).
0375-9601/$ – see front matter © 2012 Elsevier B.V. All rights reserved.
http://dx.doi.org/10.1016/j.physleta.2012.11.018
Consider a quantum mechanical observable a and a quantum
system at a state ψ . The local value of a can be decomposed into
two local contributions:
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a(r) = ā(r) + iã(r).
(1)
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The real part ā(r) gives the expectation value of a
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ā(r)|ψ|2 dr.
aψ =
(2)
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The imaginary part ã(r) is related to the variance:
V arψ (a − ā) =
2
ã(r) |ψ|2 dr.
113
(3)
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Luo showed that
ā(r) = Re
and
aψ
ψ
(r)
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(4)
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121
ã(r) = Im
aψ
ψ
122
(r).
(5)
Bohórquez and Boyd [6] obtained the local momentum in coordinate representation. The wave function can be written in polar
form
ψ(r) = (r)1/2 e ib(r) ,
(6)
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where (r) = |ψ(r)|2 is the density. The momentum operator
in coordinate representation is p = −i ∇ . (Atomic units are used
throughout this Letter.) Then the local momentum [6] has the form
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p(r) = ∇r b(r) −
i ∇r (r)
(r)
2
(7)
,
that is, the average value is p̄ (r) = ∇r b(r) and the fluctuation part
is p̃ (r) = −1/2(∇r (r))/(r).
We mention by passing that p̃ is proportional to Nelson’s [7]
osmotic velocity −∇r ln (r). Quantum properties of Fisher information were discussed by Hall [8]. Fisher information for central
potential was calculated by Romera et al. [9].
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3. Local functions of quantum observables in N -electron systems
in coordinate space
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20
Now we generalize the concept of local quantities to N-electron
systems. Consider a quantum mechanical observable A acting on
the wave function Ψ (r1 , . . . , r N ). Now we define a local quantity
as
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A (r) = Ā (r) + i à (r).
(8)
The real part Ā (r) gives the expectation value of a
A Ψ =
Ā (r)(r) dr,
(9)
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(r) is defined as
2
(r1 ) = N Ψ (r1 , . . . , rN ) dr2 . . . drN .
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Ā (r1 ) =
(r1 )
à (r1 ) =
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AΨ
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VarΨ ( A − Ā ) =
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à (r)2
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P=
(12)
N
p j.
(14)
The wave function can be written in coordinate representation in
polar form as
Ψ (r1 , . . . , r N ) = K (r1 , . . . , r N )e i F (r1 ,...,rN ) ,
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(15)
where K (r1 , . . . , r N ) and F (r1 , . . . , r N ) are real functions. Then we
immediately obtain that
Re
p jΨ
Im
= ∇r j F
Ψ
q (r) = −
P̃(r) =
2
p jΨ
Ψ
=−
∇r j K
(16)
(20)
K
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q (r).
(21)
In a recent paper [12] we established relationships between the
local wave-number vector, the local Shannon and Fisher information. The Shannon information is defined in coordinate space as
s (r) dr
(22)
with the local Shannon information
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(23)
i (r) dr
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(24)
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[∇r (r)]2
i (r) =
.
(r)
(25)
q = ∇r
s
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(26)
(17)
The real part of the local quantity P in state Ψ (r1 , . . . , r N ) takes
the form
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and
q2 =
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i
(27)
.
P̃(r) =
1
2
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That is, the local wave vector q is the gradient of the Shannon
information per particle and the square of the local wave vector is
the Fisher information per particle.
Using these results we obtain that the imaginary part of the total local momentum is the half of the gradient of the local Shannon
information per particle:
∇r
s
.
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122
(28)
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VarΨ (P − P̄) =
=
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123
.
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∇r (r)
,
(r)
has the dimension of wave-number. Comparing Eqs. (19) and (20)
we obtain that the imaginary part is the half of the local wave
vector:
1
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Therefore the variance of the total momentum can be written as
and
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(19)
.
Some years ago Nagy and March [10] introduced the ratio
of the density gradient to the electron density as a local wavenumber to characterize the ground state of atoms and molecules.
Independently, Kohout, Savin and Preuss [11] also investigated the
role of the quantity |∇r /| in the shell structure of atoms. The
local wave vector
(13)
j =1
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(r)
2
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We showed that
(r) dr.
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1 ∇r (r)
with the local Fisher information
Consider the total momentum operator P as a sum of oneelectron operators p j ,
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P̃(r) = −
I =
2
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(11)
It is related to the variance:
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(18)
The Fisher information, on the other hand, has form:
.
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,
s (r) = −(r) ln (r).
Im( Ψ )|Ψ | dr2 . . . dr N
.
(r1 )
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(r1 )
The imaginary part à (r) can be written as
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Re( AΨΨ )|Ψ |2 dr2 . . . dr N
K 2 ∇r1 F dr2 . . . dr N
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(10)
The real part Ā (r) has the form
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N
while the imaginary part is
S =
where the density
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P̄(r1 ) =
2
P̃(r)
1
4
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(r) dr
2
s
(r) ∇r
dr.
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(29)
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4. Local functions of quantum observables in N -electron systems
in momentum space
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Consider the total coordinate operator R as a sum of oneelectron operators r j = i ∇ p j ,
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γ (p1 ) = N
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∇ p γ (p)
qγ (p) = −
.
γ (p)
(30)
(31)
R=
as
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r j.
Φ(p1 , . . . , p N ) = L (p1 , . . . , p N )e iG (p1 ,...,pN ) ,
sγ (p) dp
(32)
with the local Shannon information
r jΦ
(45)
Im
r jΦ
=
Φ
R̄(p1 ) = −
∇p j L
L
N
L 2 ∇ p 1 Gdp2 . . . dp N
γ (p1 )
,
(34)
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(35)
1 ∇ p γ (p)
qγ = ∇ p
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sγ
(36)
γ
iγ
(37)
.
γ
That is, the momentum-space local wave vector qγ is the gradient of the momentum-space Shannon information per particle
and the square of the momentum-space local wave vector is the
momentum-space Fisher information per particle.
Consider a quantum mechanical observable B acting on the
momentum-space wave function Φ(p1 , . . . , p N ) and define a local
quantity as
B (p) = B̄ (p) + i B̃ (p).
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(38)
B̄ (p1 ) =
Re( BΦΦ )|Φ|2 dp2 . . . dp N
γ (p1 )
.
(39)
B̃ (p1 ) =
Im( BΦΦ )|Φ|2 dp2 . . . dp N
γ (p1 )
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.
(40)
B̄ (p)γ (p) dp,
(41)
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while the imaginary part is related to the variance:
V arΦ ( B − B̄ ) =
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sγ
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(50)
.
γ
That is, the imaginary part of the total local coordinate is minus
the half of the gradient of the local Shannon information per particle. Therefore the variance can be written as
VarΨ (R − R̄) =
=
1
4
R̃(p)2 γ (p) dp
2
sγ
γ (p) ∇ p
dp.
γ
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110
(51)
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5. Local Shannon and Fisher momentum densities in neutral
atoms
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Now, the spherically averaged local Shannon and Fisher informations
B̃ (p)2 γ (p) dp.
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119
(52)
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121
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and
The average of B is given by the real part
B Φ =
(49)
sγ ( p ) = −γ ( p ) ln γ ( p )
The imaginary part B̃ (p) can be written as
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The real part B̄ (p) has the form
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R̃(p) = − ∇ p
2
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or with the local Shannon information
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γ (p)
2
R̃(p) = − qγ (p)
2
and
q2γ =
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(48)
.
1
After elementary calculation we are led to the relationship
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(47)
2
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The latter can also be expressed with the local wave vector
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while the imaginary part is
[∇ p γ (p)]
i γ (p) =
.
γ (p)
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i γ (r) dr
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R̃(p) =
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with the local Fisher information
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(46)
.
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(33)
The Fisher information, on the other hand, takes form:
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The real part of the local quantity R in state Φ reads
sγ (p) = −γ (p) ln γ (p).
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= −∇ p j G
Φ
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30
(44)
where L (p1 , . . . , p N ) and G (p1 , . . . , p N ) are real functions. Then we
immediately obtain that
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and
Sγ =
(43)
The wave function can be written in momentum representation as
Re
Iγ =
N
j =1
The Shannon information in momentum space can be written
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Φ(p1 , . . . , p N )2 dp2 . . . dp N ,
where Φ(p1 , . . . , p N ) is the momentum-space wave function. The
momentum-space local wave vector is defined as
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All results obtained in the previous section in coordinate space
can be derived in the momentum space. The momentum-space
density γ (p) reads as
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(42)
(γ ( p ))2
iγ ( p) =
γ ( p)
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123
(53)
are analyzed. The spherically averaged momentum density γ ( p ) is
calculated with the ground state momentum densities for neutral
atoms with nuclear charge 1 Z 103 by means of the accurate Roothan–Hartree–Fock wave functions of Koga and co-workers
[13]. We have shown that attending to the number of local extrema and the zeros in the open interval (0, ∞), sγ ( p ) can be
classified into three types:
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Fig. 1. Examples of each type of sγ : Z = 10 (Type I), Z = 14 (Type II), Z = 17 (Type III) and Z = 46 (Type IV). Atomic units are used.
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Fig. 2. Examples of each type of the density i γ : Z = 24 (Type I), Z = 30 (Type II) and Z = 36 (Type III). Atomic units are used.
• Type I: sγ has a local maximum for Z = 1, 2, 7–10. Fig. 1
shows the local Shannon entropy for one of these atoms, Ne
( Z = 10).
• Type II: sγ has a local maximum and a zero for Z = 11–14,
19–32, 37–45, 47–50, 55, 82, 87–103. Fig. 1 shows the local
Shannon entropy for Si ( Z = 14).
• Type III: sγ has a minimum, a zero and a maximum for Z =
15–18, 33–36, 51–54, 83–86. Fig. 1 shows the local Shannon
entropy for Ar ( Z = 17).
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There are two exceptions to the above classification: sγ has two
relative minima and a relative maximum for Pd ( Z = 46) (Fig. 1
shows the local Shannon entropy for this element) and sγ has just
a minimum and a maximum for Ar Z = 18.
We can point out that Type II atoms correspond to groups
1–14 except H and Pd, Type II atoms are the norm for groups
15–18 (except He N, O, F, and Ne), and the rest of atoms cor-
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responds to Type I (expect Pd and Ar as we have mentioned
above).
Analysis of the local fisher density i γ in terms of the extrema
and zeros shows a complex schedule (up to ten types of densities).
In this case we can do the classification attending to the number of
zeros (always in the open interval (0, ∞)), emerging three different
types of densities:
• Type I: The 65 neutral atoms with Z = 1–7, 11–25, 31, 37–42,
49, 50, 55–74 and 87–103 have an i γ with no zeros.
• Type II: The 21 neutral atoms with Z = 8–10, 15–18, 32–36,
46, 51–54, 83–86 have a density with a zero.
• Type III: The 16 neutral atoms with Z = 26–30, 43, 45, 47 and
48 have an i γ with two zeros.
Fig. 2 presents i γ ( p ) for three atoms, one of each type: Z = 24
(Type I), Z = 30 (Type II) and Z = 36 (Type III). Obviously, the
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emergence of zeros in the local fisher information in connected
with the existence of a relative maximum or minimum in the momentum density, and we have checked that this classification is
in agreement with the classification of the spherically averaged
momentum density studied in [14–18]. As in the case of the momentum density, these types of structures are determined by the
relative contribution of the outermost s, p, and d subshells (see
[14–18] for a detailed analysis of the momentum density structure).
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6. Summary
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In summary, we generalized the formalism of Lou for Nelectron systems both in coordinate and momentum spaces. We
studied two local quantum observables: the total local momentum
and the total local coordinate. It turned out that the imaginary part
of the total local momentum is the half of the local wave vector (or
the half of the gradient of the local Shannon information per particle) and the imaginary part of the total local coordinate is the
half of the local momentum-space wave vector (or the half of the
gradient of the local momentum-space Shannon information per
particle). Finally, we analyzed the local Shannon and Fisher momentum densities of neutral atoms.
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Acknowledgements
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29
This work is supported by the Projects FIS2011-24149 and
CEI-BIOTIC-20F12.41. The work is also supported by the TAMOP
4.2.1/B-09/1/KONV-2010-0007 and TAMOP 4.2.2/B-10/1-2010-0024
5
projects. The project is co-financed by the European Union and the
European Social Fund. Grant OTKA No. K 100590 is also gratefully
acknowledged.
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JID:PLA
AID:21677 /SCO Doctopic: Atomic, molecular and cluster physics
[m5Gv1.5; v 1.86; Prn:14/11/2012; 10:18] P.6 (1-5)
Physics Letters A ••• (••••) •••–•••
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Contents lists available at SciVerse ScienceDirect
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Physics Letters A
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www.elsevier.com/locate/pla
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Local coordinate, wave vector, Fisher and Shannon information in momentum representation
Physics Letters A ••••, •••, •••
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Á. Nagya,∗ , E. Romerab , S.B. Liuc
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a
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Department of Theoretical Physics, University of Debrecen, H–4010 Debrecen, Hungary
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Departamento de Física Atómica, Molecular y Nuclear and Instituto Carlos I de Física Teórica y Computacional, Universidad de Granada,
Fuentenueva s/n, 18071 Granada, Spain
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Research Computing Center, University of North Carolina, Chapel Hill, NC 27599-3420, USA
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Highlights
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We studied total local momentum and coordinate for N-electron systems. The total local momentum’s imaginary part is the half of the local
wave vector. The total local coordinate’s imaginary part is half of Shannon entropy’s gradient.
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