4.3 The Topological Analysis of Non-Coulombic Systems:
The Harmonic Trap Model
The topological analysis of the one-particle density yields the topological structure,
through identifying critical points (CPs) and the boundaries between real-space
subsystems. Particularly, the local zero-flux equation, ~
rq ~ q
ð Þ Á~ n ~ q
ð Þ ¼ 0, is used to
determine the zero-flux surfaces that act as inter-atomic boundaries [1]. However,
these surfaces are just a small subset of the zero-flux surfaces emerging from the
equation [40, 46, 47]. It has been demonstrated that the zero-flux surfaces that are
not acting as the boundaries of topological atoms may found interesting applications; the “morphologies” of the real-space subsystems which they are shaping are
different from the topological atoms [48–55], and even more exotic (from the
viewpoint of their morphology) real-space subsystems emerge from the net
zero-flux equation,
R
X d~ qr
2
q ~ q
ð Þ ¼ 0, as demonstrated recently [47, 56]. All these
studies point to the fact that even for the Coulombic systems the topological
analysis may yield a wide spectrum of real-space subsystems apart from the
topological atoms. Accordingly, it is tempting to consider what kind of real-space
subsystems may emerge from the topological analysis of non-Coulombic systems.
In the rest of this section the harmonic trap model is considered for this purpose.
The model of N quantum particles confined in a harmonic trap has been widely
used to model the Bose-Einstein condensation in trapped dilute gases [57–68], and
more recently in trapped Fermi gases [69–73]. A simplified model of the trap may be
constructed assuming a non-interacting system of quantum particles in an external
isotropic harmonic trap, as a homogeneous potential, n ¼ 2, with the following
Hamiltonian: ^
H ¼
P N
k¼1
^ h k ¼ À h
2
2m
À
ÁP N
k¼1 r
2
k À a
2 x
2
k þ y
2
k þ z
2
k
À
Á
È
É
, where
a ¼ 2pfm= h and f is the frequency of mechanical vibration of the particle in the trap
[44]. The spectrum of the eigenfunctions and eigenvalues of the one-particle
Hamiltonian, ^ h K / v 1 v 2 v 3 ¼ e v 1 v 2 v 3 / v 1 v 2 v 3 , is well-known (v 1 ; v 2 ; v 3 are the quantum
numbers) [44], e.g. / 000 ðx; y; zÞ ¼ a=p
ð
Þ
3=4 Exp À a=2
ð
Þ x
2
þ y
2
þ z
2
ð
Þ
½
, e 000 ¼ 3p hf
and / 100 ðx; y; zÞ ¼ 4a
5
p
3
À
Á 1=4 xExp À a=2
ð
Þ x
2
þ y
2
þ z
2
ð
Þ
½
, e 100 ¼ 5p hf . The
wavefunction of the system may be constructed based on the statistics of the trapped
particles. In the ground state of the system filled with non-interacting bosons all
particles are at the lowest one-particle energy state, E
Boson
0
¼ Ne 000 ¼ 3Np hf , and
neglecting the spin variable, the spatial part of the wavefunction is a simple product of
the one-particle eigenfunctions associated to the lowest one-particle energy state:
W Boson ¼
Q N
k¼1 / 000 x k ; y k ; z k
ð
Þ¼ a=p
ð
Þ
3N=4 Exp À a=2
ð
Þ
P N
k¼1 x
2
k þ y
2
k þ z
2
k
À
Á
Â
Ã
. On
the other hand, if the trap is filled with fermions then the spin variable is of pivotal
importance and the spin-eigenfunctions, instead of the spatial eigenfunctions, must be
used to construct the fermionic wavefunction,
w v 1 v 2 v 3 ¼ / v 1 v 2 v 3 a
w v 1 v 2 v 3 ¼ / v 1 v 2 v 3 b
&
(a and b are the
spin eigenfunctions). The Pauli Exclusion Principle dictates a N Â N determinant,
composed of the spin-eigenfunctions, as the ground state wavefunction of the system:
4 Extending the Topological Analysis and Seeking the Real-Space …
95
The Harmonic Trap Model
The topological analysis of the one-particle density yields the topological structure,
through identifying critical points (CPs) and the boundaries between real-space
subsystems. Particularly, the local zero-flux equation, ~
rq ~ q
ð Þ Á~ n ~ q
ð Þ ¼ 0, is used to
determine the zero-flux surfaces that act as inter-atomic boundaries [1]. However,
these surfaces are just a small subset of the zero-flux surfaces emerging from the
equation [40, 46, 47]. It has been demonstrated that the zero-flux surfaces that are
not acting as the boundaries of topological atoms may found interesting applications; the “morphologies” of the real-space subsystems which they are shaping are
different from the topological atoms [48–55], and even more exotic (from the
viewpoint of their morphology) real-space subsystems emerge from the net
zero-flux equation,
R
X d~ qr
2
q ~ q
ð Þ ¼ 0, as demonstrated recently [47, 56]. All these
studies point to the fact that even for the Coulombic systems the topological
analysis may yield a wide spectrum of real-space subsystems apart from the
topological atoms. Accordingly, it is tempting to consider what kind of real-space
subsystems may emerge from the topological analysis of non-Coulombic systems.
In the rest of this section the harmonic trap model is considered for this purpose.
The model of N quantum particles confined in a harmonic trap has been widely
used to model the Bose-Einstein condensation in trapped dilute gases [57–68], and
more recently in trapped Fermi gases [69–73]. A simplified model of the trap may be
constructed assuming a non-interacting system of quantum particles in an external
isotropic harmonic trap, as a homogeneous potential, n ¼ 2, with the following
Hamiltonian: ^
H ¼
P N
k¼1
^ h k ¼ À h
2
2m
À
ÁP N
k¼1 r
2
k À a
2 x
2
k þ y
2
k þ z
2
k
À
Á
È
É
, where
a ¼ 2pfm= h and f is the frequency of mechanical vibration of the particle in the trap
[44]. The spectrum of the eigenfunctions and eigenvalues of the one-particle
Hamiltonian, ^ h K / v 1 v 2 v 3 ¼ e v 1 v 2 v 3 / v 1 v 2 v 3 , is well-known (v 1 ; v 2 ; v 3 are the quantum
numbers) [44], e.g. / 000 ðx; y; zÞ ¼ a=p
ð
Þ
3=4 Exp À a=2
ð
Þ x
2
þ y
2
þ z
2
ð
Þ
½
, e 000 ¼ 3p hf
and / 100 ðx; y; zÞ ¼ 4a
5
p
3
À
Á 1=4 xExp À a=2
ð
Þ x
2
þ y
2
þ z
2
ð
Þ
½
, e 100 ¼ 5p hf . The
wavefunction of the system may be constructed based on the statistics of the trapped
particles. In the ground state of the system filled with non-interacting bosons all
particles are at the lowest one-particle energy state, E
Boson
0
¼ Ne 000 ¼ 3Np hf , and
neglecting the spin variable, the spatial part of the wavefunction is a simple product of
the one-particle eigenfunctions associated to the lowest one-particle energy state:
W Boson ¼
Q N
k¼1 / 000 x k ; y k ; z k
ð
Þ¼ a=p
ð
Þ
3N=4 Exp À a=2
ð
Þ
P N
k¼1 x
2
k þ y
2
k þ z
2
k
À
Á
Â
Ã
. On
the other hand, if the trap is filled with fermions then the spin variable is of pivotal
importance and the spin-eigenfunctions, instead of the spatial eigenfunctions, must be
used to construct the fermionic wavefunction,
w v 1 v 2 v 3 ¼ / v 1 v 2 v 3 a
w v 1 v 2 v 3 ¼ / v 1 v 2 v 3 b
&
(a and b are the
spin eigenfunctions). The Pauli Exclusion Principle dictates a N Â N determinant,
composed of the spin-eigenfunctions, as the ground state wavefunction of the system:
4 Extending the Topological Analysis and Seeking the Real-Space …
95