In the local density approximation one assumes that the chemical potential is
given by sum of the local chemical potential , which is the chemical
potential of the uniform system, and the external field:
The local chemical potential is defined by the equation of state in absence of the external field and accounts for the interaction between particles and partially for the kinetic energy.
The value of the chemical potential is fixed by the normalization condition
Once the chemical potential is known a lot of useful information can be inferred: the density profile, energy, size of the cloud, density moments , etc.
In the following we will always consider a harmonic external confinement:
(1.128) |
The normalization condition (1.127) becomes:
The sizes of the cloud in three directions is fixed by the value of
the chemical potential and corresponding frequencies of the harmonic confinement
through relation:
We express the distances in the trap in units of the size of the cloud:
and in front of the integral (1.129) we have the
geometrical average
appearing. It means that the trap frequencies
(even if the trap is not spherical) enter only through combination
and the oscillator lengths correspondingly through
parameter
. Now the integral is to be taken inside a
sphere of radius and is symmetric in respect to . It follows
immediately, that the normalization condition (1.129) in general can be
written as
From the Eq. 1.131, which is basically a dimensionless version of Eq. 1.129 we discover there is a scaling in terms of the characteristic parameter . In other words systems having different number of particles and oscillator frequencies will have absolutely the same density profile and other LDA properties (once expressed in the correct units as discussed above) if they have equal values of parameter (1.132).
A similar procedure can be carried in a one-dimensional case (we choose the
axis), where the normalization condition reads as
Its dimensionless form is obtained by measuring the energies in the trap in units
of