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Trapped system

The presence of an external harmonic confinement removes the translational invariance. We will restrict ourselves to one dimensional case as the relevant to the study presented in Chapter 4. The one-body density matrix (1.11) $g_1(z_1, z_2)$ depends on both arguments:

$\displaystyle n \left(\frac{z'+z''}{2}\right) g_1(z',z'') =
\frac{N\int \psi^*(...
...(z'',..., z_N)\,dz_2...dz_N}{\int \vert\psi(z_1,..., z_N)\vert^2\,dz_1...dz_N},$     (1.32)

where $n(z)$ is the density profile.

The momentum distribution of a trapped system is obtained from the OBDM by the Fourier transformation with the respect to the relative distance

$\displaystyle n(k) = \int\!\!\!\!\int g_1 \left(Z+\frac{z'}{2}, Z-\frac{z'}{2}\right) n(Z)~e^{ikz'}\,dZdz',$     (1.33)



G.E. Astrakharchik 15-th of December 2004