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Homogeneous system

Many simplifications can be done in a homogeneous system due to the presence of the translational symmetry. The correlation functions discussed above depend only on the relative distance between two coordinates $\vert\r-\r'\vert$. The diagonal element of $G_1$ is simply a constant $G_1({\vec r},\r) = n$. The non-diagonal element of the normalized one-body density matrix (in following we will address it as OBDM) is written as

$\displaystyle g_1(r) =
\frac{N}{n}\frac{\int \psi^*({\vec r}, {\vec r_2}, ..., ...
...t \vert\psi^*({\vec r_1}, ..., {\vec r}_N)\vert^2\,{d\vec r}_1 ... {d\vec r}_N}$     (1.18)

The normalized two-body density matrix (pair distribution function) is then given by

$\displaystyle g_2(r) =
\frac{N(N-1) \int \vert\psi({\vec r}, 0, {\vec r}_3,...,...
...\int\vert\psi({\vec r_1}, ..., {\vec r}_N)\vert^2\,{d\vec r}_1 ... {d\vec r}_N}$     (1.19)

At the zero temperature some of the properties of the pair distribution can be easily understood. At large distances the correlation between particles becomes weaker and weaker and we can approximate the field operator $\hat{\Psi}(r)=\sqrt{\hat{n}(r)}e^{i~\hat{\phi}(r)}\approx $ $\sqrt{\hat{n}(r)},
r\rightarrow \infty$ and at zero temperature one has $g_2(r)\rightarrow
1-\frac{1}{N}$ and $g_2(r)$ approaches unity in the thermodynamic limit. On the contrary at short distances particles ``feel'' each other and the value at zero can be very different from the value in the bulk. In case of impenetrable particles, two particles are not allowed to overlap, thus $g_2(0)=0$. For purely repulsive interaction $g_2(0)<1$ and for purely attractive $g_2(0)>1$.

In the average of a two-body operator (1.8) it is possible to integrate out the dummy variable and get a simple expression

$\displaystyle \left\langle F^{(2)}\right\rangle =\frac{n^{2}}{2}\int\!\!\!\!\in...
...1},{\vec r_2})\,{d\vec r}_1{d\vec r}_2=\frac{Nn}{2}\int
f^{(2)}(r)~g_{2}(r)\,dr$     (1.20)

For a particular case of a contact potential $V_{int}(r) = g \delta(r)$ the potential energy is directly related to the value of the pair distribution function at zero:

$\displaystyle \frac{E_{int}}{N} = \frac{1}{2}g n g_2(0)$     (1.21)

We will also give definition of the three-body density matrix1.2


$\displaystyle g_3(0) = \frac{N(N-1)(N-2)}{n^3}
\frac{\int\vert\psi(0, 0, 0, {\v...
...\int\vert\psi({\vec r_1}, ..., {\vec r}_N)\vert^2\, {d\vec r}_1... {d\vec r}_N}$     (1.22)

Its value at zero gives the probability of finding three particles in the same point.


next up previous contents
Next: Momentum distribution and static Up: Correlation functions and related Previous: Correlation functions: first quantization   Contents
G.E. Astrakharchik 15-th of December 2004