One body density matrix in a harmonic trap

While in a homogeneous system the OBDM depends only on the relative distance, for a system in external potential it is no longer true (1.32). Instead one define the OBDM in a convenient way by integrating out the center of the mass motion.

$$\overline g_1(\r) = \int g_1\left({\vec R}+\frac{\r}{2}, {\vec R}-\frac{\r}{2}\right)\,d{\vec R},$$ (2.146)

Here the standard notation for the center of the mass variables is used ${\vec R}=({\vec r_1}+{\vec r_2})/2$, $\r={\vec r_1}-{\vec r_2}$. The point in the definition (2.146) is that the momentum distribution can be obtained by the Fourier transform with respect to $\r$

$$n(\vec k) = \frac{1}{(2\pi)^3} \int\overline g_2(\r) e^{i\vec k\r}\,d\r$$ (2.147)

For practical purposes it is convenient to change the notation

$$\left\{ \begin{array}{lll} {\vec R}&=& {\vec r_1}\\ \r&=& {\vec r_1}-{\vec r_2} \end{array}\right.$$ (2.148)

Using this notation the mixed OBDM becomes

$$g_1^{mixed} ({\vec R}, \r) = N \frac{\int...\int \psi^*_T({\vec R... .....,{\vec r}_N)\phi_0({\vec r_1}, ...,{\vec r}_N)\,d{\vec r_1} ... d{\vec r}_N},$$ (2.149)

which reminds us the expression for the OBDM of a homogeneous system (1.18).

The function $\overline g_2$ can be measured in the QMC simulation

$$\overline g_1^{mixed}(\r) = \frac{\int...\int [\psi^*_T({\vec R}+... ... r}_N} {\int...\int f({\vec r_1}, ..., {\vec r}_N)d{\vec r_1} ... d{\vec r}_N},$$ (2.150)

by taking an average of the following quantity

$$\frac{\psi_T({\vec R}+{\vec r}, ...,{\vec r}_N)}{\psi_T({\vec R},... ...ert{\vec r_1}+\r-{\vec r}_j\vert)-u_2(\vert{\vec r_1}-{\vec r}_j\vert)\right\},$$ (2.151)

where $u_1(\r)$ stands for the one-body exponent in the Jastrow-Bijl wave function (see Eq. 2.38), which for harmonic confinement is taken to be equal to (2.41).