Correlation functions: first quantization form

The meaning of the correlation functions (1.11,1.12) is best of all understood in terms of the field operators as discussed in the previous Section. Instead for the implementation of the Monte-Carlo technique it is necessary to express the correlation functions in terms of the wave function $\psi({\bf R})$ of the system. The easiest way to do SO is to find an expression of the operator average in form similar to (1.7,1.8).

The mean value of a one-body operator in the first quantization Is written as

$$\langle F^{(1)} \rangle = \frac{\int \psi^*({\bf R}) F^{(1)}({\bf... ...{\vec r_1}, ...., {\vec r}_N)\,{\bf dR}}{\vert\psi({\bf R})\vert^2\,{\bf dR}} =$$

$$= \frac{N \int\!\!\!\!\int F^{(1)}({\vec r_1},{\vec r_1}') \vert\... ...\int\!\!\!\!\int f^{(1)}({\vec r},\r') G_1({\vec r}, \r')\,{d\vec r}{d\vec r}',$$ (1.13)

where $G_1({\vec r},\r')$ stands for

$$G_1({\vec r}, \r') = \frac{N \int \psi^*({\vec r}, {\vec r_2}, ..... ..., {\vec r}_N) \psi^*({\vec r_1}, ..., {\vec r}_N)\,{d\vec r}_1 ... {d\vec r}_N}$$ (1.14)

The expression for the two-body correlation function (1.10) can be deduced from the average of a two-body operator (1.8):

$$\langle F^{(2)} \rangle = \frac{\int \psi^*({\bf R}) F^{(2)}({\bf... ... \psi({\vec r_1}, ...., {\vec r}_N)\,{\bf dR}}{\vert\psi({\bf R})\vert^2\,dR} =$$ (1.15)

$$= \frac{N(N-1) \int f^{(2)}({\vec r_1}, {\vec r_2}) \vert\psi({\v... ...)}({\vec r_1}, {\vec r_2}) g_2({\vec r_1}, {\vec r_2})\,{d\vec r}_1{d\vec r}_2,$$ (1.16)

where the first quantization expression for the two body correlation function is

$$G_2(\r', \r'') = \frac{N(N-1) \int \vert\psi(\r', \r'', {\vec r}_... ...\int\vert\psi({\vec r_1}, ..., {\vec r}_N)\vert^2\,{d\vec r}_1 ... {d\vec r}_N}$$ (1.17)