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In an n-electron system,

The second order reduced DM is defined as

[tex]\Gamma (x_{1},x_{2}) = \frac{N(N-1)}{2}\int{\psi(x_{1},x_{2}...,x_{n})\psi^{*}(x_{1},x_{2}...,x_{n})}dx_{3}...dx_{n}[/tex]

It can be intepreted as the probability of finding two electrons at [itex]x_{1}[/itex] and [itex]x_{2}[/itex] respectively for all possible configurations of the other electrons because of the probabilistic interpretation of [itex]|\psi\psi*[/itex]

The single particle density matrix is defined as :

[tex]\gamma (x_{1},x'_{1}) = N\int{\psi(x_{1},x_{2}...,x_{n})\psi^{*}(x'_{1},x_{2}...,x_{n})}dx_{2}...dx_{n}[/tex]

Where the prime variable appears only in the complex conjugate of the function.

Can this be interpreted in probabilistic terms? why is there a prime? What is the meaning of the product of a wave function with its complex conjugate and having a different variable.

I know that for the diagonal terms it reduces to [itex]\rho (r)[/itex].

Any replies are much appreciated.

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# Single particle Density Matrix meaning

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