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In quantum chemistry, there are concepts about one-particle reduced density matrix (1-RDM) and similarly, the two-particle reduced density matrix (2-RDM). Generally, for an $n$ particle wavefunction $|\Psi\rangle$ and the corresponding density matrix is ${}^nD=|\Psi\rangle\langle\Psi|$. Then there are two definitions of the 1-RDM and 2-RDM. The first is: \begin{equation}\tag{1} {}^1D= \begin{pmatrix} 1 \\ n \end{pmatrix} tr_{2,3,\cdots,n} {}^nD, \end{equation} in which will resulting $C_n^1=n$ density matrix. Another method is to define the element of 1-RDM as: \begin{equation}\tag{2} {}^1D_{ij} = \langle \Psi|a_i^{\dagger} a_j|\Psi\rangle, \end{equation} in which $i$ and $j$ run over $1$ to $n$ and the matrix will be a $n\times n$ one. Is there any connection or difference between the two representations?

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  • $\begingroup$ Can you specify what you mean by $a^{\dagger}_{i}$ and $a_{j}$? Are they ladder operators? $\endgroup$
    – JSdJ
    Dec 14, 2020 at 9:59
  • $\begingroup$ There are creation and annihilation operators. $\endgroup$
    – 刘环宇
    Dec 14, 2020 at 10:47
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    $\begingroup$ You'll find more people that know what you're talking about here: mattermodeling.stackexchange.com $\endgroup$ Dec 14, 2020 at 19:53
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    $\begingroup$ If you are interested in moving the question to Matter Modeling, I'm a mod there and we would be open to having it migrated. I can contact the mods here to have it moved. $\endgroup$
    – Tyberius
    Dec 15, 2020 at 23:12
  • $\begingroup$ OK. Thanks a lot. $\endgroup$
    – 刘环宇
    Dec 16, 2020 at 8:12

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