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I am working through Pathria's statistical mechanics book; on page 114 I found the following definition for the density operator:

[tex] \rho_{mn}= \frac{1}{N} \sum_{k=1}^{N}\left \{ a(t)^{k}_m a(t)^{k*}_n \right \}, [/tex]

where N is the number of systems in the ensemble and the a(t)'s are expansion coefficents.

Now my question is: what does this definition mean? Especially the term [tex] a(t)^{k}_m a(t)^{k*}_n. [/tex] I do not get it.

Any help would be greatly appreciated!