B.3.d. Matrix Elements
Consider an N-particle operator that is a sum of N 1-particle operators, i.e.,
. Now, consider the matrix element
Thus, the matrix elements of
are expected to be a sum of matrix elements of
Indeed, after some tedious counting of
states [see e.g., chapter 1 of Fetter & Walecka], it can be shown that if
Note that (B53) is valid for both bosons
and fermions, as well as for all N.
Similarly, for an N-particle operator that is a sum of 2-particle operators, i.e.,
the n-representation version is
Note that in
the states associated with the 1st (2nd) particle are a, a' (b, b ' ).
The particular order of the operators in
should be noted.