B.3.d. Matrix Elements

Consider an N-particle operator that is a sum of N 1-particle operators, i.e.,

        O ˆ ( N ) ( x 1 x N )= j=1 N O ˆ ( 1 ) ( x j )                            (B.52)

where x j =( q j , p j , σ j ) .  Now, consider the matrix element

                α 1 α j α N | O ˆ ( 1 ) ( x j )| α 1 α j α N

        = α j | O ˆ ( 1 ) ( x )| α j α 1 α N | α 1 α N

Thus, the matrix elements of O ˆ ( N )  are expected to be a sum of matrix elements of O ˆ ( 1 ) , i.e.,

        α 1 α j α N | ( S/A ) O ˆ ( N ) ( x 1 x N ) | α 1 α j α N ( S/A )

        jj' α j | O ˆ ( 1 ) ( x )| α j α 1 α N | ( S/A ) a α + a α' | α 1 α N ( S/A )

Indeed, after some tedious counting of states [see e.g., chapter 1 of Fetter & Walecka], it can be shown that if

        | α 1 α j α N ( S/A ) =| n 0 n α n

        | α 1 α j α N ( S/A ) =| n 0 n α n

then

        α 1 α j α N | ( S/A ) O ˆ ( N ) ( x 1 x N ) | α 1 α j α N ( S/A )

        = n 0 n α n | O ˆ | n 0 n α n

provided

        O ˆ = α=0 α'=0 α| O ˆ ( 1 ) ( x )| α    a α + a α'                                (B53,72)

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.,

        O ˆ ( N ) ( x 1 x N )= 1 2 i=1 N j=1i N O ˆ ( 2 ) ( x i , x j )   = i=1 N j=1 i1 O ˆ ( 2 ) ( x i , x j )

where O ˆ ( 2 ) ( x i , x j )= O ˆ ( 2 ) ( x j , x i ) , the n-representation version is

        O ˆ = 1 2 α=0 β=0 α'=0 β'=0 αβ | O ˆ ( 2 ) ( x 1 , x 2 )| α β    a α + a β + a β' a α'              (B.58,73)

Note that in αβ | O ˆ ( 2 ) ( x, x )| α β , the states associated with the 1st (2nd) particle are a, a' (b, b ' ).  The particular order of the operators in a α + a β + a β' a α'  should be noted.