A Method of Calculating Ground State Properties of Many Particle Systems Using R

Claude Garrod

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This restriction can be relaxed at the end. It is not difficult to verify [l2] that the destruction operator for any such two-particle state may be written in some representation in the form: n= -S Hence the quantity we want to investigate is: + S (3. 57) J\==^~ A^r(n, -n|m, -m)A^ n, m =-S Now equation (3. 5^) is equivalent to (in fact was derived from) the equation; true for any values of x, and Xp : - 20 - (3. 58) x^G^^^l, !' |l, l' )+X2G^^^2, 2' |2, 2' )^2x^X2G ^ ^ ^ 1, 1 ' |2, 2' ) We now c...hoose: x, = A, ^p = '^■^ and: k-, = -m, k' = -n, kp = n, k' = m.
Then (3 '58) can be written: (3-59) A^+A^>2A A \v ^y n -n -m -m -n m m n n m — n m -n -m n m Summing over n and m and using the commutation relations we obtain: 2 S A^ (1+5 )- + n nm -n -n -m n, m (3. 60) + ^^ A^ (1+6 )- > 2 X~ A A (5, /(-nim) + ^> m nm m m n — ^ n m n+m^ ' n, m n, m + P'(n, -n |m, -m) ) Using the facts that: ^^ 1 = 2S and: V~ = N, 2i ^ -n -m -n ' m m we obtain: (3.

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