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6.1.5. Number Representation: Operators Consider a 1-P operator A p, x . Given the complete orthonormal basis , we can write A p, x A A where the matrix elements A A where are numbers. Using â , is the vacuum, we can write  A aˆ â Note that the vacuum projector serves to confine  to the 1-particle  0 if the number of particles in either or is not one. subspace, i.e., By removing this restriction, we obtain the desired many body version  A â â Next, we consider the 2-P potential N 1 N V V xi , x j V xi , x j 2 i j 1 i j 1 , a basis vector for the 2-P Hilbert Given the complete orthonormal 1-P basis space is 1 2 1 2 1 2 where, to avoid ambiguity, we have used subscript to indicate the particle occupying the state. Taking the hermitian conjugate, we obtain the adjoint basis vector 1 2 2 1 2 1 1 2 where the order of the factors is to be noted with care. condition I , 1 2 2 1 1 2 , 1 2 we can write the 2-P potential (1st quantized) operator as Using the completeness V 1 2 1 2 2 V 1 1 2 2 1 1 12 12 V 1 2 1 2 2 Using 1 2 1 2 2 1 1 2 â â 1 2 â â â â we can write the 2nd quantized version of V as 1 Vˆ â aˆ V aˆ â 2 where V d 3x1 d 3x 2 * x1 * x 2 V x1 , x 2 x 2 x1 As before, the vacuum projector serves to confine Vˆ to the 2-particle subspace. Removing this restriction then gives 1 Vˆ V â â â â 2 (a) Note that , and , refer to the particle at x1 and x2, respectively. Thus, in eq(a), the first and last operators refer to particle 1.