By Ruth Colvin Clark; Richard E Mayer

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**Example text**

55) = 0, ∓ ˆ that follow similarly. It is worth noting that the d-operators obey the same (anti)commutation relations as the ﬁeld operators with the index n playing the role of x. This is a very important observation since the results of the previous section relied only on the (anti)commutation relations of ψˆ and ψˆ† , and hence remain valid in this more general basis. To convince the reader of this fact we derive some of the results of the previous section directly from the (anti)commutation relations.

The (N − 1)-particle density matrix can be obtained by integrating out one coordinate. Setting yN = xN we have dxN ΓN (y1 , . . , yN −1 , xN ; x1 , . . , xN ) = Ψ|ψˆ† (x1 ) . . ψˆ† (xN −1 ) ˆ N ) ψ(y ˆ 1 )|Ψ ˆ N −1 ) . . ψ(y dxN ψˆ† (xN )ψ(x ˆ N ˆ† ˆ† ˆ N −1 ) . . ψ(y ˆ 1 )|Ψ = Ψ|ψ (x1 ) . . ψ (xN −1 )ψ(y = ΓN −1 (y1 , . . , yN −1 ; x1 , . . , xN −1 ), ˆ acts on a one-particle state. We can continue where we use the fact that the operator N this procedure and integrate out coordinate yN −1 = xN −1 .

8. 91). 92) j = x, y, z, σσ ′ with σx = 0 1 1 0 , σy = 0 −i i 0 σz = , 1 0 0 −1 the Pauli matrices. 7 7 The Levi–Civita tensor is zero if at least two indices are equal and otherwise εP (1)P (2)P (3) = (−)P , where P is an arbitrary permutation of the indices 1, 2, 3. 7. 7 35 Density matrices and quantum averages We have already stressed several times that if we know the ket |Ψ , which means if we know the wavefunction Ψ(x1 , . . , xN ) = x1 . . xN |Ψ , then we can use the (anti)commutation ˆ ˆ is an rules of the ﬁeld operators to calculate any quantum average Ψ|O|Ψ , where O operator in second quantization.