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By Seeger R. J.

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63). The rule is that the wave functions corresponding to the states γ and β are attributed to the same particle, and so are the ones corresponding to the states α and δ. This “natural” way of grouping the labels α, β, γ , δ is commonly referred to as direct pairing. More complicated observables – acting on groups of three or more particles at a time – can also be handled in a similar way. 3 Construction of the second-quantized hamiltonian Let us apply the formalism of second quantization to the construction of the jellium model hamiltonian of Eq.

N . 33) of the basis states, and going through the following chain of transformations: † aˆ β aˆ αi (aˆ α† 1 aˆ α† 2 . . aˆ α† i . . aˆ α† N )|0 † = aˆ α† 1 aˆ α† 2 . . aˆ β aˆ αi aα† i . . aˆ α† N |0 † = aˆ α† 1 aˆ α† 2 . . aˆ β [1 − aα† i aˆ αi ] . . aˆ α† N |0 † = aˆ α† 1 aˆ α† 2 . . aˆ β . . aˆ α† N |0 = |α1 , α2 , . . , αi → β, . . , α N . 56) † The first equality follows from the fact that aˆ β aˆ αi commutes with all the aˆ α† ’s with α < αi . 45), and, in the third one, of the fact that aˆ αi anticommutes with all the aˆ α† ’s with α > αi , and thus annihilates the vacuum.

Returning to Eq. 61), we now see that the action of the operator Vˆ (2) is identical with † that of the operator 12 γ δ i= j Vγ δαi α j aˆ γ† aˆ δ aˆ αi aˆ α j . We can extend the sum over occupied states αi , α j to a sum over all states, since the additional terms give zero. We thus obtain the formula expressing a two-particle operator as a sum of products of four creation and destruction operators: 1 Vˆ (2) = 2 i= j 1 † Vˆi j = Vαβγ δ aˆ α† aˆ β aˆ γ aˆ δ . 63) The schematic diagram of Fig. 8 provides a helpful visual guide to the construction of the matrix elements in Eq.

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