Commutator Relation between Creation and Annihilation Operators

5 Commutator Relation between Creation and Annihilation Operators 

Application of the second quantized formalism in quantum chemistry is merely an appHcation of simple algebraic rules followed by creation and annihilation operators. We have already been acquainted with one rule of this kind the anticommutator relation for creation operators of Eq. (2.11). The mutual commutator properties of creation and annihilation operators will be studied below. Again, the true annihilation operators will be considered as introduced above leaving open the question how a is related to a.  

Consider an N-electron one-determinantal wave function V in the particle number representation  

Let us rearrange this wave function so that two selected orbitals, say i and k (with i k) form the first two columns  

According to the rules given above, this expression vanishes if either n = 1 (because no more electron can be created), or if nj = 0 (because no electron can be annihilated from an empty orbital). So one must have n, = 0 and nj = 1 otherwise Eq. (2.29) vanishes. In this case, indicating only the orbitals i and k, the wave function may be abbreviated as i , and Eq. (2.29) becomes  

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We have seen that the creation operators anticommute among themselves (1.2.11) and that the same is true for the annihilation operators (1.2.12). We shall now establish the commutation relations between creation and annihilation operators. Combining (1.2.5) and (1.2.16), we obtain... 

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