Quantum Mechanics, Volume 3. Claude Cohen-Tannoudji

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two equations yields zero, hence proving that the anticommutator is zero:

      (A-42)

      In the case where i = j, the limitation on the occupation numbers (0 or 1) leads to:

      (A-43)

Equalities (A-37) and (A-39) are still valid if i and j are equal. We are now left with the computation of the anticommutator of ai and . Let us first examine the product ; it yields zero if applied to a ket having an occupation number ni = 1, but leaves unchanged any ket with ni = 0, since the particle created by is then annihilated by ai. We get the inverse result for the product where the order has been inverted: it yields zero if ni = 0, and leaves the ket unchanged if ni = 1. Finally, whatever the occupation number ket is, one of the terms of the anticommutator yields zero, the other 1 , and the net result is always 1 . Therefore:

      (A-44)

All the previous results valid for fermions are summarized in the following three relations, which are for fermions the equivalent of relations (A-32) for bosons:

      (A-45)

      A-5-c. Common relations for bosons and fermions

      To regroup the results valid for bosons and fermions in common relations, we introduce the notation:

(A-46)

      with:

      (A-47)

so that (A-46) is the commutator of A and B for bosons, and their anticommutator for fermions. We then have:

      (A-48)

      and the only non-zero combinations are:

(A-49)

A-6. Change of basis

      What are the effects on the creation and annihilation operators of a change of basis for the individual states? The operators and aui have been introduced by their action on the Fock states, defined by relations (A-7) and (A-10) for which a given basis of individual states {|ui〉} was chosen. One could also choose any another orthonormal basis {|vs〉} and define in the same way bases for the Fock state and creation and annihilation avs operators. What is the relation between these new operators and the ones we defined earlier with the initial basis?

      For creation operators acting on the vacuum state |0〉, the answer is quite straightforward: the action of on |0〉 yields a one-particle ket, which can be written as:

(A-50)

      This result leads us to expect a simple linear relation of the type:

(A-51)

      with its Hermitian conjugate:

(A-52)

Equation (A-51) implies that creation operators are transformed by the same unitary relation as the individual states. Commutation or anticommutation relations are then conserved, since:

      (A-53)

which amounts to (as expected):

      (A-54)

      Furthermore, it is straightforward to show that the creation operators commute (or anticommute), as do the annihilation operators.

       Equivalence of the two bases

We have not yet shown the complete equivalence of the two bases, which can be done following two different approaches. In the first one, we use (A-51) and (A-52) to define the creation and annihilation operators in the new basis. The associated Fock states are defined by replacing the by the in relations (A-17) for the bosons, and (A-18) for the fermions. We then have to show that these new Fock states are still related to the states with numbered particles as in (A-18) for bosons, and (A-10) for fermions. This will establish the complete equivalence of the two bases.

We shall follow a second approach where the two bases are treated completely symmetrically. Replacing in relations (A-7)

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