Читать книгу Quantum Mechanics, Volume 3 - Claude Cohen-Tannoudji - Страница 30

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) and (A-10) the ui by the vs, we construct the new Fock basis. We next define the operators by transposing relations (A-17) and (A-18) to the new basis. We then must verify that these operators obey relation (A-51), without limiting ourselves, as in (A-50), to their action on the vacuum state.

(i) Bosons

Relations (A-7) and (A-17) lead to:

(A-55)

where, on the right-hand side, the ni first particles occupy the same individual state ui the following nj particles, numbered from ni + 1 to ni + nj, the individual state uj, etc. The equivalent relation in the second basis can be written:

(A-56)

with:

(A-57)

Replacing on the right-hand side of (A-56), the first ket |vs〉 by:

(A-58)

we obtain:

(A-59)

Following the same procedure for all the basis vectors of the right-hand side, we can replace it by:

(A-60)

or else⁶, taking into account (A-55):

(A-61)

We have thus shown that the operators .. act on the vacuum state in the same way as the operators defined by (A-51), raised to the powers ps, pt, ..

When the occupation numbers ps, pt, .. can take on any values, the kets (A-56) span the entire Fock space. Writing the previous equality for ps and ps + 1, we see that the action on all the basis kets of and of yields the same result, establishing the equality between these two operators. Relation (A-52) can be readily obtained by Hermitian conjugation.

(ii) Fermions

The demonstration is identical, with the constraint that the occupation numbers are 0 or 1 . As this requires no changes in the operator or state order, it involves no sign changes.