Quantum Mechanics, Volume 3. Claude Cohen-Tannoudji

Quantum Mechanics, Volume 3 - Claude Cohen-Tannoudji


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else6, taking into account (A-55):

      (A-61)image

      We have thus shown that the operators image.. 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 image and of image 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.

      Using creation and annihilation operators makes it much easier to deal, in the Fock space, with physical operators that are thus symmetric (§ C-4-a-β of Chapter XIV). We first study the simplest of such operators, those which act on a single particle and are called “one-particle operators”.

      Consider an operator image defined in the space of individual states; image acts in the state space of particle q. It could be for example the momentum of the q-th particle, or its angular momentum with respect to the origin. We now build the operator associated with the total momentum of the N-particle system, or its total angular momentum, which is the sum over q of all the image associated with the individual particles.

      A one-particle symmetric operator acting in the space S(N) for bosons - or A(N) for fermions - is therefore defined by:

      We choose a basis {|ui〉} for the individual states. The matrix elements fkl of the one-particle operator image are given by:

      (B-3)image

      They can be used to expand the operator itself as follows:

      B-2-a. Action of F(N) on a ket with N particles

      (B-6)image

      with coefficients fkl. Let us use (A-7) or (A-10) to compute this ket for given values of k and l. As the operator contained in the bracket is symmetric with respect to the exchange of particles, it commutes with the two operators SN and AN (§ C-4-a-β of Chapter XIV)), and the ket can be written as:

      (B-7)image


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