Читать книгу Quantum Mechanics, Volume 3 - Claude Cohen-Tannoudji - Страница 130

All the average values can be expressed in terms of the projector PN onto the subspace of the space the individual states spanned by the N individual states |θ₁〉, |θ₂〉, ….|θN〉, which means, according to (50), in terms of the one-particle reduced density operator . Hence it is this operator that is the pertinent variable to optimize rather than the set of individual states: certain variations of those states do not change PN, and are meaningless for our purpose.

Furthermore, the choice of the trial ket is equivalent to that of PN. In other words, the variational ket built in (1) does not depend on the basis chosen in the subspace : if we choose in this subspace any orthonormal basis {|uj〉} other than the {|θj〉} basis, and if we replace in (1) the by the , the ket will remain the same (to within a non-relevant phase factor) as we now show. As seen in § A-6 of Chapter XV, each operator is a linear combination of the , so that in the product of all the (j = 1, 2, ..N) we will find products of N operators . Relation (A-43) of Chapter XV however indicates that the squares of any creation operators are zero, which means that the only non-zero products are those including once and only once each of the N different operators . Each term is then proportional to the ket built from the . Consequently, the two variational kets built from the two bases are necessarily proportional. As definition (1) ensures they are also normalized, they can only differ by a phase factor, which means they are equivalent from a physical point of view. It is thus the operator that best embodies the trial ket .