Читать книгу Quantum Mechanics, Volume 3 - Claude Cohen-Tannoudji - Страница 89

Consider a variation of |θ〉:

(35)

where |δα〉 is an arbitrary infinitesimal ket of the individual state space, and χ an arbitrary real number. To ensure that the normalization condition (6) is still satisfied, we impose |δα〉 and |θ〉 to be orthogonal:

(36)

so that 〈θ|θ〉 remains equal to 1 (to the first order in |δα〉). Inserting (35) into (34) to obtain the variation of the variational energy, we get the sum of two terms: the first one comes from the variation of the ket |θ〉 and is proportional to eiχ the second one comes from the variation of the bra 〈θ| and is proportional to e–χ. The result has the form:

(37)

The stationarity condition for must hold for any arbitrary real value of χ. As before (§ 2-b-α), it follows that both δc₁ and δc₂ are zero. Consequently, we can impose the variation to be zero as just the bra 〈ι| varies (but not the ket |θ〉), or the opposite.

Varying only the bra, we get the condition:

(38)

As the interaction operator W₂ (1, 2) is symmetric, the last two terms within the bracket in this equation are equal. We get (after simplification by N):

(39)