Читать книгу Quantum Mechanics, Volume 3 - Claude Cohen-Tannoudji - Страница 139

Consider an arbitrarily given Hamiltonian H(t). We assume the state vector |Ψ(t)〉 to have any time dependence, and we note the ket physically equivalent to |Ψ(t)〉, but with a constant norm:

(6)

The functional S of is defined as¹:

(7)

where t₀ and t₁ are two arbitrary times such that t₀ < t₁. In the particular case where the chosen is equal to a solution of the Schrödinger equation:

(8)

the bracket on the first line of (7) obviously cancels out and we have:

(9)

Integrating by parts the second term² of the bracket in the second line of (7), we get the same form as the first term in the bracket, plus an already integrated term. The final result is then:

(10)

where we have used in the second line the fact that the norm of always remains equal to unity. This expression for S is similar to the initial form (7), but without the real part.