Читать книгу Quantum Mechanics, Volume 3 - Claude Cohen-Tannoudji - Страница 98

Let us introduce the functional of |Ψ(t)〉:

(4)

It can be shown that this functional is stationary when |Ψ(t)〉 is solution of the exact Schrodinger equation (an explicit demonstration of this property is given in § 2 of Complement F_XV. If |Ψ(t)〉 belongs to a variational family, imposing the stationarity of this functional allows selecting, among all the family kets, the one closest to the exact solution of the Schrodinger equation. We shall therefore try and make this functional stationary, choosing as the variational family the set of kets written as in (1) where the individual ket |θ(t)〉 is time-dependent.

As condition (3) means that the norm of remains constant, the second bracket in expression (4) must be zero. We now have to evaluate the average value of the Hamiltonian H(t) that, actually, has been already computed in (34) of Complement C_XV:

(5)

The only term left to be computed in (4) contains the time derivative.

This term includes the diagonal matrix element:

(6)

For an infinitesimal time dt, the operator is proportional to the difference , hence to the difference between two creation operators associated with two slightly different orthonormal bases. Now, for bosons, all the creation operators commute with each other, regardless of their associated basis. Therefore, in each term of the summation over k, we can move the derivative of the operator to the far right, and obtain the same result, whatever the value of k. The summation is therefore equal to N times the expression:

(7)

Now, we know that:

(8)

Using in (6) the bra associated with that expression, multiplied by N, we get:

(9)

Regrouping all these results, we finally obtain:

(10)