Читать книгу Quantum Mechanics, Volume 3 - Claude Cohen-Tannoudji - Страница 88

To compute the average energy value , we use a basis {|θ_k〉} of the individual state space, whose first vector is |θ₁〉 = |θ〉.

Using relation (B-12) of Chapter XV, we can write the average value as:

(29)

Since is a Fock state whose only non-zero population is that of the state |θ₁〉, the ket is non-zero only if l = 1; it is then orthogonal to if k ≠ 1. Consequently, the only term left in the summation corresponds to k = l = 1. As the operator multiplies the ket by its population N, we get:

(30)

With the same argument, we can write:

(31)

Using relation (C-16) of Chapter XV, we can express the average value of the interaction energy as³:

(32)

In this case, for the second matrix element to be non-zero, both subscripts m and n must be equal to 1 and the same is true for both subscripts k and l (otherwise the operator will yield a Fock state orthogonal to ). When all the subscripts are equal to 1, the operator multiplies the ket by N (N — 1). This leads to:

(33)

The average interaction energy is therefore simply the product of the number of pairs N(N —1)/2 that can be formed with N particles and the average interaction energy of a given pair.

We can replace |θ₁〉 by |θ〉, since they are equal. The variational energy, obtained as the sum of (30), (31) and (33), then reads:

(34)