Читать книгу Quantum Mechanics, Volume 3 - Claude Cohen-Tannoudji - Страница 119

Let us introduce a complete orthonormal basis {|θs} of the one-particle state space by adding to the set of states |θi (i = 1, 2, N) other orthonormal states; the subscript s now ranges from 1 to D, dimension of this space (D may be infinite). We can then expand Ĥ₀ as in relation (B-12) of Chapter XV:

(10)

where the two summations over r and s range from 1 to D. The average value in of the kinetic energy can then be written:

(11)

which contains the scalar product of the ket:

(12)

by the bra:

(13)

Note however that in the ket, the action of the annihilation operator aθs yields zero unless it acts on a ket where the individual state is already occupied; consequently, the result will be different from zero only if the state |θS〉 is included in the list of the N states |θ₁〉, |θ₂〉, ….|θN〉. Taking the Hermitian conjugate of (13), we see that the same must be true for the state |θr〉, which must be included in the same list. Furthermore, if r ≠ s the resulting kets have different occupation numbers, and are thus orthogonal. The scalar product will therefore only differ from zero if r = s, in which case it is simply equal to 1. This can be shown by moving to the front the state |θr〉 both in the bra and in the ket; this will require two transpositions with two sign changes which cancel out, or none if the state |θr〉 was already in the front. Once the operators have acted, the bra and the ket correspond to exactly the same occupied states and their scalar product is 1. We finally get:

(14)

Consequently, the average value of the kinetic energy is simply the sum of the average kinetic energy in each of the occupied states |θi〉.

For spinless particles, the kinetic energy operator is actually a differential operator –ħ² Δ/2m acting on the individual wave functions. We therefore get:

(15)