Читать книгу Quantum Mechanics, Volume 3 - Claude Cohen-Tannoudji - Страница 36

A first very simple example is the operator , already described in (A-29), and corresponding to the total number of particles:

(B-15)

As expected, this operator does not depend on the basis {|ui〉} chosen to count the particles, as we now show. Using the unitary transformations of operators (A-51) and (A-52), and with the full notation for the creation and annihilation operators to avoid any ambiguity, we get:

(B-16)

which shows that:

(B-17)

For a spinless particle one can also define the operator corresponding to the probability density at point r₀:

(B-18)

Relation (B-12) then leads to the “particle local density” (or “single density”) operator:

(B-19)

The same procedure as above shows that this operator is independent of the basis {|ui〉} chosen in the individual states space.

Let us assume now that the chosen basis is formed by the eigenvectors |K_i〉 of a particle’s momentum ħk_i, and that the corresponding annihilation operators are noted a_ki. The operator associated with the total momentum of the system can be written as:

(B-20)

As for the kinetic energy of the particles, its associated operator is expressed as:

(B-21)