Читать книгу Quantum Mechanics, Volume 3 - Claude Cohen-Tannoudji - Страница 117

We choose as the trial family for the state of the N-fermion system all the states that can be written as:

(1)

where are the creation operators associated with a set of normalized individual states |θ₁〉, |θ₂〉, |θ_N〉, all orthogonal to each other (and hence distinct). The state is therefore normalized to 1. This set of individual states is, at the moment, arbitrary; it will be determined by the following variational calculation.

For spinless particles, the corresponding wave function can be written in the form of a Slater determinant (Chapter XIV, § C-3-c-β):

(2)

The system Hamiltonian is the sum of the kinetic energy, the one-body potential energy and the interaction energy:

(3)

The first term, Ĥ₀, is the operator associated with the fermion kinetic energy, sum of the individual kinetic energies:

(4)

where m is the particle mass and P_q, the momentum operator of particle q. The second term, , is the operator associated with their energy in an applied external potential V₁:

(5)

where R_q is the position operator of particle q. For electrons with charge qe placed in the attractive Coulomb potential of a nucleus of charge —Zqe positioned at the origin (Z is the nucleus atomic number), this potential is attractive and equal to:

(6)

where ε₀ is the vacuum permittivity. Finally, the term corresponds to their mutual interaction energy:

(7)

For electrons, the function W₂ is given by the Coulomb repulsive interaction:

(8)

The expressions given above are just examples; as mentioned earlier, the Hartree-Fock method is not limited to the computation of the electronic energy levels in an atom.