Quantum Mechanics, Volume 3. Claude Cohen-Tannoudji

      Complement F_XV Fermions, time-dependent Hartree-Fock approximation

      The Hartree-Fock mean field method was introduced in Complement E_XV for a time-independent problem: the search for the stationary states of a system of interacting fermions (the search for its thermal equilibrium will be discussed in Complement G_XV. In this complement, we show how this method can be used for time-dependent problems. We start, in § 1, by including a time dependence in the Hartree-Fock variational ket (time-dependent Fock state). We then introduce in § 2 a general variational principle that can be used for solving the time-dependent Schrödinger equation. We then compute, in § 3, the function to be optimized for a Fock state; the same mean field operator as the one introduced in Complement E_XV will here again play a very useful role. Finally, the time-dependent Hartree-Fock equations will be obtained and discussed in § 4. More details on the Hartree-Fock methods in general can be found, for example, in Chapter 7 of reference [5], and especially in its Chapter 9 for time-dependent problems.

1. Variational ket and notation

      We assume the N-particle state vector to be of the form:

(1)

      where the are the creation operators associated with an arbitrary series of orthonormal individual states |θ₁(t)〉, |θ₂(t)〉, …, |θN(t)〉 which depend on time t. This series is, for the moment, arbitrary, but the aim of the following variational calculation is to determine its time dependence.

As in the previous complements, we assume that the Hamiltonian Ĥ is the sum of three terms: a kinetic energy Hamiltonian, an external potential Hamiltonian, and a particle interaction term:

      (2)

      with:

      (3)

      (m is the particles’ mass, P_q the momentum operator of particle q), and:

      (4)

      and finally:

      (5)

2. Variational method

      Let us introduce a general variational principle; using the stationarity of a functional S of the state vector Ψ(t)〉, it will yield the time-dependent Schrödinger equation.

2-a. Definition of a functional

      Consider an arbitrarily given Hamiltonian H(t). We assume the state vector |Ψ(t)〉 to have any time dependence, and we note the ket physically equivalent to |Ψ(t)〉, but with a constant norm:

(6)

The functional S of is defined as¹:

(7)

      where t₀ and t₁ are two arbitrary times such that t₀ < t₁. In the particular case where the chosen is equal to a solution of the Schrödinger equation:

(8)

the bracket on the first line of (7) obviously cancels out and we have:

      (9)

Integrating by parts the second term² of the bracket in the second line of (7), we get the same form as the first term in the bracket, plus an already integrated term. The final result is then:

(10)