Quantum Mechanics, Volume 3. Claude Cohen-Tannoudji

is still present for a single particle, since making g = 0 in (57) does not change this potential. For g = 0, the Gross-Pitaevskii equation simply reduces to the standard Schrödinger equation, valid for a single particle.

      3 ³ or not at all, if we suppose the functions ul(r, z) and ul′(r, z) to be equal.

      4 ⁴ When several relaxation channels are present, the one associated with the lowest barrier mainly determines the time evolution.

      Complement E_XV Fermion system, Hartree-Fock approximation

      1  1 Foundation of the method 1-a Trial family and Hamiltonian 1-b Energy average value 1-c Optimization of the variational wave function 1-d Equivalent formulation for the average energy stationarity 1-e Variational energy 1-f Hartree-Fock equations

      2  2 Generalization: operator method 2-a Average energy 2-b Optimization of the one-particle density operator 2-c Mean field operator 2-d Hartree-Fock equations for electrons 2-e Discussion

      Introduction

Computing the energy levels of a system of N electrons, interacting with each other through the Coulomb force, and placed in an external potential V₁(r) is a very important problem in physics and chemistry. It is encountered in the determination of the energy levels of atoms (in which case the external potential for the electrons¹ is the Coulomb potential created by the nucleus –Zq²/4πε₀r), or of molecules as well, or of electrons in a solid (submitted to a periodic potential), or in an aggregate or a nanocristal, etc. It is a problem where two ingredients simultaneously play an essential role: the fermionic character of the electrons, which forbids them to occupy the same individual state, and the effects of their mutual interactions. Ignoring the Coulomb repulsion between electrons would make the calculation fairly simple, and similar to that of § 1 in Complement C_XIV, concerning free fermions in a box; the free plane wave individual states would have to be replaced by the energy eigenstates of a single particle placed in the potential V₁(r). This would lead to a 3-dimensional Schrodinger equation, which can be solved with very good precision, although not necessarily analytically.

However, be it in atoms or in solids, the repulsion between electrons plays an essential role. Neglecting it would lead us to conclude, for example, that, as Z increases, the size of atoms decreases due to the attractive effect of the nucleus, whereas the opposite occurs²! For N interacting particles, even without taking the spin into account, an exact computation would require solving a Schrödinger equation in a 3N-dimensional space; this is clearly impossible when N becomes large, even with the most powerful computer. Hence, approximation methods are needed, and the most common one is the Hartree-Fock method, which reduces the problem to solving a series of 3-dimensional equations. It will be explained in this complement for fermionic particles.

      The Hartree-Fock method is based on the variational approximation (Complement E_XI), where we choose a trial family of state vectors, and look for the one that minimizes the average energy. The chosen family is the set of all possible Fock states describing the system of N fermions. We will introduce and compute the “self-consistent” mean field in which each electron moves; this mean field takes into account the repulsion due to the other electrons, hence justifying the central field method discussed in Complement A_XIV. This method applies not only to the atom’s ground state but also to all its stationary states. It can also be generalized to many other systems such as molecules, for example, or to the study of the ground level and excited states of nuclei, which are protons and neutrons in bound systems.

      This complement presents the Hartree-Fock method in two steps, starting in § 1 with a simple approach in terms of wave functions, which is then generalized in § 2 by using Dirac notation and projector operators. The reader may choose to go through both steps or go directly to the second. In § 1, we deal with spinless particles, which allows discussing the basic physical ideas and introducing the mean field concept keeping the formalism simple. A more general point of view is exposed in § 2, to clarify a number of points and to introduce the concept of a one-particle (with or without spin) effective Hartree-Fock Hamiltonian. This Hamiltonian reduces the interactions with all the other particles to a mean field operator. More details on the Hartree-Fock methods, and in particular their relations with the Wick theorem, can be found in Chapters 7 and 8 of reference [5].

1. Foundation of the method

      Let us first expose the foundation of the Hartree-Fock method in a simple case where the particles have no spin (or are all in the same individual spin state) so that no spin quantum number is needed to define their individual states, specified by their wave functions. We introduce the notation and define the trial family of the N-particle state vectors.

1-a. Trial family and Hamiltonian

      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