Читать книгу Quantum Mechanics, Volume 3 - Claude Cohen-Tannoudji - Страница 76

1  1 Notation, variational ket 1-a Hamiltonian 1-b Choice of the variational ket (or trial ket)

2  2 First approach 2-a Trial wave function for spinless bosons, average energy 2-b Variational optimization

3  3 Generalization, Dirac notation 3-a Average energy 3-b Energy minimization 3-c Gross-Pitaevskii equation

4  4 Physical discussion 4-a Energy and chemical potential 4-b Healing length 4-c Another trial ket: fragmentation of the condensate

The Bose-Einstein condensation phenomenon for an ideal gas (no interaction) of N identical bosons was introduced in § 4-b-β of Complement B_XV. We show in the present complement how to describe this phenomenon when the bosons interact. We shall look for the ground state of this physical system within the mean field approximation, using a variational method (see Complement E_XI). After introducing in § 1 the notation and the variational ket, we study in § 2 spinless bosons, for which the wave function formalism is simple and the introduction of the creation and annihilation operators does not lead to any major computation simplifications. This will lead us to a first version of the Gross-Pitaevskii equation. We will then come back in § 3 to Dirac notation and the creation operators, to deal with the more general case where each particle may have a spin. Defining the Gross-Pitaevskii potential operator, we shall obtain a more general version of that equation. Finally, some properties of the Gross-Pitaevskii equation will be discussed in § 4, as well as the role of the chemical potential, the existence of a relaxation (or “healing”) length, and the energetic consequences of “condensate fragmentation” (these terms will be defined in § 4-c).