The limit where while the density remains constant is often called the “thermodynamic limit”.6 6 As above, we assume periodic conditions on the box walls. Another choice would be to impose zero values for the wave functions on the walls: the numerical coefficients of the individual energies would be changed, but not the line of reasoning.
Complement CXV Condensed boson system, Gross-Pitaevskii equation
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 BXV. 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 EXI). 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).
1. Notation, variational ket
We first define the notation and the variational family of state vectors that will lead to relatively simple calculations for a system of identical interacting bosons.
1-a. Hamiltonian
The Hamiltonian operator Ĥ we consider is the sum of operators for the kinetic energy
(1)
The first term
(2)
where:
(Pq is the momentum of particle q). Similarly,
(4)
Finally,
(5)
(this summation can also be written as a sum over q < q′, while removing the prefactor 1/2).
1-b. Choice of the variational ket (or trial ket)
Let us choose an arbitrary normalized quantum state |θ〉:
and call
where |θ〉 can vary, only constrained by (6). Consider a basis {|θk〉} of the individual state space whose first vector is |θ1〉 = |θ〉. Relation (A-17) of Chapter XV shows that this ket is simply a Fock state whose only non-zero occupation number is the first one:
