Claude Cohen-Tannoudji - Quantum Mechanics, Volume 3

As already mentioned, we shall see in § 4-a that μ is simply the chemical potential.

We now go back to the previous line of reasoning, but in a more general case where the bosons may have spins. The variational family is the set of the N -particle state vectors written in (7). The one-body potential may depend on the position r, and, at the same time, act on the spin (particles in a magnetic field gradient, for example).

To compute the average energy value , we use a basis {| θ k〉} of the individual state space, whose first vector is | θ 1〉 = | θ 〉.

Using relation (B-12) of Chapter XV, we can write the average value as:

(29)

Since is a Fock state whose only non-zero population is that of the state | θ 1〉, the ket is non-zero only if l = 1; it is then orthogonal to if k ≠ 1. Consequently, the only term left in the summation corresponds to k = l = 1. As the operator multiplies the ket by its population N , we get:

(30)

With the same argument, we can write:

(31)

Using relation (C-16) of Chapter XV, we can express the average value of the interaction energy as 3 :

(32)

In this case, for the second matrix element to be non-zero, both subscripts m and n must be equal to 1 and the same is true for both subscripts k and l (otherwise the operator will yield a Fock state orthogonal to ). When all the subscripts are equal to 1, the operator multiplies the ket by N ( N — 1). This leads to:

(33)

The average interaction energy is therefore simply the product of the number of pairs N ( N —1)/2 that can be formed with N particles and the average interaction energy of a given pair.

We can replace | θ 1〉 by | θ 〉, since they are equal. The variational energy, obtained as the sum of (30), (31)and (33), then reads:

(34)

Consider a variation of | θ 〉:

(35)

where | δα 〉 is an arbitrary infinitesimal ket of the individual state space, and χ an arbitrary real number. To ensure that the normalization condition (6)is still satisfied, we impose | δα 〉 and | θ 〉 to be orthogonal:

(36)

so that 〈 θ | θ 〉 remains equal to 1 (to the first order in | δα 〉). Inserting (35)into (34)to obtain the variation of the variational energy, we get the sum of two terms: the first one comes from the variation of the ket | θ 〉 and is proportional to eiχ the second one comes from the variation of the bra 〈 θ | and is proportional to e–χ . The result has the form:

(37)

The stationarity condition for must hold for any arbitrary real value of χ . As before (§ 2-b- α ), it follows that both δc 1and δc 2are zero. Consequently, we can impose the variation to be zero as just the bra 〈 ι | varies (but not the ket | θ 〉), or the opposite.

Varying only the bra, we get the condition:

(38)

As the interaction operator W 2(1, 2) is symmetric, the last two terms within the bracket in this equation are equal. We get (after simplification by N ):

(39)

To deal with equation (39), we introduce the Gross-Pitaevskii operator , defined as a one-particle operator whose matrix elements in an arbitrary basis are {| u i〉} given by:

(40)

which leads to:

(41)