Читать книгу Quantum Mechanics, Volume 3 - Claude Cohen-Tannoudji - Страница 92

Since the ket |φ〉 is normalized, multiplying (44) by the bra 〈φ| and by N, we get:

(49)

We recognize the first two terms of the left-hand side as the average values of the kinetic energy and the external potential. As for the last term, using definition (41) for , we can write it as:

(50)

which is simply twice the potential interaction energy given in (33) when |θ₁〉 = |φ〉. This leads to:

(51)

To find the energy , note that N μ/2 is the sum of and of half the kinetic and external potential energies. Adding the missing halves, we finally get for :

(52)

An advantage of this formula is to involve only one- (and not two-) particle operators, which simplifies the computations. The interaction energy is implicitly contained in the factor μ.

The quantity μ does not yield directly the average energy, but it is related to it, as we now show. Taking the derivative, with respect to N, of equation (34) written for |θ〉 = |φ〉, we get:

(53)

For large N, one can safely replace in this equation (N — 1/2) by (N — 1); after multiplication by N, we obtain a sum of average energies:

(54)

Taking relation (51) into account, this leads to:

(55)

We know (Appendix VI, § 2-b) that in the grand canonical ensemble, and at zero temperature, the derivative of the energy with respect to the particle number (for a fixed volume) is equal to the chemical potential. The quantity μ introduced mathematically as a Lagrange multiplier, can therefore be simply interpreted as this chemical potential.