Читать книгу Quantum Mechanics, Volume 3 - Claude Cohen-Tannoudji - Страница 83

Let us vary the function θ(r) by a quantity:

(17)

where δf(r) is an infinitesimal function and χ an arbitrary number. A priori, δf(r) must be chosen to take into account the normalization constraint (6), which forces the integral of the θ(r) modulus squared to remain constant. We can, however, use the Lagrange multiplier method (Appendix V) to impose this constraint. We therefore introduce the multiplier μ (we shall see in § 4-a that this factor can be interpreted as the chemical potential) and minimize the function:

(18)

This allows considering the infinitesimal variation δf(r) to be free of any constraint. The variation of the function is now the sum of 4 variations, coming from the three terms of (16) and from the integral in (18). For example, the variation of yields:

(19)

which is the sum of a term proportional to e^–iχ and another proportional to e^iχ. This is true for all 4 variations and the total variation can be expressed as the sum of two terms:

(20)

the first being the δf^*(r) contribution and the second, that of δf(r). Now if is stationary, must be zero whatever the choice of χ, which is real. Choosing for example χ = 0 imposes δc₁ + δc₂ = 0, and the choice χ = π/2 leads (after multiplication by i) to δc₁—δc₂ = 0. Adding and subtracting the two relations shows that both coefficients δc₁ and δc₂ must be zero. In other words, we can impose to be zero as just θ^*(r) varies but not θ(r) - or the opposite¹.