Читать книгу Quantum Mechanics, Volume 3 - Claude Cohen-Tannoudji - Страница 101

Let us see which excitations can propagate in this physical system, whose wave function is no longer the function (19), uniform in space. We assume:

(22)

where δφ(r, t) is sufficiently small to be treated to first order. Inserting this expression in the right-hand side of (16), and keeping only the first-order terms, we find in the interaction term the first-order expression:

(23)

We therefore get, to first-order:

(24)

which shows that the evolution of δφ(r, t) is coupled to that of δφ*(r, t). The complex conjugate equation can be written as:

(25)

We can make the time-dependent exponentials on the right-hand side disappear by defining:

(26)

This leads us to a differential equation with constant coefficients, which can be simply expressed in a matrix form:

(27)

where we have used definition (20) for μ to replace 2gn₀ — μ by gn₀. If we now look for solutions having a plane wave spatial dependence:

(28)

the differential equation can be written as:

(29)

The eigenvalues ħw(k) of this matrix satisfy the equation:

(30)

that is:

(31)

The solution of this equation is:

(32)

(the opposite value is also a solution, as expected since we calculate at the same time the evolution of and of its complex conjugate; we only use here the positive value). Setting:

(33)

relation (32) can be written:

(34)

The spectrum given by (32) is plotted in Figure 1, where one sees the intermediate regime between the linear region at low energy, and the quadratic region at higher energy. It is called the “Bogolubov spectrum” of the boson system.