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equation (16) simply becomes:

(18)

       Comment:

      It can be shown that this time evolution does conserve the norm of |φ(t)〉, as required by (3). Without the nonlinear term of (16), it would be obvious since the usual Schrödinger equation conserves the norm. With the nonlinear term present, it will be shown in § 2-a that the norm is still conserved.

1-c. Phonons and Bogolubov spectrum

      Still dealing with spinless bosons, we consider a uniform system, at rest, of particles contained in a cubic box of edge length L. The external potential V₁(r) is therefore zero inside the box and infinite outside. This potential may be accounted for by forcing the wave function to be zero at the walls. In many cases, it is however more convenient to use periodic boundary conditions (Complement C_XIV, § 1-c), for which the wave function of the individual lowest energy state is simply a constant in the box. We thus consider a system in its ground state, whose Gross-Pitaevskii wave function is independent of r:

(19)

      with a μ value that satisfies equation (16):

(20)

where n₀ = N/L³ is the system density. Comparing this expression with relation (58) of Complement C_XV allows us to identify μ with the ground state chemical potential. We assume in this section that the interactions between the particles are repulsive (see the comment at the end of the section):

(21)

       α. Excitation propagation

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.

       β. Discussion

Let us compute the spatial and time evolution of the particle density n(r, t) when δφ(r, t) obeys relation Скачать книгу