By Vasko F.T., Raichev O.E.

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10) with u replace es (q)/ Ms by eql / Mc , where the index l numbers the acoustic modes and the mass of the cell stands here due to the normalization conditions s |es (q)|2 = 1 and |eql |2 = 1. As a result, we obtain the operator ˆ ac (r) = u 2ρωql V ql eql eiq·r ˆbql + ˆb+ −ql . (29) The relative change of the volume at the point r, generated by the long-wavelength acoustic vibrations in the isotropic medium, is equal to ∇ · uac (r). Only the longitudinal vibrations contribute to this quantity.

22) dz −d/2 Outside the well region, for |z| > d/2, we use the wave function (18) with κ = 2m(U0 − ε0 ), and the energy of the level becomes ε0 = U0 − mU02 . 2( /d)2 (23) 32 QUANTUM KINETIC THEORY This expression veriﬁes the shallow level condition. The wave function is formed mostly by the tails outside the well; see Eq. (18). Low-dimensional states. The dynamics of electrons is modiﬁed essentially due to the above-described conﬁnement eﬀect. Apart from the widely known case of attractive three-dimensional potentials describing the states in atoms or the localized states on the impurity centers in solids, there exist low-dimensional systems realized in solid-state nanostructures.

2 (9) Substituting the expression for Qql into Eq. (5), we obtain the secondquantized displacement operator ˆ ns = (N Ms )−1/2 u ql 2ωql es (ql)eiq·Rn (ˆbql + ˆb+ −ql ) (10) 38 QUANTUM KINETIC THEORY written through the linear combination of creation and annihilation operators. Therefore, the displacements us satisfy the oscillatory equations (2) analogous to the equations for the electromagnetic ﬁeld in the absence of free charges. To quantize the lattice vibrations, one should express the displacements in terms of the generalized normal coordinates and momenta and, further, introduce the elementary excitations, phonons.