We consider a cylindrical quantum wire with the radius

*ρ*_{0}. Let the quantum wire be surrounded with compositionally identical barriers. In order to separate the effect of exactly dielectric irregularities, we assume that the potential well for electrons is rather deep, so that the penetration of the wave functions under the barrier can be disregarded. In this case, the interaction of charged particles with barrier phonons is weak. We write the Hamiltonian of the system as

Here,

is the electron Hamiltonian for which the interaction of the electron with phonons is disregarded. The Hamiltonian is given by

where

is the quantum wire potential and

*M* is the electron effective mass. If the interaction of an electron with polar optical phonons is strong, the polaron binding energy can be determined with the use of adiabatic approximation. In so doing, the electron subsystem is fast and phonon subsystem is slow. The adiabatic parameter here is the ratio of the quantum wire radius

*ρ*_{0} to the polaron radius

*a*_{0}:

The exact expression for polaron radius

*a*_{0} is obtained below. The condition (Eq.

13) implies that the main contribution to the polaron binding energy is given by small values of the wave vector

*k* such that

If condition (Eq.

13) is satisfied, the wave function of an electron localized in the n-th size-quantization level can be represented as:

where the wave function

describes the two-dimensional electron motion not disturbed by electron–phonon interaction. This motion occurs inside the quantum wire. The wave function

represents the electron localization in the self-consistent potential well created by phonons. The quantum numbers

*n*^{(e)},

*m*^{(e)} define not disturbed electron state in the quantum wire. In the case of total electron localization in the cylindrical quantum wire, the wave function

has the form:

Here
is *n*^{(e)}-th root of *m*^{(e)}-th order Bessel function. The wave function
is to be obtained by solving self-consistent problem. In so doing, the total wave function from Eq. 15 is perceived to be normalized.

The procedure of polaron binding energy determination is similar to that used in [

7]. We average the total Hamiltonian of the system from expression (Eq.

11) with yet unknown electron wave function from formula (Eq.

15). The Hamiltonian

from (Eq.

12) takes the form after this procedure:

Here

is the energy of an electron on relevant size-quantization level,

*M* is the electron mass inside the quantum wire. The form of phonon Hamiltonian

from Eq.

11 remains unchanged. Averaged Hamiltonian of electron–phonon interaction

can be written as:

Here,

and

are the coefficients

and

*α*_{
m
}(

*k*) from Eq.

5 averaged with the electron wave function from formula (Eq. 15). We obtain average Hamiltonian

It can be brought to the form diagonal in phonon variables by the unitary transformation

where

The unitary transformation application gives the following equation:

From expression (Eq. 21), we can see that, in the adiabatic approximation used here, the bulk phonon spectrum and the interface phonon spectrum remain unchanged. The last summand in expression (Eq. 21) presents the energy of a large radius polaron. In the general case, the energy

involved in (Eq. 21) depends on the dielectric properties of the materials of both the quantum wire and the barriers. In the general case, the polaron binding energy

*ΔE*_{
e
} depends on electron size-quantization level number and on optical-phonon spectrum properties. These phonons are localized in the quantum wire and at the heteroboundary. After the procedure of angle averaging which is expressible in explicit form, we obtain this energy

*ΔE*_{
e
} as:

The energy (Eq. 22) is defined by the electron interaction with phonon modes correspond to *m* = 0 only. This equation (Eq. 22) contains the contribution to polaron energy for all size-quantization levels. This contribution is caused by the interaction of localized electron with confined and interface phonons. It can be used for numerical analysis of electron–phonon interaction characteristic properties. However, the electron energy and wave function can be obtained analytically on condition the unequality (Eq. 14) is satisfied.