Recently

with the rapid development of epitaxial techniques such as molecular beam epitaxy MBE and metal-organic chemical vapor deposition MOCVD

more and more quasi-one-dimensional electronic systems have been fabricated. In such quantum well wires

the electron motion along the length of the wire is free

but it is quantized in the two dimensions perpendicular to the wire. These quantum structures present some special physical properties and have been applied to many semiconductor devices

such as high-electron-mobility transistors. In recent year

a great deal of experimental and theoretical interest has been concentrated on the study of the electronic properties of these quasi-one-dimensional semiconductor systems. With the use of variational solutions to the effective mass equations

Dagani and Hipolito have calculated the energy and effective mass in a rectangular quantum wires; Yeung have studied the cyclotron resonance of a magetopo-(laron) in parabolic quantum wire by using the variational-perturbation expansion method. Guo et al. discussed the binding energy of polaron in quantum wires by means of density-matrix treatment. Taking into account the interaction of the electron with optical phonon modes

Chuu et al. have studied the energies of the ground state and the excited state in cylindrical quantum wires using Pekar alternative approach and perturbation-variational method. Zhou and Gu have presented the properties of polaron and magetopolaron in cylindrical quantum wires by using variational solutions and linear combination operators methods. The properties of polaron in rectangular quantum wires and in cylindrical quantum wires have been studied by the present authors and co-workers.Considering the interaction of the electron with optical phonon modes

we investigated effective mass of strong-coupling polaronand the mean number of optical phonon in a parabolic quantum wire by using the Tokuda’s improved linear combination operators

the Lagrange multiplier and the variational method.The numerical results showed that

with the increasing of confinement strength ω

0

of quantum wire and the Lagrange multiplier u

effective mass m of polaron and the mean number of optical phonon and vibration frequency λ of polaron will increase.