Lagrangian of the Electron in a Quasi-1D Quantum Wire

We write the Lagrangian corresponding to the Hamiltonian in 18.184:

Here the electron coordinate is z, the longitudinal optical phonon generalized coordinates are q_г for a frequency mode со_f .The amplitude of the electron-phonon coupling interaction y* (r) for the ic moving wave:

From the Feynman variational principle, the interaction of the electron with the polarized vibrations of the crystal is modelled by an elastic coupling of the electron and a hypothetical (fictitious) particle that attracts the electron to itself. The complex pair can now move in the direction of the wire axis. So, the effect of the polarized crystal lattice on the electron is approximated to an elastic attraction of the second particle. From these analyses, the model Lagrangian may be selected in the one oscillatory approximation:

Here M and ay are, respectively, the mass of a fictitious particle and the frequency of the elastic coupling serving as variational parameters; Z_fis the coordinate of the fictitious particle and t = - ihr with . _ dz dr

Partition function of the Electron in a Quasi-1D Quantum Wire

In order to find the polaron energy we proceed to find the following quantities:

where

The model Lagrangian may be conveniently represented in quadratic form by doing a change of variables:

and

with

The quantities p_x and p₂ are, respectively, the coordinates of the relative motion and of the center of mass. The parameter p is a reduced mass and v is a scaled frequency.

So,

The coordinate p₂ describes the free motion of the particle with mass M + m, while the coordinate p_v the harmonic oscillator with frequency v. Hence the partition function in 18.192 now becomes

or

Here, l_z is the length of the quantum wire and the thermal de Broglie wavelength:

From the Lagrangian of the full system we have where

is the lattice partition function and the functional of the electron-phonon interaction influence phase

is:

and

We may now write the full action functional of the full system:

We may now write the influence phase of the full system:

Consider

The influence phase of the electron-phonon interaction in 18.206 can then be rewritten

The action functional of the model system is obtained via the partition function of the model system:

Here

and

So,

and