# Bipolaron Characteristics in a Cylindrical Quantum Dot

We consider the motion of the electron in the z -axis direction to be bounded by an infinite high rectangular potential well and bounded on the oxy-plane by a transversal parabolic potential.

## System Hamiltonian

The Hamiltonian of the system is written in the form:

Here, *V(z)* is the confinement potential (infinite high rectangular potential well) in the direction of the oz-axis. The state of the electron is described by the variational wave function that has a large spread compared to the ground state wave function:

Averaging the Lagrangian 18.534 by the wave function 18.535 we have

ч /

Here,

The model Lagrangian of the transversal motion:

### Model System Action Functional

From the model Lagrangian 18.538 we have the model action functional

Following the procedure of path integration seen earlier then

### Equation of Motion / Normal Modes

The equations of motion for the transversal motion are

For normal modes we substitute the following into the equation of motion and letting, then

Solving for the eigenmodes we have

with

The **bipolaron effective mass **is conveniently obtained from the eigenmode equations:

### Lagrangian Diagonalization

To diagonalize the model Lagrangian, we move to normal coordinates:

Substituting these equations into the equation of motion and also considering the conservation of the kinetic energy in any representation then

and the model diagonalized Lagrangian in normal coordinates:

It shows that in the motion of the bipolaron we have four oscillators indicating four internal motions.

### Bipolaron Partition Function

The bipolaron partition function can be obtained from the equation:

or

The Coulomb interaction can be written in the form:

### Bipolaron Generating Function

We find now (Svia the generating function:

Letting,

then

where

### Bipolaron Energy

We again observe all the formulae in the bipolaron problem have the same dependence on the quantity |r -

where,

and

with

For Л_{0}» 1 then

**18.19 Polaron Characteristics in a Quasi-OD Cylindrical**