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Bipolaron Characteristics in a Cylindrical Quantum DotTable of Contents:
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 oxyplane by a transversal parabolic potential. System HamiltonianThe 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 ozaxis. 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 FunctionalFrom the model Lagrangian 18.538 we have the model action functional Following the procedure of path integration seen earlier then Equation of Motion / Normal ModesThe 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 DiagonalizationTo 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 FunctionThe bipolaron partition function can be obtained from the equation: or
The Coulomb interaction can be written in the form: Bipolaron Generating FunctionWe find now (Svia the generating function:
Letting,
then
where Bipolaron EnergyWe 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 QuasiOD Cylindrical 
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