Bipolaron Equation of Motion
The model Lagrangian 18.289 of the system considers the variables of the electronic as well as those of the fictitious particle system that are coupled to each other. Path integration will be done with ease if we decouple these variables by diagonalizing the Lagrangian. To do this we find the eigenmodes of the electronic and fictitious particle system via the Lagrange equation of motion
Considering 18.289, we have
Suppose
where s is the square of the frequency. We introduce new coordinates:
the center of mass of two electrons:
the relative distance between the electrons:
the centre of mass of two fictitious particle and
the relative distance between the fictitious particles.
This change of variables transforms the system of equations in 18.294 into two pairs of equations:
and
Solving the system of equations in 18.303 yields
This permits us to obtain the equations for the normal frequencies
The first equation yields the solution
while the second equation of 18.305 yields the solution:
where
Summing the two equations in 18.302 we have
This confirms one free motion with
The normal coordinate
is the center of mass of all four particles moving freely. This gives the reason why the frequency of this motion is zero. The square of the second mode of the oscillator, s2, does not have the Coulomb variable К . This is because it describes only the relative oscillations of the electronic and fictitious particle subsystems. The Coulomb variable К relates only to the internal and inter-electronic interaction as described by the square of the modes s314. To conclude, we have one free motion with s, = 0 and three internal oscillations with square of frequencies, s2 and s3 4.
Transformation into Normal Coordinates
With the help of the frequency eigenmodes in 18.306 and 18.307 we can now diagonalize the Lagrangian in 18.289 via a unified technique that will be used throughout this book by a continuous coordinate transformation that involves spatial translations as well as rotations. In this technique, even if one selects a direction for the transformation, there is always the continuous coordinate and, in addition, this technique conserves all quantities. It is instructive to note that such a continuous coordinate transformation has no effect on the particle state since the particle state is an abstract object and makes no reference to a given coordinate system.
Diagonalization of the Lagrangian
One now applies the continuous coordinate transformation transforming the non-harmonic variables and £2/ via two harmonic coordinates dependent on the following frequency eigenmodes:
and
This will help to diagonalize the model Lagrangian in 18.289. From this transformation, equation 18.304 can then be rewritten:
We consider the continuous coordinate transformation that involves spatial translations as well as rotations for £, and £2/>n 18.318. For brevity, the transformation via normal variables x, can be represented as follows:
Hie explicit expressions of the coefficients C, and of the functions F,(a»3) and F2{co4) can be obtained from the equation of motion 18.304 by substituting first f2(&>3) and £2/а>3) into the second equation of 18.304:
and, similarly, £,,(«,) and into ^le first equation of 18.304:
Considering the conservation of the kinetic energy in any of the representations then
and
The model Lagrangian now has the new form
where,
It is obvious from the Lagrangian 18.321 that we have one free motion and three oscillatory motions. The form of the Lagrangian can be a proof of an extraordinary transformation transforming real to normal coordinates. This permits an elegant diagonalization of the Lagrangian and renders it perfectly quadratic. So, the Feynman path integral will be a Gaussian one and computable exactly. This transformation of real to normal coordinates applied to the polaron problem will be observed to be an elegant and effective way of computing problems of particles interacting with the environment.
