Green Function SHO - Reading Materials & Math Background

[clumps tim]

hi, I need some reading materials on green function for SHO. my instructor provided a GF frequency and wanted us to find the deformation of poles , boundary conditions for the function. I need to know which mathematical background should I have to solve this. any useful material suggestion will be most welcome.
regards

 

[Greg Bernhardt]

Thanks for the post! Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?

 

[vanhees71]

I don't exactly know, which approach you are looking for. I know at least three. I've written out one in my QFT lecture notes using the method of path integrals. It's one of the few examples where one can calculate the lattice-version of the path integral exactly and then take the continuum limit. It's not the most convenient approach, but it helps to understand path integrals. The first chapter of my QFT lecture notes is about non-relativistic quantum theory to introduce path integrals. Perhaps this helps you:

http://fias.uni-frankfurt.de/~hees/publ/lect.pdf

The other two approaches are:

(a) use the Heisenberg picture of the time evolution, solve for the operator-equations of motion and then evaluate the propagator as
U(t,x;t_0,x_0)=\langle t,x|t_0,x_0 \rangle,
where |t,x \rangle are the time-dependent position eigenvectors in the Heisenberg picture.

(b) use the energy eigen functions and resum the corresponding series for the propagator. This is a quite tricky business using the integral representation of the Hermite polynomials.

You find these two approaches in my German QM manuscript:

http://theory.gsi.de/~vanhees/faq/quant/node18.html
http://theory.gsi.de/~vanhees/faq/quant/node49.html

Perhaps you can follow the calculations even without understanding the German text. There are quite a lot of steps written out in formulas :-).