Green Function SHO - Reading Materials & Math Background

In summary, the first approach is using the Heisenberg picture of the time evolution, and the second is using the energy eigen functions.
  • #1
clumps tim
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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
 
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  • #2
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?
 
  • #3
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
[itex]U(t,x;t_0,x_0)=\langle t,x|t_0,x_0 \rangle,[/itex]
where [itex]|t,x \rangle[/itex] 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 :-).
 

1. What is a Green Function in the context of a Simple Harmonic Oscillator?

A Green Function is a mathematical tool used to solve differential equations in physics. In the context of a Simple Harmonic Oscillator, the Green Function is a function that describes the response of the oscillator to an external force. It can be thought of as a "transfer function" that connects the input (external force) to the output (motion of the oscillator).

2. How is the Green Function for a Simple Harmonic Oscillator derived?

The Green Function for a Simple Harmonic Oscillator can be derived using the method of variation of parameters. This involves solving the homogeneous equation of the oscillator and then finding a particular solution by varying the parameters in the solution. The Green Function is then obtained by taking the derivative of the particular solution with respect to the initial conditions of the oscillator.

3. What is the significance of the Green Function in solving problems involving Simple Harmonic Oscillators?

The Green Function allows us to solve for the motion of a Simple Harmonic Oscillator in response to any arbitrary external force. This is useful in many physical systems, such as in mechanical engineering, where we may want to calculate the displacement or velocity of a mass-spring system subject to different forces.

4. What is the relationship between the Green Function and the Transfer Function in a Simple Harmonic Oscillator?

The Green Function and the Transfer Function are closely related in a Simple Harmonic Oscillator. The Transfer Function is the Fourier transform of the Green Function, which means they are two different representations of the same function. This allows us to switch between the time and frequency domains when solving problems involving Simple Harmonic Oscillators.

5. What mathematical background is needed to understand the concept of Green Function in the context of Simple Harmonic Oscillators?

To understand the concept of Green Function in the context of Simple Harmonic Oscillators, one needs to have a good understanding of differential equations, specifically second-order homogeneous and non-homogeneous equations. Some knowledge of linear algebra and Fourier transforms is also helpful in understanding the derivation and applications of the Green Function.

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