Physics problems related to green function ?

In summary, the conversation discusses examples of problems that involve green function and solving differential equations using programmed numerical solutions. Suggestions are given, such as problems governed by Laplace's equation and simulating a classical damped, driven harmonic oscillator. The connection with quantum mechanics is also mentioned, with the Schrödinger equation being a useful linear PDE to know Green's functions for. Examples of problems in quantum mechanics, such as a 1-dimensional "particle in a box" and quantum harmonic oscillator, are suggested for further exploration.
  • #1
lofaif
2
0
hello all !

my teacher told me to do a research on examples of problems that has connection with green function on solving differential equations (with programmed numerical solutions) in my final year project , can you give me such problems to work on as an undergraduate ? , thank you !
 
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  • #2
Just about any problem governed by Laplace's equation, Del^2 Phi = 0. This equation is used in 2-D elasticity and fluid flow.
 
  • #3
Try simulating a classical damped, driven harmomic oscillator. I'm assuming you're working with Fourier transforms, yes?
 
  • #4
sudu.ghonge said:
Try simulating a classical damped, driven harmomic oscillator. I'm assuming you're working with Fourier transforms, yes?

yes , how about its connection with quantum mechanics ? , can i do that or is it hard for non physics student ?
 
  • #5
SteamKing said:
Just about any problem governed by Laplace's equation, Del^2 Phi = 0. This equation is used in 2-D elasticity and fluid flow.
sudu.ghonge said:
Try simulating a classical damped, driven harmomic oscillator. I'm assuming you're working with Fourier transforms, yes?
lofaif said:
yes , how about its connection with quantum mechanics ? , can i do that or is it hard for non physics student ?

The Schrödinger equation* is a linear PDE with a ##\nabla^2## in it:

##
\imath \hbar \partial_t \Psi(\mathbf{r},t) = \frac{-\hbar^2}{2m}\nabla^2 \Psi(\mathbf{r},t) + V(\mathbf{r},t) \Psi(\mathbf{r},t)
##

So it can be useful in QM to know Green's functions for the Helmholtz equation, which is closely related to what SteamKing mentioned. Probably the simplest example would be to look up a 1-dimensional "particle in a box" problem. Pick some initial wavefunction ##\Psi(x,0)## and use Green's functions and convolution to find future wavefunctions ##\Psi(x,t)##.

If you want to really show off, combine this with sudu.ghonge's idea and do the same for a 1-dimensional quantum harmonic oscillator instead of a particle-in-a-box. But it might be a good idea to practice on something other than quantum mechanics because QM is often counterintuitive and hard to visualize.

* in position representation, if anyone asks.
 

1. What is a green function in physics?

A green function, also known as a propagator, is a mathematical tool used in quantum mechanics and other areas of theoretical physics. It describes the propagation of a particle or field from one point to another in space and time.

2. How are green functions used in physics problems?

Green functions are used to solve differential equations and other mathematical problems in physics. They allow for the calculation of physical quantities such as energy levels, scattering amplitudes, and correlation functions.

3. What is the significance of the term "green" in green functions?

The term "green" comes from the name of the mathematician George Green, who first introduced the concept of a green function in the 19th century. It has no physical meaning and is simply a historical reference.

4. Can green functions be used in classical physics?

Yes, green functions can also be applied in classical physics, particularly in the study of wave propagation and boundary value problems. However, they are most commonly used in quantum mechanics.

5. Are there any limitations or drawbacks to using green functions in physics?

One limitation of green functions is that they can be difficult to calculate for complex systems, leading to approximations and simplifications. They also do not provide a complete physical picture and must be combined with other mathematical techniques to fully understand a system.

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