Physics problems related to green function ?

[lofaif]

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 !

 

[SteamKing]

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]

Try simulating a classical damped, driven harmomic oscillator. I'm assuming you're working with Fourier transforms, yes?

 

[lofaif]

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 ?

 

[NegativeDept]

Just about any problem governed by Laplace's equation, Del^2 Phi = 0. This equation is used in 2-D elasticity and fluid flow.

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 ?

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.