Green fuktion on the wave operator in 1 dimension

[nobody2100]

I have a Homework here I'm trying to solve like the last 3-4 hours but somehow I'm stuck so I ask you guys for help:

They gave me the Green Function like this:
http://idefix.physik.uni-freiburg.de/~aufgabe/THEO204/theo9/img50.png
with http://idefix.physik.uni-freiburg.de/~aufgabe/THEO204/theo9/img53.png for y >= 0 and http://idefix.physik.uni-freiburg.de/~aufgabe/THEO204/theo9/img55.png for y<0
Now I'm supposed to solve this:
http://idefix.physik.uni-freiburg.de/~aufgabe/THEO204/theo9/img61.png
with
http://idefix.physik.uni-freiburg.de/~aufgabe/THEO204/theo9/img36.png and http://idefix.physik.uni-freiburg.de/~aufgabe/THEO204/theo5/img39.png and http://idefix.physik.uni-freiburg.de/~aufgabe/THEO204/theo9/img62.png

Would be so nice if someone could help me! thanks in advance!

 

[dextercioby]

So you're basically saying to solve
$$box_{1}[I_{G_{1}}(\phi)]$$

That means $box_{1} \int G_{1}(x,t)\phi (x) dx$
with he "box"/d'alembertian taken wrt to "y" and "t'"("t" prime)
Then u can insert the d'alembertian under the integral and use the differential eq.verified by the propagator.

It think the integration will be immediate...

Daniel.

 

[nobody2100]

thx for your quick reply! But I don't quite following that one, could you explain it to me with more details? and phi is also a function of t ;)
I Expect to get phi(0,0) at the end. I tried some more and got something similar except a minus in front of it but i really don't think that it was mathematically correct what I did there over those 2 pages x_X have to get this one done by tomorrow morning so i would appreciate any kind of help! thx!
What i basically did until now is that I made my integration invervals smaller:
for all t < 0 Teta is 0 anyway so my integral over dt goes from 0 to infinite, now with my t >0 I started to change the interval for dx by only integrating over the support of Teta that means my Interval changed to [-ct , ct]
Now my first problem was that my Integration Interval is a function of t. I did get one differentation to t inside the Integral over dx and got 3 terms and well ... here I'm not sure anymore if that was right x_X
This is the last thing i have where I'm quite sure that it could be right:

$((1/c^2) \partial_{t}^2 - \partial_{x}^2) \int_{0}^{\infty} dt \int_{-ct}^{ct} dx ( (c/2)\phi (x,t) )$

 

[nobody2100]

anyone? still can't solve it x_X

 

[dextercioby]

Okay,first question:I find kind of awkward the shape of the Green's function.
It should always look like this:
$$G_{1}(x,x',t,t')$$ and should verify identically the differential equation
$$\hat{O}_{x,t} G(x,x',t,t')=\delta(x-x')\delta(t-t')$$(1)

U can make the substitution
$$x-x' \rightarrow R$$(2)
$$t-t' \rightarrow \tau$$ (3)

and the new propagator would be
$$G_{1}(R,\tau})$$,which would check the LPDE

$$\hat{O}_{R,\tau} G_{1}(R,\tau)=\delta(R)\delta(\tau)$$(4)

If u say that your propagator is
$$G_{1}(R,\tau)=\theta(R-c\tau)$$(5)
and u want to compute
$$I_{G_{1}}=\int [O_{R,\tau}G_{1}(R,\tau)]\phi(R,\tau)=\int \delta(R)\delta(\tau)\phi(R,\tau) =\phi(0,0)$$

Daniel.

 

[dextercioby]

BTW,your problem is set up wrongly.
If u were to do exactly what was indicated,u'd end up with a dalembertian applied to a number (the value of the integral) which would be identically zero.

Daniel.

 

[nobody2100]

yes, it should look like G(t,t',x,x') but t' and x' are set 0 we kinda did it in the couse that we subsitute it.
$$G_{1}(R,\tau)=\theta(R-c\tau)$$
Is practically the same
and this:
$$I_{G_{1}}=\int [O_{R,\tau}G_{1}(R,\tau)]\phi(R,\tau)=\int \delta(R)\delta(\tau)\phi(R,\tau) =\phi(0,0)$$
is exactely what I'm supposed to show
Yes the notation looks like ****, but the "box" operator is kinda a differentiation of a distribution here:
$$\partial I_{G_{1}}( \phi(t,x)) = I_{\partial G_{1}}( \phi(t,x)) = - I_{G_{1}}( \partial \phi(t,x))$$
could you just help me how you got the last line? or how to solve this one (hope it's right this time):
$\int_{0}^{\infty} dt \int_{-ct}^{ct} dx ( ((1/2c) \partial_{t}^2 - (c/2)\partial_{x}^2)\phi (x,t))$

 

[dextercioby]

Well,i just I've given a proof.Remember that the Green function/propagator is a solution of this eq.
$$\hat{O}G(R,\tau)=\delta(R)\delta\tau$$
which means
$$\hat{O}\theta(R-c\tau)=\delta(R)\delta\tau$$
,which is just i used to get the integral involving the product of delta functionals and the 'phi'.

Daniel.

 

[nobody2100]

I know, that this is the solution but in this homework they say I have to show explicit, that this given G actually is a Green function to that Operator by solving that equation. So I'm not allowed to use that equation above with the delta distribution because that is exactely what i have to proof ^^
I Just cleared up that integral and finally got a $$\phi(0,0)$$ term. still there are some ugly integrals left which i hope will result to 0 somehow ...