#### AwesomeTrains

Gold Member

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**1. The problem statement, all variables and given/known data**

Determine the differential equation of [itex] G(\vec{r},\vec{r}',\omega) [/itex]

**2. Relevant equations**

I've been given the Fourier transform for the case where the Hamiltonian is time independent:

[itex] G(\vec{r},\vec{r}',t-t')=\int \frac{d\omega}{2\pi} G(\vec{r},\vec{r}',\omega)e^{-i\omega(t-t')} [/itex]

and the DE for [itex] G(\vec{r},\vec{r}',t,t')[/itex]:

[itex] (i\hbar\partial_t -\hat{H})G(\vec{r},\vec{r}',t,t')=\delta(\vec{r}-\vec{r}')\delta(t-t')[/itex]

**3. The attempt at a solution**

I thought I would just plug the Fourier Transform into the given DE and get:

[itex] (i\hbar\partial_t -\hat{H})\int \frac{d\omega}{2\pi} G(\vec{r},\vec{r}',\omega)e^{-i\omega(t-t')}=\delta(\vec{r}-\vec{r}')\delta(t-t')[/itex]

I couldn't come up with anything else to do.

Any hints are very appreciated :)