[Physics] Green’s Function in the Lippmann Schwinger Equation

greens-functionsquantum mechanicsscatteringscattering-cross-section

When deriving the scattering cross section using the Lippmann-Schwinger equation we need to calculate the Green's function defined by $$G(\mathbf{r},\mathbf{r'},E)=\langle\mathbf{r}|\frac{1}{E-H+i\epsilon}|\mathbf{r'}\rangle$$ where $H=\frac{p^2}{2m}$ represents the free particle Hamiltonian. This can be calculated as $$G(\mathbf{r},\mathbf{r'},E)=-\frac{2m}{\hbar^2}\frac{e^{ik|\mathbf{r}-\mathbf{r'}|}}{4 \pi |\mathbf{r}-\mathbf{r'}|}$$ and by definition should satisfy $$(E-H)G(\mathbf{r},\mathbf{r'},E)=\delta^3(\mathbf{r}-\mathbf{r'})$$ I am trying to verify the latter expression by writing $H=-\frac{\hbar^2}{2m}\nabla^2$ and working through the derivatives using $\mathbf{R}=\mathbf{r}-\mathbf{r'}$ to simplify things a little. My method is to use the identity $$\nabla^2(fg)=f\nabla^2g+2\nabla g.\nabla f+g\nabla^2f$$
to calculate the action of $\nabla^2$ on the Green's function before adding everything together using $E=\frac{\hbar^2k^2}{2m}$ to get $$(E-H)G(\mathbf{r},\mathbf{r'},E)=e^{ik|\mathbf{r}-\mathbf{r'}|}\delta^3(\mathbf{r}-\mathbf{r'})$$ which is slightly wrong. When I think about it I see no way in which the mathematical operations performed by $E-H$ can make this extra unwanted factor disappear from the final result. Is there a flaw in anything I've said or is there an obvious place in which I have picked up this factor? Thanks for any help.

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I haven't work out the math but looks like expression you get is correct, you just need to use the following identity $$ f(x)\delta^n(x)=f(0)\delta^n(x) $$ Therefore $$ (E-H)G=e^{ik|r-r'|}\delta(r-r')=\delta(r-r') $$

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Green’s Function for Wave Equation – How to Derive It

In Mathematica:

Refine[-(c^2/(4 π c r)) Integrate[
  DiracDelta[r - c t] Exp[I ω t], {t, -∞, ∞}], 
    Assumptions -> {r ∈ Reals, c > 0}]

The output is

$$-\frac{e^{\frac{i r \omega }{c}}}{4 \pi r}=-\frac{e^{irk_0}}{4 \pi r}$$ The extra factor of $c$ is eliminated since the $\delta$ function has an argument of $ct$.

[Physics] Conditions to determine the Green’s function for scattering phenomena

"My problem here is the following: to find G we need boundary conditions of the problem. I can't understand, though, what boundary conditions we should impose here.

So to solve scattering problems using Green's functions like that, what are the boundary conditions we need to impose to compute the Green's function?"

Why do you want to set a boundary condition? Clearly here your problem has no boundary since you can detect some particles very far away from the place where the interaction with the potential takes place. However, surely you will need an intitial condition provided by $\psi^0_\textbf{k}(\textbf{r})$.

I feel (tell me if I'm wrong) that what you mean by "boundary condition" refers actually to what kind of solution you are interested in. As you pointed out, the equation $$ (\nabla^2+k^2)G(\textbf{r},\textbf{r}')=-4\pi\delta(\textbf{r}-\textbf{r}') $$ has two solution called retarded $G^+$ and advanced $G^-$ Green functions. This gives rise to two different kind of solutions to your differential equation :

The causal solution : $$ \psi_\textbf{k}(\textbf{r})=\psi^{\text{(in)}}_\textbf{k}(\textbf{r})-\frac{1}{4\pi}\int\mathrm{d}\textbf{r}'\,G^+(\textbf{r}-\textbf{r}')\,U(\textbf{r}')\,\psi_\textbf{k}(\textbf{r}') $$ where $\psi^{\text{(in)}}_\textbf{k}\equiv \psi^{\text{0}}_\textbf{k}$ is interpreted as the incoming field, i.e. the asymptotic field you obtain when taking the limit $t\rightarrow -\infty$ and which evolves freely with time. This is usually the solution people are interested in because it describes the scattering event as a result of the interaction with the potential.
The anti-causal solution : $$ \psi_\textbf{k}(\textbf{r})=\psi^{\text{(out)}}_\textbf{k}(\textbf{r})-\frac{1}{4\pi}\int\mathrm{d}\textbf{r}'\,G^-(\textbf{r}-\textbf{r}')\,U(\textbf{r}')\,\psi_\textbf{k}(\textbf{r}') $$ where $\psi^{\text{(out)}}_\textbf{k}$ is interpreted as the outcoming field, i.e. the asymptotic field you obtain when taking the limit $t\rightarrow +\infty$ and which had evolved freely with time from the past.

In both of these cases, the limits $t\rightarrow\pm\infty$ allow you to get rid of the integral terms and give rise to different interpretations of the solutions.

You may also be only interested in the radiated field which is defined as : $$ \psi^{\text{(rad)}}_\textbf{k}(\textbf{r})=\psi^{\text{(out)}}_\textbf{k}(\textbf{r})-\psi^{\text{(in)}}_\textbf{k}(\textbf{r})=\frac{1}{4\pi}\int\mathrm{d}\textbf{r}'\,\mathcal{G}(\textbf{r}-\textbf{r}')\,U(\textbf{r}')\,\psi_\textbf{k}(\textbf{r}') $$ with $$ \mathcal{G}(\textbf{r})=G^+(\textbf{r})-G^-(\textbf{r}) $$

"I've seem this method in some notes and as things are written it seems the only imposed condition is that $G(\textbf{r},\textbf{r}')=G(\textbf{r}-\textbf{r}')$."

It seems to me that this is not directly related to your initial problem. One can show independently that the property $$ G(\textbf{r},\textbf{r}')=G(\textbf{r}-\textbf{r}') $$ is just a consequence of the fact that the dispersion relation $\langle\textbf{k}| H|\textbf{k}\rangle =E(\textbf{k})$ only depends on the magnitude of the impulsion, i.e. $E(\textbf{k})\equiv E(k)$, which is a consequence of the fact that your system is invariant under translation.

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[Physics] Conditions to determine the Green’s function for scattering phenomena

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