Fourier Transform of Photon Emission Hamiltonian

[thatboi]

Hey all,
I just wanted to double check my logic behind getting the Fourier Transform of the following Hamiltonian:
$$H(x) = \frac{ie\hbar}{mc}A(x)\cdot\nabla_{x}$$
where $$A(x) = \sqrt{\frac{2\pi\hbar c^2}{\omega L^3}}\left(a_{p}\epsilon_{p} e^{i(p\cdot x)} + a_{p}^{\dagger}\epsilon_{p} e^{-i(p\cdot x)}\right)$$
and $\epsilon_{p}$ is the polarization vector and $a_{p},a_{p}^{\dagger}$ are the photon creation/annihilation operators for a photon with momentum $p$. Also, we can treat the $p$ and $x$ as 4-vectors. To transform $H(x)$ to the momentum basis, I insert an integral of $d^{4}x$ and multiply by $e^{ik\cdot x}$. Doing this leaves me with $$H(k) \propto \delta(p-k)k$$ since the $\nabla_{x}$ drags down a factor of $k$ and the Fourier Transform of an exponential function is just a Dirac delta. This result seems too simple to me so I was wondering if I made a mistake somewhere.
Thanks.

 

[PeterDonis]

What happens to $A(x)$ when you do the integral? It seems like you're just leaving it out.

 

[thatboi]

What happens to $A(x)$ when you do the integral? It seems like you're just leaving it out.

I thought the $\delta(p-k)$ factor came from the $e^{(ip\cdot x)}$ term in $A(x)$, in which case I also forgot a $\delta(p+k)*k$ factor in my above response.

 

[Haborix]

Let me try to say what @PeterDonis said but slightly more direct. Where are $a_p$ and $\epsilon_p$? Also, the expression you give for $H(k)$ is proportional to $k$, and $k$ is generally a vector. Your sanity check alarms should be going off.

 

[hutchphd]

Perhaps you should fill in the steps (using, say, mathematics

A(x)=2πℏc2ωL3(apϵpei(p⋅x)+ap†ϵpe−i(p⋅x))
and ϵp is the polarization vector and ap,ap† are the photon creation/annihilation operators for a photon with momentum p. Also, we can treat the p and x as 4-vectors. To transform H(x) to the momentum basis, I insert an integral of d4x and multiply by eik⋅x. Doing this leaves me with

Hand waving is not enough. It is good that your radar went off. It is bad that you didn't do the physics.