- #1
thatboi
- 121
- 18
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.
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.