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I am trying to understand the fundamentals of Quantum Electrodynamics through the simple example of refraction. Let's start with a plane wave coherent state (or a number state) impinging on a boundary along positive $x$ axis with momentum $\hbar k_0$. The initial state is then given by $|\psi\rangle=|\alpha\rangle_{k_x=k_0,k_y=0,k_z=0}|0\rangle|0\rangle|0\rangle\dots$

For the free field case, we have the Hamiltonian given simply by $\hat{H}=\hbar \omega (\hat{a}^{\dagger}\hat{a}+1/2)$ so in the Schrodinger picture the initial state simply gains becomes $|\psi(t)\rangle=|\alpha e^{-i\omega t}\rangle$ due to time evolution under this Hamiltonian.

Now, using interaction picture for convenience, what would the Hamiltonian look like for the case when we have another medium with refractive index, say $n$, which is at an angle to the $x$ axis, as shown below? Using this Hamiltonian how can we see that the light is bent/scattered into a different $\vec{k}$ mode?enter image description here

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