Photoemission from a non-interacting electron gas

[SQUB]

Homework Statement

Calculate the intensity of photoelectrons escaping a semi-infinite metallic crystal as a function of energy and momentum in the dipole approximation.
Follow these steps:
. Write down the corresponding Hamiltonian in the operator notation and in the second-quantization notation.
. Discuss the possible ways of treating the surface of the crystal.
. Derive the expression for the photocurrent in the three-step model.
. What approximations were necessary to arrive at this expression, and when might they break down?

Relevant Equations

H=?

I had another exercise of the long list of the same topic (solid state physic) where I need a bit of help. All other exercise where about interband transition, dispersion relation, refracting and absorption coefficient, x-rays and so on, and I managed to solve them or I think I will be able to do so, for this one I have a lot of questions:

-The Hamiltonian on the operator notation should be just the hamiltonian of a dipole?
-The Hamiltonian with the second quantization notation should have an anhilation operator for the photon and the electron or should have also a creation operator for the ejected electron?
-I have no idea on how to treat the surface of the crystal

I'm still blocked at the first three points, but if you have suggestion about the other two (or how to search for explanation about them) every help it's welcome

 

[member 553083]

The Hamiltonian on the operator notation should be just the hamiltonian of a dipole. This would include the potential energy of the dipole and any external forces acting on it. The Hamiltonian with the second quantization notation should include an annihilation operator for the photon and the electron, as well as a creation operator for the ejected electron.To treat the surface of the crystal, you need to consider the boundary conditions at the surface. This includes the electrostatic and electromagnetic properties of the surface, such as the surface charges and currents, and how they interact with the incident wave. You also need to consider the refractive index of the material, which determines how much of the wave is reflected and transmitted. Finally, you need to consider the surface roughness, which affects the scattering of the wave.