PHY 742 -- Assignment #12

Consider a bound electron having eigenstates with energies En (n=0,1,2,3...). Suppose a plane polarized classical electromagnetic field with electric field with amplitude E₀ and frequency w in induces transitions from the ground state (n=0) to excited states.

1. Write two expressions for the interaction Hamiltonian -- one involving a scalar potential to represent E₀, and the other involving a vector potential representation of E₀.
2. Write the Fermi Golden Rule for the two representations.
3. Show that if the matrix elements are taken between exact eigenstates of the unperturbed Hamiltonian, the two results are identical.
4. Discuss what differences might be introduced between the two representations if approximate eigenstates are used.