# Adiabatic theorem:
If the hamiltonian depends on a parameter that changes very slowly,
the an eigenstate at time t evolves into an eigenstate of the new hamiltonian at time t'.
Caveat: nondegenerate eigenvalues, rate of change slow compared to Bohr frequencies.

Our computation with the exponential turn-on of a perturbation exp(eta t) V
established the adiabatic theorem at first order, by showing that the first order
transition probability from |0> to |n> is equal to the square of the overlap between
the perturbed eigenstate and |n>.

Good reference for a general proof and analysis of the correction terms:
A. Messiah, Quantum Mechanics, vol. 2.
At a more elementary level, see D. Griffiths, Introduction to Quantum Mechanics.

Berry's phase: Under a cyclical adiabatic change of the Hamiltonian, the state (if nondegenerate)
must come back to its starting point...up to a phase. This phase is called Berry's phase,
and is observable if a system is coherently split into parts, one of which undergoes the cyclical
change and the other does not. The book by Griffiths has a nice discussion of this.

# Coupling of charges to radiation using  Coulomb gauge... all this material is in Baym.
The gauge business: can choose gauge so that div A = 0. Then div E = 4 pi rho
becomes del^2 phi = -4 pi rho, which can be solved at each t for phi in terms of the
charge density. So phi is a "slave" field, not an independently propagating one.
Meanwhile, curl B = 1/c d_t E => box A = -1/c d_t grad phi. So a changing charge density
drives A non-locally as a source term for the A wave eqn. But note this is suppressed
by 1/c, so in nonrelativistic settings is not very important.

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