Hamiltonian of an electron under EM radiation

[blue_leaf77]

I might have learned what I am going to ask during my electrodynamics class long time ago but just that do not remember it now.
I always wonder why does an electron moving in space with EM radiation have Hamiltonian of the form
$H = \left( \mathbf{p}-e\mathbf{A}/c \right)^2/2m +e\phi$ where $\mathbf{A}$ and $\phi$ are vector and scalar potentials, respectively? I want to study it myself and now I'm having the EM book by Griffith, in case you know that such derivation exists in that book I would prefer that you tell me which chapter it is, otherwise I'm fine if you want to explain it here instead.

 

[robphy]

Write down the Lagrangian of the charged particle in an EM field and compute the canonical momentum. It differs from the usual momentum expression.
Then construct the Hamiltonian.

 

[blue_leaf77]

Ok I guess I need to go to classical mechanics.
After some reading, I found that in the case of electron in an EM field it seems it's the canonical momentum $p_i$ that enters in the usual commutation relation with $x_i$, not the kinematic momentum. Why is this so? The commutation relation between p and x originally follows from the definition of momentum as the translation operator. But in the derivation process, at least in the book I read, the author didn't made any reference as to whether canonical or kinematic momentum that defines the translation operator. The fact that in the case of electron's Hamiltonian in an EM field it's the canonical momentum that enters in the commutation relation with x, do we define it simply by an analogy with the classical Poisson bracket? I was just guessing though that the $p_i$ in classical Poisson bracket is the canonical one, I haven't checked myself.

 

[blue_leaf77]

Any idea?

 

[robphy]

That's why it is distinguished and called the "canonical momentum"
http://en.m.wikipedia.org/wiki/Canonical_coordinates
http://en.m.wikipedia.org/wiki/Translation_operator_(quantum_mechanics)#Momentum_as_generator_of_translations

While the free particle might have been motivation for the equations of lagrangian and Hamiltonian mechanics, one eventually learns that canonical momentum is not always m x-dot and the Lagrangian is not always kinetic energy minus potential energy, and so on