Path integral for a particle coupled to a magnetic field

[IRobot]

Hi all,

I am currently having trouble with an exercise: writing the propagator of a particle coupled to a magnetic field.

So the lagrangian is $L_A (\vec{x},\dot{\vec{x}}^2) = \frac{m}{2}\dot{\vec{x}} + e\vec{A}.\dot{\vec{x}}$
And it says that I should solve it in two different ways:
-by writing $e\vec{A}.\dot{\vec{x}}$ has a derivative relative to t
-by completing the square in the Lagrangian, ie something like $L_A (\vec{x},\dot{\vec{x}}) = \frac{m}{2}(\dot{\vec{x}}+\frac{e}{m}\vec{A})^2 - \frac{e^2}{2m}\vec{A}^2$ I guess.

But still, I develop a few lines more of calculus but I am stuck. I would be really happy if someone could explain how to proceed.

 

[jfy4]

I think I'm going to try and solve this with you :)

I can't do it tonight cause I have a test tomorrow that I have to study for, but this looks cool and full of learning.

let me get this straight we have
$$L=\frac{m}{2}\dot{x}^{2}_{i} +eA_j \dot{x}_j$$
and we should put this in two different forms:
$$L=\frac{1}{2m}(m\dot{x}_j+e A_i )^2-e^2 A_{i}^2$$
and
$$L=\frac{m}{2}\dot{x}_{i}^2-e\dot{A}_j x_j +e\frac{\partial}{\partial t}(A_i x_i)$$
Do those look ok to you?

 

[IRobot]

Yes, it does look good. But in the second lagrangian, because we don't want to have an electrical field, we have $\dot{\vec{A}}=0$ so the second terms vanishes.