Phy5645/harmonicoscinmagneticfield/problem2

Problem 2:

Find the equations of (three dimensional) motion in the Heisenberg picture for the position and momentum operators for a system with Hamiltonian

$H={\frac {1}{2m}}\left(\mathbf {p} -{\frac {e}{c}}{\mathbf {A}}({\mathbf {r}},t)\right)^{2}+e\phi ({\mathbf {r}},t)$

Solution:

The Heisenberg equations of motion are:

${\begin{aligned}{\frac {d}{dt}}A_{H}={\frac {1}{i\hbar }}\left(\left[A,H\right]\right)_{H}={\frac {1}{i\hbar }}U^{\dagger }\left[A,H\right]U\end{aligned}}$

So, we substitut $r_{i}$ and $p_{i}$ into the equation above instead of A.

Finally,we could get:

$({\frac {d}{dt}}r_{i})_{H}=({\frac {p_{i}-eA_{i}}{m}})_{H}$

$({\frac {d}{dt}}p_{i})_{H}=-e({\frac {\partial \phi }{\partial x_{i}}})_{H}+{\frac {e}{2m}}(p_{j}-eA_{j})_{H}({\frac {\partial A_{j}}{\partial x_{i}}})_{H}+{\frac {e}{2m}}({\frac {\partial A_{j}}{\partial x_{i}}})_{H}(p_{j}-eA_{j})_{H}$