# Hamiltonian for Electron in Magnetic Field with Symmetric Gauge in Polar Coordinates

I am new on the board and have a question about how to write the Hamiltonian for an electron in a magnetic field rotating at a fixed radius. I would like to write the hamiltonian using the symmetric gauge transformation, but in cylindrical polar coordinates.

The standard for of the hamiltonian is $H=\frac{1}{2m}|\mathbf{p}−e\mathbf{A}|^2$. The symmetric gauge is $\mathbf{A}=\frac{1}{2}(−By,Bx,0)$.

Written in polar coordinates, the gauge to the form for $(r,\theta,z)$ of $\mathbf{A}=\frac{1}{2}(0,Br,0)$.

Does this allow me to write my hamiltonian as $H=\frac{1}{2m}\left[p_r^2+(p_\theta-\frac{e}{2}Br)^2\right]$ or does it need to be $H=\frac{1}{2m}\left[(p_\theta-\frac{e}{2}Br)^2\right]$?

I'm confused mainly because I'm not sure if P is 0 when the radius is constant. I'm not even sure if that is relevant to how the hamiltonian is written. I assumed it would be, since there shouldn't be a P term in it, which should be 0.

Thank you for your help.

• Hello, and welcome to the community. Please be sure to make your notation clear so we can understand correctly your question. Use LaTeX to write the formulas, so it's easier for us to read. What is capital $P$? – dpravos Oct 12 '14 at 9:48
• Hello, thank you for your help with my first post. I cleaned it up and, hopefully, now it is clearer. Please let me know if I should edit it further. – PhysicsStudent Oct 12 '14 at 17:12