Orbital Spin and External Magnetic Fields

[ourio]

Homework Statement

A Hydrogen atom is subjected to a magnetic field strong enough to overwhelm the spin-orbit coupling. Into how many levels would the 2p level split? What would the spacing be in the terms of B_ext, e, m_e, and \hbar

Homework Equations

I know that I have to use the g_Lande equation:
U=g_Lande\frac{e}{2(m_e)}m_j\hbarB_ext

The Attempt at a Solution

I have a feeling that the 2p level will split into 3 levels, but I honestly don't know how to show it.

Any help will be greatly appreciated!

 

[ourio]

OK... I think I may have got some of it...
The 2p state of hydrogen has \ell=1 with m_\ell=-1,0,1
So the 2p state splits into three levels by the external B.

Calculating the g_Lande factor for the 2p state as \frac{4}{3} and the 1s state as 2, I find that the splitting of the 2p state is only \frac{2}{3} that of the 1s state


SO... the spacing between the levels is:
\frac{e\hbar}{2m}B_ext\frac{2}{3}

Does that look right to anybody out there? Help! :-(