Understanding basic quantum mechanics notation I was talking with a guy about energy levels of an atom in a magnetic field. 
He said that energy levels are shifted and that, if you want know how much, you have to analyze this: 
for 1s state:
$$\left<n=1; l=0; m_l=0, m_s', m_i'\ \big|\ a |I \cdot S| + w_0 (L_z+2S_z)\ \big|\  n=1; l=0; m_l=0, m_s, m_i\right>$$
I got curious about the notation, but had to go and I haven't understood very much.. I have knowledge in Analytical Mechanics, but not in Quantum Mechanics. Could you explain me something about the notation employing plain words? Thank you!
 A: It's unclear precisely which notation you're asking about, but I'm going to guess it's about the bra-ket notation. The things next to the bra (which is $\langle \text{this}|\ $) and the ket (which is this $|\text{this}\rangle\ $) are typically either complex numbers or quantum mechanical operators. The bras and kets themselves represent quantum mechanical states.
Anything beyond this you'll probably have to ask as a separate question.
A: If you are interested in this beyond the specific question you asked and would like to see this developed more thoroughly as a combination of math (notation) and the relevant physics, you might take a look at the first few of these great (free) video lectures by James Binney (Oxford).
What makes them particularly appealing to me is most texts treat the math as a separate section from the physics - which I found to be a bit dry. Binney integrates them providing motivation and an intuitive understanding.
http://www.youtube.com/playlist?list=PLE73AA240E8655D16
