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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!

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closed as too broad by DanielSank, Kyle Kanos, Qmechanic Aug 12 '15 at 9:21

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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.

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  • $\begingroup$ which is the algebric meaning of doing bra and ket on the middle term? what is the middle term (the term between | and |)? a matrix? what do I get in the end? the value of energy due to the magnetic effect? $\endgroup$ – Surfer on the fall Oct 8 '14 at 18:32
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    $\begingroup$ Braket notation is just the physics notation for an inner product on a Hilbert space. Are you familiar with the mathematical formalism of Hilbert spaces? In that context, just interpret $\langle \psi |\hat{O}|\phi \rangle$ as $(\psi, O\phi)$, where $(\cdot,\cdot)$ denotes the inner product on the Hilbert space of choice. $\endgroup$ – Danu Oct 8 '14 at 18:41
  • $\begingroup$ @Danu the only thing that I know about Hilbert space is that it is a generalization of the euclidean space.. Doing bra and ket on O is equal to doing a matrix product? Thanks for your help! $\endgroup$ – Surfer on the fall Oct 8 '14 at 18:48
  • $\begingroup$ @Surferonthefall - No, it's not that simple. When physicists use the term "Hilbert space" they are often talking about an infinite dimensional space. The dot product becomes an integral, and the space is the set of all square integrable functions. $\endgroup$ – David Hammen Oct 8 '14 at 21:52
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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

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