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I am having issues in giving correct physical meaning to the associated sigma matrices: \begin{equation} \sigma_+ = \begin{pmatrix} 0&1\\ 0&0\end{pmatrix} \text{ and } \sigma_-=\begin{pmatrix} 0&0\\ 1&0\end{pmatrix}\end{equation} Is the $\sigma_+$ related to emission and $\sigma_-$ to absorption or is it the opposite? In Dirac notation I have always written $\sigma_+=|1\rangle\langle 0|$ thinking intuitively about absorption, but given the matrix form above (which I am pretty sure it's right) and defining $|0\rangle = \begin{pmatrix} 1\\ 0\end{pmatrix} $ and $|1\rangle = \begin{pmatrix} 0\\ 1\end{pmatrix} $ I don't get $\sigma_+ |0\rangle = |1\rangle$.

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Your vector definitions of $|0\rangle$ and $|1\rangle$ should be switched based on the form you gave for the $\sigma_+$ and $\sigma_-$ matrices. $\sigma_+$ acts on the spin-down state $|0\rangle$ to turn it into the spin-up state $|1\rangle$. $\sigma_-$ does the opposite: it takes spin up to spin down. These operators do not represent observable quantities because they are not Hermitian (they're not even symmetric). There is also no clear association with a physical process. Emission and adsorption require some thing (like electromagnetic radiation) to carry the extra energy away, so you would need to add extra things into your theory before you could even talk about these processes.

If you remember the creation and annihilation operators for the quantum harmonic oscillator, or the raising and lowering operators for angular momentum, the $\sigma_\pm$ matrices are just the spin-1/2 versions of these operators.

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  • $\begingroup$ Thank you, so is it common use switching the definitions for the vectors I wrote? Or is it arbitrary? Of course I know you can make up your own conventions as long as they are coherent in your work, but I was wondering which one is the most common $\endgroup$
    – Hub One
    Commented Sep 26, 2020 at 8:35
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    $\begingroup$ It is conventional to use $(1, 0)$ for spin up and $(0,1)$ for spin down. This is arbitrary, but there needs to be consistency between how you define spin up/down and how you define the sigma matrices. $\endgroup$ Commented Sep 26, 2020 at 15:55

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