I'm given a spin state: $|s\rangle$ = some linear combination of $|\uparrow\rangle + |\downarrow\rangle$ possibly with an imaginary component.

$\hat{\mu}_e = g\mu_B\hat{\sigma}$

$g$ is the gyrmoagnetic factor and is approximately 2.0023.

$\mu_B =\frac{e\hbar}{2m_o}$ is the Bohr magneton.

I'm asked to find the expectation value of the electron spin magnetic moment. Which I believe is $\langle s| \hat{\mu}_e |s\rangle$

Yet the problem states that "Note: the result is a vector"

How and why would an expectation value be a vector?


In that case, $\hat{\sigma}$ here refers to a vector formed by $\hat{\sigma}_x$, $\hat{\sigma}_y$ and $\hat{\sigma}_z$ as its Cartesian components. The individual components of the expectation value of the magnetic moment vector would then be obtained using the corresponding components of the Pauli spin operators.


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