I'm given a spin state: |s⟩ = some linear combination of |↑⟩+|↓⟩ possibly with an imaginary component.
ˆμe=gμBˆσ
g is the gyrmoagnetic factor and is approximately 2.0023.
μB=eℏ2mo is the Bohr magneton.
I'm asked to find the expectation value of the electron spin magnetic moment. Which I believe is ⟨s|ˆμe|s⟩
Yet the problem states that "Note: the result is a vector"
How and why would an expectation value be a vector?
Answer
In that case, ˆσ here refers to a vector formed by ˆσx, ˆσy and ˆσz 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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