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When does $\vec{a} \cdot \vec{\sigma}$ define a measurement in x-y, y-z, and x-z planes?

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    $\begingroup$ Did you mean projective measurement, i.e., $|\vec{a}|=1$? If so I think the answer is trivial. If not, can you give some references or a more detailed description of your problem? $\endgroup$
    – narip
    Jan 15 at 10:32
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$\vec{a}\cdot\vec{\sigma}$ defines an operator $$ a_xX+a_YY+a_ZZ $$ where $X,Y,Z$ are the Pauli matrices. So, for example of $\vec{a}=(0,a_Y,a_Z)$ then it has no component in the X plane, and we say it defines a measurement in the $Y-Z$ plane.

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  • $\begingroup$ Then a measurement in $x-y$ plane would correspond to $\vec{a}=(a_X, a_Y,0) = (\sin\theta \cos\phi, \sin\theta \sin\phi, 0)$. But this depends on both $\theta$ and $\phi$, while as we know the measurements in $x-y$ plane should only depend on $\phi$. $\endgroup$
    – User101
    Jan 14 at 16:32
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    $\begingroup$ But remember that the vector has length 1. So the $\sin\theta$ disappears in normalisation (or, you're say that you set $\cos\theta=0$, so $\sin\theta=\pm 1$). Equally, just because a vector has been parametrised in a certain way doesn't mean that's a good parametrisation for the context. For example, the y-z plane, you're best to just parametrise it as $(0,\cos\gamma,\sin\gamma)$, automatically capturing the length 1 property. $\endgroup$
    – DaftWullie
    Jan 17 at 7:09

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