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Is a particle where $\vec{p} = (p^1,0,0)$ an eigenstate of the helicity operator?

First, can I determine this without doing the math? Second, I also wanna prove it mathematically but doing the math I could not prove this.

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Any plane wave may be an eigenstate of helicity; it does not depend on what direction the momentum points. For the $x$-momentum eigenstate described in the question, the helicity operator is $h=\vec{\sigma}\cdot\hat{p}=\sigma_{x}$. So if the spin part of the wave function is in an eigenstate of $\sigma_{x}$, then the whole state is an eigenstate of helicity. However, it is also perfectly possible to have a spin state that is a linear combination of the up and down $x$-spin states; in that case, the state is not an eigenstate of helicity. If the momentum pointed in another direction, the analysis would be essentially the same, except the spin operator would be a different linear combination of Pauli matrices.

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