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## Homework Statement

How to prove that for any representation of the spin, the state [tex] e^{-i{\pi}J_x/\hbar}|j,m\rangle[/tex]

is proportional to [tex]|j,-m\rangle[/tex]

The exponential term is the rotation operator where [itex] J_x [/itex] is the x-component of the total angular momentum operator,

and [itex]|j,m\rangle[/itex] is an eigenket.

## Homework Equations

[itex] J_x=\frac{1}{2}(J_+ + J_-)[/itex] where [itex]J_+ [/itex] and [itex] J_- [/itex] are the ladder operators.

[itex] J_±|j,m\rangle=\sqrt{(j{\mp}m)(j±m+1)}|j,m±1> [/itex]

## The Attempt at a Solution

Taylor series expansion of the exponential term?

[itex]e^{-i{\pi}J_x/\hbar}=1-i\frac{{\pi}J_x}{\hbar} - \frac{1}{2}(\frac{{\pi}J_x}{\hbar})^2 +... [/itex]