- #1
Rovello
- 3
- 0
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]