# Time-reversal operator with spin-1/2 system

In Sakurai's Modern Quantum Mechanics section 4.4, while deriving the explicit expression of the time-reversal operator for spin-1/2 systems, it uses (eq. 4.4.62 and 4.4.64 in 2nd ed.): $$\left| \hat{n}, + \right> = e^{-\frac{i S_z \alpha}{\hbar}}e^{-\frac{i S_y \beta}{\hbar}} \left|+\right>$$ and $$\left|-\right> = e^{-\frac{i S_y \pi}{\hbar}} \left|+\right>$$ However, we can get the same state with a different set of rotation: $$\left|\hat{n}, +\right> = e^{-\frac{i S_z \alpha'}{\hbar}}e^{-\frac{i S_x \beta'}{\hbar}} \left|+\right>$$ and $$\left|-\right> = e^{-\frac{i S_x \pi}{\hbar}} \left|+\right>$$ By using this we will get a different time-reversal operator. Does that mean we can have a different time-reversal operator even under the same representation, or did I make a mistake?

Actually, what only matters is how the operators relate to one another. In this case, any anti-unitary operator $$T$$ is valid as long as it satisfies: $$TS_k=-S_kT\\ T^2=-1$$ (the anti-unitary condition can be lifted since it can be deduced from these conditions)
This means you could have many possible $$T$$ operators. Actually, in this case, all the possible $$T$$ differ only by a phase factor. They are therefore similar via conjugation by a square root of the differing phase factor (which preserves the linear spin operators).