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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?

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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).

Any choice is equally valid, and physical results do not depend on the choice you make. Might as well take the one that makes computations easiest.

Hope this helps.

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