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Sdakouls
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In Modern Quantum Mechanics (2nd ed.) by J.J. Sakurai, in section 4.4 on 'The Time-Reversal Discrete Symmetry' he derives the time-reversal operator, \Theta, for the spin-$\frac{1}{2}$ case as (pg.: 277, eq. (4.4.65)):
\Theta = \eta e^{\frac{-i \pi S_{y}}{\hbar}}K = -i \eta \left( \frac{2S_{y}}{\hbar} \right) K
where \eta is some arbitrary unit magnitude complex number, S_{y} is the y-component of the spin operator and K is the complex conjugation operator.
Now, I can follow everything he does, except this last equality. I don't know how/why he is able to write down this last equality (I know it's not some kind of Taylor expansion because of the absence of \pi on the RHS). If anyone could shed any light on this, it'd be most appreciated.
\Theta = \eta e^{\frac{-i \pi S_{y}}{\hbar}}K = -i \eta \left( \frac{2S_{y}}{\hbar} \right) K
where \eta is some arbitrary unit magnitude complex number, S_{y} is the y-component of the spin operator and K is the complex conjugation operator.
Now, I can follow everything he does, except this last equality. I don't know how/why he is able to write down this last equality (I know it's not some kind of Taylor expansion because of the absence of \pi on the RHS). If anyone could shed any light on this, it'd be most appreciated.
