Time-reversal operator for fermions (Sakurai)

[Sdakouls]

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, [tex]\Theta[/tex], for the spin-[tex]$\frac{1}{2}$[/tex] case as (pg.: 277, eq. (4.4.65)):

[tex]\Theta = \eta e^{\frac{-i \pi S_{y}}{\hbar}}K = -i \eta \left( \frac{2S_{y}}{\hbar} \right) K[/tex]

where [tex]\eta[/tex] is some arbitrary unit magnitude complex number, [tex]S_{y}[/tex] is the y-component of the spin operator and [tex]K[/tex] 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 [tex]\pi[/tex] on the RHS). If anyone could shed any light on this, it'd be most appreciated.

 

[Avodyne]

Work in a basis where [itex]S_y/\hbar[/itex] is diagonal with eigenvalues [itex]\pm1/2[/itex]. You can check that the last equality is valid for each eigenvalue; therefore it is valid as a matrix equation.