The time evolution operator (QM) Algebraic properties

[knowlewj01]

Homework Statement

The hamiltonian for a given interaction is

$H=-\frac{\hbar \omega}{2} \hat{\sigma_y}$

where

$\sigma_y = \left( \begin{array}{cc} 0 & i \\ -i & 0 \end{array} \right)$

the pauli Y matrix

Homework Equations

The Attempt at a Solution

So from the time dependent Schrödinger equation we, can take the time dependence and put it into the time evolution operator U(t)

$HU(t)\left|\Psi(r,0)\right>=i\hbar \frac{d}{dt}U(t)\left|\Psi(r,0)\right>$

becomes

$i\hbar\frac{d}{dt}U(t) = HU(t)$

so for a non time dependent Hamiltonian H, this means:

$U(t) = e^{-\frac{i}{\hbar}H t}$

so we have then:

$U(t) = e^{\frac{i\omega t}{2}\hat{\sigma_y}}$

How do you treat this? Is there any particular identity that allows you to move the operator out of the exponent?

 

[knowlewj01]

edit: changed the matrix to the correct form

 

[dextercioby]

Do you know how the exponential of a finite matrix is defined? If so, use the definition.