# Propagator with Pauli matrices

## Homework Statement

Consider a 1/2-spin particle. Its time evolution is ruled by operator $U(t)=e^{-i\Omega t}$ with $\Omega=A({\vec{\sigma}}\cdot {\vec{L}})^{2}$. A is a constant. If the state at t=0 is described by quantum number of ${\vec{L}}^2$, $L_{z}$ and $S_{z}$, $l=0$, $m=0$ and $s_{z}={1/2}$, determinate the state at a generic time t.

## Homework Equations

$({\vec{\sigma}}\cdot {\vec{L}})^{2}={{\vec{L}}^{2}}-{{\vec{\sigma}}\cdot{\vec{L}}}$
and
$(\vec{\sigma}\cdot\vec{u})^{2n}=1$ with $2n$ even number and $\vec{u}$ a unitary vector

## The Attempt at a Solution

I've used the relations I've written above to write the propagator as $e^{-iA{\vec{L}}^{2}t}$$e^{iA{{{\vec{\sigma}}\cdot{\vec{L}}}}t}$ and I've found out

$e^{-iA{\vec{L}}^{2}t}[{cos(At)+i{({\vec{\sigma}}\cdot\vec{L})}sin(At)]}$. But I don't think it is correct because L is not a versor.

Thanks