Q:

Matrix of any special unitary transformation in two dimensions

I want to show that every special unitary transformation in two dimensions can be written as the matrix
$$ U = \left(\begin{array}{cc} e^{i(\delta + \varphi)}\cos\theta & i~e^{i(\delta - \varphi)} \sin\theta \\ i~e^{-i(\delta - \varphi)}\sin\theta & e^{-i(\delta + \varphi)}\cos\theta\end{array}\right),$$
where $\delta,\varphi,\theta \in\mathbb{R}$. I want to understand neutrino oscillations and my professor just wrote down this matrix without deriving it. Unfortunately, I cannot find the right way to do it myself. Any help is appreciated.

A:

For simplicity, suppose $U$ is special unitary so its determinant is $+1$.  If you want full unitary simply multiply $U$ by an overall phase $e^{i\zeta}$.  
Write
$$
U=\left(\begin{array}{cc}
a&b\\
c&d\end{array}\right)\, ,\qquad 
U^{-1}=\left(
\begin{array}{cc}
 d & -b \\
 -c & a \\
\end{array}
\right)\, ,\qquad U^\dagger = 
\left(\begin{array}{cc}
a^*&c^*\\
b^*&d^*\end{array}\right)
$$
where $\hbox{Det}(U)=1$ has been used.  Since $U^\dagger=U^{-1}$ this immediately shows
that $d=a^*$, $c=-b^*$ so
$$
U=\left(\begin{array}{cc}
a&b\\
-b^*&a^*\end{array}\right)
$$
The condition on the determinant is now $aa^*+bb^*=1$.  In your case you chose
$$
a=e^{i(\delta+\varphi)}\cos\theta\, ,\qquad ie^{i(\delta-\varphi)}\sin\theta.
$$
The factor of $i$ in the off-diagonal terms and the general form of your solution is typical of the parametrization used in optics.
Of course this is not the only solution, indeed this is not the "standard" solution. A special unitary $2\times 2$ matrix is often written in the factorized form 
$$
e^{-i\alpha\sigma_z/2}e^{-i\beta\sigma_y/2}e^{-i\gamma\sigma_z/2}
=\left(
\begin{array}{cc}
 e^{-\frac{1}{2} i (\alpha +\gamma )} \cos \left(\frac{\beta }{2}\right) & -e^{-\frac{1}{2} i (\alpha -\gamma )} \sin \left(\frac{\beta }{2}\right) \\
 e^{\frac{1}{2} i (\alpha -\gamma )} \sin \left(\frac{\beta }{2}\right) & e^{\frac{1}{2} i (\alpha +\gamma )} \cos \left(\frac{\beta }{2}\right) \\
\end{array}
\right)\, .
$$

