Eigenvectors of dressed states Given the following hamiltonian: (basis $|1 \rangle ; |2 \rangle) $
$$H = \frac{\hbar}{2}\begin{bmatrix}
    \Delta      & - \Omega \\
    - \Omega     &  \Delta\\
\end{bmatrix}$$
I found the eigenvalues: $\lambda_{\pm} = \frac{\hbar}{2}\left(\Delta \pm \Omega\right)$. Now I'm asked to find the eigenvectors corresponding to the dressed states and rewrite them as:
$|+ \rangle = \begin{bmatrix}
    \cos(\theta) \\
    - \sin(\theta)
\end{bmatrix}$ and $|- \rangle = \begin{bmatrix}
    \sin(\theta) \\
    \cos(\theta)
\end{bmatrix}$. The eigenvalues that've found are:
$|u_{1} \rangle = \frac{1}{\sqrt{2}}\left[\frac{\Delta + \Omega}{\Omega}\right]^{1/2} \left(|1 \rangle + |2 \rangle \right)$
$|u_{2} \rangle = \frac{1}{\sqrt{2}}\left[\frac{\Delta - \Omega}{\Omega}\right]^{1/2} \left(|1 \rangle - |2 \rangle \right)$
Thus, I cannot make any identification between my expressions and exercise. What am I missing?
P.S.
According to the book, we should have:
$\sin(\theta) = \left[\frac{\Delta - \Omega}{\Omega}\right]^{1/2}$ and
$\cos(\theta) = \left[\frac{\Delta + \Omega}{\Omega}\right]^{1/2}$
 A: 
$$H = \frac{\hbar}{2}\begin{bmatrix}
    \Delta      & - \Omega \\
    - \Omega     &  \Delta\\
\end{bmatrix}$$


I found the eigenvalues: $\lambda_{\pm} = \Delta \pm \Omega$.

The eigenvalues of the Hamiltonian are actually half of the ones listed above (times hbar):
$$
\lambda_{\pm} = \frac{\hbar}{2}\left(\Delta \pm \Omega\right)
$$

$|u_{1} \rangle = \frac{1}{\sqrt{2}}\left[\frac{\Delta + \Omega}{\Omega}\right]^{1/2} \left(|1 \rangle + |2 \rangle \right)$


$|u_{2} \rangle = \frac{1}{\sqrt{2}}\left[\frac{\Delta - \Omega}{\Omega}\right]^{1/2} \left(|1 \rangle + |2 \rangle \right)$

The above eigenvectors you found are not both correct. The above eigenvectors effectively just differ in normalization factor (that is to say, they are effectively the same).
One of the eigenvectors should have a minus sign between the basis vectors (and the normalization on one is wrong).

If the Hamiltonian actually looks like this:
$$H = \frac{\hbar}{2}\begin{bmatrix}
    \Delta      & - \Omega \\
    - \Omega     &  -\Delta\\
\end{bmatrix}$$
Then the eigenvalues are $\pm \frac{1}{2}\sqrt{(\Delta^2 + \Omega^2)}$, and you should be able to solve for the eigenvalues and it looks like you will arrive at something close to what you expect.
