Eigenvalues and eigenvectors of a Hamiltonian

[astrocytosis]

Homework Statement

The Hamiltonian of a certain two-level system is:

$$\hat H = \epsilon (|1 \rangle \langle 1 | - |2 \rangle \langle 2 | + |1 \rangle \langle 2 | + |2 \rangle \langle 1 |)$$

Where $|1 \rangle, |2 \rangle$ is an orthonormal basis and $\epsilon$ is a number with units of energy. Find its eigenvalues and eigenvectors (as linear combinations of $|1 \rangle, |2 \rangle$). What is the matrix H representing $\hat H$ with respect to this basis?

Homework Equations

N/A?

The Attempt at a Solution

[/B]
This problem seems like it should be simple but I think I'm having trouble internalizing the notation. I know each bracket pair represents the 11, 12, 21, 22 components of the matrix, so I thought H should be\begin{bmatrix}
\epsilon & \epsilon \\
\epsilon & -\epsilon
\end{bmatrix}

then I tried to find the eigenvalues the usual way by subtracting $\lambda I$ and taking the determinant of\begin{bmatrix}
\epsilon - \lambda & \epsilon \\
\epsilon & -\epsilon- \lambda
\end{bmatrix}
but I ended up with an expression that implies $\lambda = 0$

$$-\epsilon^2 + \lambda^2 + \epsilon^2 = 0$$

so I must be misunderstanding something.

EDIT: I miscalculated the determinant, so my answer is actually plausible, but would still appreciate confirmation that this is correct

$$\lambda = \pm \frac{1}{\sqrt{2}} \epsilon$$

 

[haruspex]

Are you sure you wrote that last equation as you intended?

 

[astrocytosis]

I lost a negative sign in the determinant. That will teach me not to skip steps :P Also, it's just $\sqrt{2} \epsilon$