# Eigenvalues and eigenvectors of a Hamiltonian

## 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?

N/A?

## The Attempt at a Solution

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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

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

Last edited:

haruspex