Energy Eigenvalue for a Two State System

[jameson2]

Homework Statement

The Hamiltonian for a two state system is given by H=a(|1><1|-|2><2|+|1><2|+|2><1|) where a is a real number. Find the energy eigenvalue and the corresponding energy eigenstate.

Homework Equations

The Attempt at a Solution

I don't know how to start, I'm looking for a hint rather than the answer.

 

[fzero]

It's probably easier to translate to an explicit matrix form by thinking of the states as vectors

$$|1\rangle = \begin{pmatrix} 1 \\ 0 \end{pmatrix} , ~ |2\rangle = \begin{pmatrix} 0 \\ 1 \end{pmatrix}.$$

Once you've solved the problem using that familiar notation, you can go back and figure out how you could have done it using the bra-ket notation. Also note that there are 2 energy eigenvalues and 2 corresponding eigenstates.

 

[vela]

If you know the states $|1\rangle$ and $|2\rangle$ are orthonormal, calculate the matrix elements $\langle i|H|j \rangle$ and express H as a matrix.

 

[jameson2]

Thanks guys.