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.

 

[paranormal]

One approach to finding the energy eigenvalue and eigenstate for a two state system is to use the eigenvalue equation H|ψ>=E|ψ>, where H is the Hamiltonian operator, |ψ> is the energy eigenstate, and E is the energy eigenvalue. In this case, H is given by the equation provided and |ψ> can be represented as a linear combination of the two basis states |1> and |2>. From there, you can solve for the energy eigenvalue and the coefficients of the linear combination, which will give you the corresponding energy eigenstate. Another approach is to diagonalize the Hamiltonian matrix and find the eigenvalues and eigenvectors from there.