Eigenvalues and eigenvectors of observables

[Fixxxer125]

Homework Statement

Calculate the Eigenvalues and eigenvectors of
H= 1/2 h Ω ( ]0><1[ + ]1><0[ )

Homework Equations

I know H]λ> = λ]λ>

The Attempt at a Solution

I don't know if I am meant to concert my bra's and ket's into matrices, and if so how to represent these as matrices?

 

[vela]

Let me make your post look a bit prettier:

Homework Statement

Calculate the Eigenvalues and eigenvectors of
$$\hat{H} = \frac{1}{2}\hbar\Omega (|0\rangle\langle 1| + |1\rangle\langle 0| )$$

Homework Equations

I know $\hat{H}|λ\rangle = λ|λ\rangle$.

The Attempt at a Solution

I don't know if I am meant to concert my bra's and ket's into matrices, and if so how to represent these as matrices?

Are the states $| 0 \rangle$ and $| 1 \rangle$ orthonormal? If so, just calculate the matrix elements $\langle i |\hat{H}|j \rangle$.

 

[Fixxxer125]

Thanks! In the solution given the Eigenstates are given in terms of the |0⟩ and |1⟩ states in the Hamiltonian. How do I know what these states are in terms of matrices so I can write the eigenstates in terms of these? Cheers

 

[vela]

Those two states are the basis you're using, so...