Hamiltonian diagonalization

1. Nov 26, 2013

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1. The problem statement, all variables and given/known data
The exercise: https://www.physicsforums.com/attachment.php?attachmentid=64229&d=1385257430

2. Relevant equations
Are in my attempt at a solution. I am sure it would be easier to use the transformation equation for the operator and plug it into the diagonalized Hamiltonian at the bottom. The problem is I don't really understand this procedure so I have started from the basic equation for diagonalizing a matrix. The problem is now that I will need the matrix elements of H in the old basis and I know really see what basis the Hamiltonian is even written in. Which is that?

3. The attempt at a solution
How I tried to solve it: https://www.physicsforums.com/attachment.php?attachmentid=64283&d=1385405657