Eigenvectors of a 2D hermitian operator (general form)

[gboff21]

Homework Statement

Calculate the eigenvectors and eigenvalues of the two-dimensional
matrix representation of the Hermitean operator $\hat{O}$
given by

|v_k'>$\left(O$|v_k>= {{O_11,O_12},{O_21,O_22}}


where all Oij are real. What does Hermiticity imply for the o-
diagonal elements O12 and O21? For the eigenvectors you may
assume that the two base vectors read

v_1= (1) v_2= (0)
...(0)...(1)

Homework Equations

None

The Attempt at a Solution

So, I know how to work out eigenvectors/values:

|A-λI|=0

so, I end up with (O11-λ)(O22-λ)-O12O21=0 as the characteristic equation

with solution λ= 1/2(O22+O11)±1/2√((O22+O11)^2-4(O11O22-O21O12))


the matrix O, being hermitian, makes O12=O21*

but how do I progress on to an actual answer?

 

[vela]

Since the matrix is real, you know that O₂₁^* = O₂₁. You should be able to show that
$$\lambda = \frac{(O_{11}+O_{22}) \pm \sqrt{(O_{11}-O_{22})^2 + 4O_{12}^2}}{2}$$ That's about as simple as you can get it. Now you want to solve for the eigenvectors. You just have to grind through the algebra.

 

[gboff21]

But it just ends up as a horrible mess which cannot be the answer!

 

[vela]

It's not that bad. You need to adjust your threshold for pain.