(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Calculate the eigenvectors and eigenvalues of the two-dimensional

matrix representation of the Hermitean operator [itex]\hat{O}[/itex]

given by

|v_k'>[itex]\left(O[/itex]|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)

2. Relevant equations

None

3. 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?

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# Eigenvectors of a 2D hermitian operator (general form)

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