Eigenvectors of the hamiltonian

[captainjack2000]

Homework Statement

The Hamiltonian of a system has the matrix representation

H=Vo*(1-e , 0 , 0
0 , 1 , e
0 , e , 2)

Write down the eigenvalues and eigenvectors of the unperturbed Hamiltonian (e=0)

Homework Equations

when unperturbed the Hamiltonian will reduce to Vo* the 3x3 matrix with 1,1,2 along the diagonal. the eigenvalues are therefore Vo,Vo,2Vo (right??)

I am a bit confused about how to calculate the eigenvectors. I have tried looking this up but still get confused. Would they not all be zero since if you sub the eigenvalue Vo back into matrix you would get for the first row

Vo(1-Vo,0,0) * (x,y,z) = (0,0,0) where (x,y,z) is a vertical matrix?

 

[morphemera]

What is the problem then? You should know that the 0 vector is the trivial solution, so it is not count for the eigenvector cos it cannot span any solution space.

 

[captainjack2000]

So how would you calculate the other eigenvectors. Sorry I am still confused.

 

[morphemera]

for e=0,
H=
(Vo , 0 , 0
0 , Vo , 0
0 , 0 , 2Vo)
eigenvalue is Vo,Vo,2Vo as you said. So what is the standard procedure to find the eigenvector? I assume that you should take at least one linear algebra before you take QM.