Normalize both ψ1 and ψ2 and then find the energy eigenvalues of each

[joshr]

Hi there this is my first post here, I am having some trouble with a homework question in quantum.
I need to normalize both ψ1 and ψ2 and then find the energy eigenvalues of each.
The given Hamiltonian is
H0 = (1 2 )
(2 -1)

And ψ1 = ( -2 ) ψ2 = ( -2 )
(1-√5) (1+√5)

Ive had a go and i think that the normalization constant is
ψ1 : 1/(10-2√5)^(1/2) and ψ2 : 1/(10+2√5)^(1/2)

(the things in brackets are ent to be matrices and vectors)
please help!
Thanks josh

 

[joshr]

hi again just noticed that it shifted everything along as i don't no how to do a matrix so the H0 is a matrix with the next line should be the bottom line of the 2x2 matrix.
also psi1 and psi2 should be column vectors so shift all the next line along abit too lol sorry bout that.

 

[physicsboy]

Mr. Joshr,

You've got the normalization correct. If you find the eigenvalues of the Hamiltonian you'll discover that they are plus/minus root 5. You would expect these to be real since your matrix is Hermitian (which it should be since this is supposed to represent an observable - namely the energy). Use these to solve for the eigenvectors (i.e. the energy eigenstates of the system) and you'll find that root 5 corresponds to your psi1 and minus root 5 corresponds to your psi2. Thus the energies are plus/minus root 5 respectively.