I am given the Hamiltonia operator of a system in two-dimensional Hilbert space:
H = i[tex]\Delta[/tex](|w1><w2| + |w2><w1|) and am asked to find the corresponding eigenstates.
I wrote this operator as a matrix, where H11 = 0, H22 = 0, and H12= i[tex]\Delta[/tex] and H21= -i[tex]\Delta[/tex]
The eigenvalues are then, I think, the solutions to
(0-E)2 - [tex]\Delta[/tex]2= 0
so E = [tex]\Delta[/tex], -[tex]\Delta[/tex]
But when I construct eigenstates from these E values I am getting
(i , 1) and ( 1 , -i) (except with normalization factors) which seem to be orthornormal and satisfy the two eigenvalues respectively but aren't linearly independent.
Why is this? Is the matrix I'm using for the Hamiltonian correct? And if not why? I thought I can find the matrix values by <wm|H|wn>, where <wm|wn> = [tex]\delta[/tex]m,n
Edit: In addition-
What I'm ultimately trying to do is write the eigenfunctions |w1> and |w2> in terms of the eigenstates |E1> and |E2>. I have missed some class and I'm actually really lost
The homework problem states that |w1> and |w2> are eigenstates of the observable operator[tex]\Omega[/tex]. Is [tex]\Omega[/tex] just any general observable operator, so then what is the significance of its eigenstates? What exactly are |w1> and |w2> as opposed to |E1> and |E2>? Why is the Hamiltonian, an observable, written in terms of the eigenstates of [tex]\Omega[/tex]? Right now I'm trying to do this homework before the deadline purely through a shoddy understanding of linear algebra, but it would really help if I understood what is going on...