Solve Confusing Matrix Operator Equation in Quantum Mechanics

[Beer-monster]

Homework Statement

I'm new to matrix mechanics in quantum mechanics and I admit that linear algebra is the weakest part of my maths toolkit but I have an operator to convert to matrix form. I'm about 85% sure I've done it right, however I can't solve for the eigenvectors. I'm sure I'm missing something obvious and was wondering if someone could help me find it.

The operator is a Hamiltonian of the form.

$$H=c \left[ \begin{array}{cc} 1 & 1 \\ 1 & -1 \end{array} \right]$$

If I solve for the eigenvalues I get $\lambda = \pm c$

Which seems right. However, if I try to solves for the eigenvectors using $H'=(H-\lambda I)$ and $H'x = 0$ for the eigenvalue +c, I get a set of equations that don't make sense to me.

$$H'=c \left[ \begin{array}{cc} 1-1 & 1 \\ 1 & -1-1 \end{array} \right]= \left[ \begin{array}{cc} 0 & 1 \\ 1 & -2 \end{array} \right]$$

Using x=(x,y), and if I'm reading this right, this would give a set of equations where y=0 but also y=x. This would imply the answer is a null vector which makes no sense.

Can anyone see where I've gone wrong/what I'm missing?

 

[Beer-monster]

Nevermind , apparently I had forgotten how to take a determinant.

Thanks anyway.

 

[Beer-monster]

Stuck again because now I need to solve this equations:

$$(1-\sqrt{2})x+y = 0 ; x+(-1-\sqrt{2})y = 0$$I've been pounding my head against it buit can't seem to find the way, standard Gaussian techniques seem to be leading to a mess.

 

[Phys O]

Just set x or y to one, then you get the value of the other, (1, something) or (something, 1)