Quantum Operators - Eigenvalues & states

[knowlewj01]

Homework Statement

an operator for a system is given by

$\hat{H}_0 = \frac{\hbar \omega}{2}\left[\left|1\right\rangle\left\langle1\right| - \left|0\right\rangle\left\langle0\right|\right]$

find the eigenvalues and eigenstates

Homework Equations

The Attempt at a Solution

so i formulated this operator as a matrix

$\frac{\hbar \omega}{2} \left( \begin{array}{cc} -1 & 0 \\ 0 & 1 \\ \end{array} \right)$
and found the eigenvalues to be -1 and 1

but when i put them back into the eigenvalue equation Hx = λx [x is a 2 row column vector with elements x1 and x2]
i get nonsense, something like

for λ=1:
x1 = -x1
x2 = x2

for λ=-1:
x1 = x1
x2 = -x2

am i missing something important here?

thanks.

 

[fzero]

but when i put them back into the eigenvalue equation Hx = λx [x is a 2 row column vector with elements x1 and x2]
i get nonsense, something like

for λ=1:
x1 = -x1
x2 = x2

for λ=-1:
x1 = x1
x2 = -x2

am i missing something important here?

thanks.

Those look fine with no nonsense. Try solving each set of equations. One equation will fix one component, while the other doesn't fix anything. This is because, as with any eigenvector equation, a multiple of a solution is still a solution. You should just pick a convenient choice for the unknown variable in order to write down a solution.