What's Wrong with My Eigenvector Calculation?

[athrun200]

Homework Statement

Homework Equations

The Attempt at a Solution

I don't know what's wrong with my work. I can't obtain the eigenvector provided in the model answer.

My work

Model Answer

 

[vela]

Your work is fine. Remember that an eigenvector is only unique up to a multiplicative constant. The eigenvector you found can be written
$$\left\lvert \frac{\hbar}{\sqrt{2}} \right\rangle = \begin{bmatrix} \frac{1-i}{2} \\ \frac{1}{\sqrt{2}} \end{bmatrix} = \frac{1}{\sqrt{2}}\begin{bmatrix}\frac{1-i}{\sqrt{2}} \\ 1 \end{bmatrix} = \frac{1}{\sqrt{2}} \begin{bmatrix}e^{-i\pi/4} \\ 1 \end{bmatrix}.$$ If you multiply that by $e^{i\pi/8}$ and ignore the normalization factor of $1/\sqrt{2}$, you'll get the answer in the solution.

 

[athrun200]

Oh! Thank you very much.
But why do we bother to have such a complicated eigenvector?
The one with (1-i)/2 and 1/(sqrt2) is much easier to find. Why do we need to change it to the form of exp?

 

[vela]

You don't need to, but you have to admit there's certain symmetry there. And physicists like symmetry.