Exploring Quantum Physics - Final Exam Part 2 Question 1

Question:

Given:

$ s_3 | \uparrow \rangle = \frac{\hbar}{2} | \uparrow \rangle $ and $ s_3 | \downarrow \rangle = -\frac{\hbar}{2} | \downarrow \rangle $

Find

$ s_3 \frac{1}{\sqrt{2}}(| \uparrow \downarrow \rangle - | \downarrow \uparrow \rangle) $

Solution:

Disclaimer: My answer is wrong.

It seems to be a really simple question, just apply the definitions.

$ \begin{eqnarray*} & & s_3 \frac{1}{\sqrt{2}}(| \uparrow \downarrow \rangle - | \downarrow \uparrow \rangle) \\ &=& \frac{1}{\sqrt{2}} s_3 [| \uparrow \rangle - | \downarrow \rangle , | \downarrow \rangle - | \uparrow \rangle] \\ &=& \frac{1}{\sqrt{2}} [| \frac{\hbar}{2}\uparrow \rangle + \frac{\hbar}{2}| \downarrow \rangle , -\frac{\hbar}{2}| \downarrow \rangle - \frac{\hbar}{2}| \uparrow \rangle] \\ &=& \frac{\hbar}{2}\frac{1}{\sqrt{2}} [| \uparrow \rangle + | \downarrow \rangle , -| \downarrow \rangle - | \uparrow \rangle] \\ \end{eqnarray*} $

Now I am stuck, it appears the vector I get is not in the choices. For the exam, I guessed an answer, and got the wrong answer.