representation of spinors

I am trying to get from the abstract representation of Spinors, as wave functions $|\Psi \rangle$ in the base of tensors products $| S_z \rangle \otimes | x \rangle$ of eigenvectors of the spin operator $\hat S_z$ and position operator $\hat x$ to the representation in wave function form: $\begin{pmatrix}{\Psi_+(x) \\ \Psi_-(x)} \end{pmatrix}$.

I seem to be missing something key. I would really apprechiate some help with this. Here is where i go wrong:

$$\int dx \sum_{s_z} (| x \rangle \otimes | S_z\rangle) \; (\langle x| \otimes \langle S_z|) |\Psi \rangle=\int dx \sum_{s_z} (| x \rangle \otimes | S_z\rangle) \; (\langle x|\Psi \rangle \otimes \langle S_z|\Psi \rangle)$$ which i would normally read as $$= \int dx \sum_{s_z} (| x \rangle \otimes | S_z\rangle) \; ( \Psi(x) \; \langle S_z|\Psi \rangle) \\ =\int dx (| x \rangle \otimes | +\rangle) \; ( \Psi(x) \; \begin{pmatrix}{1 \\ 0} \end{pmatrix})+(| x \rangle \otimes | -\rangle) \; ( \Psi(x) \; \begin{pmatrix}{0 \\ 1} \end{pmatrix})$$

which then if i leave the summation over the base vectors out gives $\begin{pmatrix}{\Psi(x) \\ \Psi(x)} \end{pmatrix}$ which is exactly not what i want. Since it is the same wave function in both components, what am i doing wrong here?

How do i get different wave functions for the different components using the dirace formalism? I would really be glad about some Tipps.

• The components of the wave function are defined by $(\langle x| \otimes \langle s|)|\Psi\rangle= \Psi_s (x)$ where $s$ labels the spin state – Holographer Feb 7 '15 at 10:16
• Soory pressed enter by mistake. So what i still dont understand is this: $\hat S_z (\Psi_+(x) + \Psi_-(x)) = {\hbar \over 2 }((\Psi_+(x) - \Psi_-(x)) )$ how do we get to the vector then? – pindakaas Feb 7 '15 at 10:21
• Oh well i guess $\begin{pmatrix}{1 & 0 \\ 0 & -1 } \end{pmatrix} \begin{pmatrix}{\Psi_+(x) \\ \Psi_-(x)} \end{pmatrix}$ would be a valid representation of that right? (I have no idea why the latex is wrong) – pindakaas Feb 7 '15 at 10:28
• maybe someone knows what is wrong? ^^ sorry i can not for the life of me find it – pindakaas Feb 7 '15 at 10:39