Basic Probability question for a spin 1/2 Particle

[richyw]

Homework Statement

A beam of spin-1/2 particles is prepared in the state

$$\left|\psi\right\rangle=\frac{2}{\sqrt{13}}\left|+\right\rangle_x+i \frac{2}{\sqrt{13}}\left|-\right\rangle_x$$

a) What are the possible results of a measurement of the spin component Sz, and with what probabilities would they occur?

b) What are the possible results of a measurement of the spin component Sx, and with what probabilities would they occur?

Homework Equations

$$P_x=\left|\left\langle x\right| \left. \psi \right\rangle\right|^2$$

The Attempt at a Solution

So I can find out the answer to part b just by plugging it straight into the formula and using that $_x\left\langle +\right| \left. + \right\rangle_x=1$ and $_x\left\langle +\right| \left. - \right\rangle_x=0$

my problem is with the first part of the question. If I use the same method as before I end up with
$$_z\left\langle +\right| \left. + \right\rangle_x\\<br /> _z\left\langle +\right| \left. - \right\rangle_x\\<br /> _z\left\langle -\right| \left. + \right\rangle_x\\<br /> _z\left\langle -\right| \left. - \right\rangle_x$$ in my equations. And I do not understand how to know what these are.

 

[mfb]

You can express $_z\left\langle +\right |$ in terms of $_x\left\langle +\right |$ and $_x\left\langle -\right |$ (or vice versa).

Your wavefunction does not look properly normalized.