# Mathematical analysis of recombining streams in a Stern-Gerlach experiment

My quantum mechanics textbook skips some steps in its mathematical analysis of a Stern-Gerlach experiment, and I am having trouble filling in the blanks.

The experiment sends a streams of electrons with spin 1/2 which first pass through an apparatus that measures the Z component of the spin. Those with +$$\hbar/2$$ spin are then sent through an apparatus that measures the X component. The two streams from the second apparatus are recombined, at which point they are sent to an apparatus that once again measures the Z component of the spin. We see that the number of electrons that are measured to have +$$\hbar/2$$ spin by this last apparatus are the same as in the first.

When analyzing this experiment, the textbook defines the quantum state vector (in the z basis), describing the spin of the electrons exiting the second apparatus as follows:

$$|\psi_2\rangle = \frac{(P_{+x} + P_{-x})|+\rangle}{\sqrt{\langle+|(P_{+x} + P_{-x})|+\rangle}}.$$

The denominator simplifies to one, since $$P_{+x} + P_{-x}$$ is the identity operator. The textbook then goes on to expand the numerator, making it

\begin{align} |\psi_2\rangle &= (|+\rangle_{x x}\langle+| + |-\rangle_{x x}\langle-|)|+\rangle\\ &= |+\rangle_{x x}\langle+|+\rangle + |-\rangle_{x x}\langle-|+\rangle \end{align}

Finally, it uses this representation of the beam exiting the second apparatus to calculate the probability of measuring $$-\hbar/2$$ at the final appartus:

\begin{align} P_- &= |\langle-|\psi_{2}\rangle|^2 \\ &= |\langle-|+\rangle_{x x}\langle+|+\rangle + \langle-|-\rangle_{x x}\langle-|+\rangle|^2 \end{align}

Now, this is where I get confused. The textbook claims without proof that the expansion of this is

\begin{align} P_- &= |\langle-|+\rangle_{x x}\langle+|+\rangle|^2 + |\langle-|-\rangle_{x x}\langle+|+\rangle|^2\\ &+ \langle-|+\rangle^*_{x x}\langle+|+\rangle^*\langle-|-\rangle_{x x}\langle-|+\rangle\\ &+ \langle-|+\rangle_{x x}\langle+|+\rangle\langle-|-\rangle^*_{x x}\langle-|+\rangle^* \end{align}

Presumably, it is somehow using the fact that $$(a+b)^2 = a^2 + b^2 + 2ab$$, and claiming that \begin{align} &+ \langle-|+\rangle^*_{x x}\langle+|+\rangle^*\langle-|-\rangle_{x x}\langle-|+\rangle\\ &+ \langle-|+\rangle_{x x}\langle+|+\rangle\langle-|-\rangle^*_{x x}\langle-|+\rangle^* \end{align}

is somehow equivalent to $$2ab$$ in this case. Why is this true?

It also goes on to say that these two terms are referred to collectively as the "interference terms", and claims that they depend on the relative phase between the different states of a coherent superposition. I also am struggling to see how these two terms depend on the relative phase between the different states of the $$|\psi_2\rangle$$ vector. Any explanation of this would be appreciated.