# 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.

• One thing never mentioned is each measuring device also changes the spin or polarization of the electron as it goes through. Each time it goes through additional measuring devices it’s like starting over again. Commented Sep 20, 2020 at 17:47
• @Bill Aslept A Stern Gerlach Apparatus (SGA) does NOT change the spin of the particle passing through. It splits the position of the particle into two possible paths, each associated with one particular spin. The SGA does not produce information that could be used to determine which path the particle took, especially for cold magnets, so there is NOT a measurement. The Stern-Gerlach Interferometer (SGI) recombines the two beams. Quantum information science and engineering (QISE) is a rapidly emerging discipline. science.org/doi/10.1126/sciadv.abg2879 Commented Sep 20, 2022 at 4:08

I think it is straightforward if you consider the fact that any complex number verifies $$|z|^2=z^*\cdot z$$ together with $$\langle u,v \rangle ^*=\langle v,u \rangle$$