I'm having some trouble understanding something concerning sequential measurements.
Suppose we have a wave function consisting of a superposition of spin up and spin down in the z direction (spin=1/2). We measure the spin in the z direction, then in x and then in z again and ask what is the probability that the spin will be up.
I know the answer is half, as once I measure in the x direction, there's a 1/2 chance to have spin up x and 1/2 for spin down x, regardless of the original wave function, and the same when going back to measuring spin z again.
However, when I perform $S_zS_xS_z$ on the original wave function, then project it on spin up (z) and calculate the absolute value squared (the probability for spin up), I get the same probability as with the original wave function for the first time I measured spin in the z direction. So suppose I originally had 1/3 probability to have spin up in the first measurement, then I will have the same the second time even after I've measured spin in the x direction inbetween.
Obviously just multiplying the original wave function by the operators $S_zS_xS_z$ doesn't work, but why? Why is this fundamentally wrong?
Thanks in advance
(I'd write the matrices and calculate them here, but my computer is broken and doing so on the phone isn't the easiest)