Question about simultaneous measurement

[Mr J]

Greetings - I have a question about observables and Quantum Mechanics.

Let's say I want to measure the spin state of a particle.
If I measure σZ (sigma Z), I will get either +1 or -1. That will then prepare the spin state accordingly.
If I measure σX (sigma X), I will get either +1 or -1. That will then prepare the spin state accordingly.
If I measure σY (sigma Y), I will get either +1 or -1. That will then prepare the spin state accordingly.

What if I measure all three axes simultaneously? Won't that then give me three different resultant spin states, when there can only be one?

 

[Nugatory]

What if I measure all three axes simultaneously?

You can't. Two ways of seeing this:
1) Try to imagine the measuring device that would do this... We measure spin in a given direction by passing the particle through an inhomogeneous magnetic field oriented in that direction. To measure the spin in both the x and y directions we'd need a magnetic field that was oriented in both the x and y directions, but the way magnetic fields add there's no such thing. If we try to combine an x-measuring field and a y-measuring field, we end up with a single field oriented in some third direction and we'll be making a single measurement in that third direction, not a simultaneous x and y measurement.
2) Look at the mathematical formalism. A measurement of spin in the x direction corresponds to projecting the state vector onto the x-axis in the Hilbert space; a measurement of the spin in the y direction corresponds to projecting the state vector onto the y axis, and clearly we can't do both at once. Another way of thinking about it: a simultaneous measurement on the x and y axes would yield a state that is an eigenstate of both the $S_x$ and the $S_y$ operators - and there is no such state.