Can operators describe a single or many measurements simultaneously?

[cscott]

Can operators describe a single or many measurements at a single time?

i.e. a stern-gerlach atom deflection experiment where we can measure "up" deflection and "down" deflection or just "up" deflection.

I think I'm pretty confused..

 

[Galileo]

In this case you are not making many measurements at a single time. You only make a single measurement. It's just that the outcome of that measurement can take different values, either "up" or "down" here.

Let's say you where measuring where the z-component of the spin is "up" or "down", but you also want to know whether the x-component is "up" or "down" (in the x-direction).
Now that would require two measurements, and in this case it is impossible to know them both at the same time. As soon as you measure whether S_z is up or down you change the state of the particle.

In general, you can perform two measurements at the same time (or rather, know the values of two physical properties simultaneously) if their operators commute. Otherwise, no go.