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.

 

[denian]

I can provide some clarification on this topic. In the context of scientific experiments and measurements, operators refer to mathematical tools used to describe the physical properties of a system. In quantum mechanics, operators are used to describe the observable quantities of a particle, such as its position, momentum, and spin.

In the case of a Stern-Gerlach experiment, the operator would be used to describe the spin of the particle being measured. The operator can indeed describe both "up" and "down" deflection simultaneously, as these are two possible outcomes of the measurement. However, the measurement itself can only yield one of these outcomes at a single time.

In other words, the operator can describe multiple measurements at once, but the actual measurement can only give one result at a time. This is a fundamental principle of quantum mechanics known as superposition, where a particle can exist in multiple states simultaneously until it is observed or measured.

I hope this helps to clarify your confusion. It is important to note that operators are just mathematical tools used to describe the physical properties of a system, and the actual measurement is what determines the specific outcome.