B Confusion over spin measurements

[etotheipi]

TL;DR Summary

Sorry if this question has popped up before.

I will refer to the spin example outlined in the opening chapters of the Theoretical minimum.

Suppose we prepare a spin with a z component of +1. If we rotate the apparatus about 180 degrees, the ‘classical component’ of the prepared spin vector along the new axis of the detector is -1, so every time we make this new measurement the apparatus will show -1.

However, does this still mean that the z component of spin is +1, since a negative spin relative to a detector oriented in the negative z direction of our arbitrarily defined coordinate system seems to imply that the spin vector is pointing in the positive z direction?

I ask since he later goes on to say that the up and down states are orthogonal since if we prepare a spin with say a +1 (up) z component, there is a 0% probability of measuring a -1 (down) z component.

This only makes sense to me if we say that, assuming a +1 z spin has been prepared, however we measure the z component immediately after (regardless of whether the apparatus points in the positive or negative z directions) we will get the original spin with 100% probability, so if we flip the apparatus over 180 degrees we will always measure -1, which consequently corresponds to +1 in the positive z direction?

Sorry if I’m messing anything up here!

 

[PeroK]

Summary: Sorry if this question has popped up before.

I will refer to the spin example outlined in the opening chapters of the Theoretical minimum.

Suppose we prepare a spin with a z component of +1. If we rotate the apparatus about 180 degrees, the ‘classical component’ of the prepared spin vector along the new axis of the detector is -1, so every time we make this new measurement the apparatus will show -1.

However, does this still mean that the z component of spin is +1, since a negative spin relative to a detector oriented in the negative z direction of our arbitrarily defined coordinate system seems to imply that the spin vector is pointing in the positive z direction?

I ask since he later goes on to say that the up and down states are orthogonal since if we prepare a spin with say a +1 (up) z component, there is a 0% probability of measuring a -1 (down) z component.

This only makes sense to me if we say that, assuming a +1 z spin has been prepared, however we measure the z component immediately after (regardless of whether the apparatus points in the positive or negative z directions) we will get the original spin with 100% probability, so if we flip the apparatus over 180 degrees we will always measure -1, which consequently corresponds to +1 in the positive z direction?

Sorry if I’m messing anything up here!

I think the answer is yes!

Theoretically, when you turn your apparatus upside-down you are still measuring the same thing. The only thing you are doing is reversing the directions you call $\pm$.

This actually relates to the fundamental idea of symmetry of the universe. There is no intrinsic orientation of a z-axis. Up and down are the same physical axis, with only a different sign convention.

A rough analogy is if you measure (classically) the z-coordinate of position. It's the same measurement, whether your z-axis is up or down. But, it is different from measuring the coordinate along any other axis.

 

[etotheipi]

I think the answer is yes!

Theoretically, when you turn your apparatus upside-down you are still measuring the same thing. The only thing you are doing is reversing the directions you call $\pm$.

This actually relates to the fundamental idea of symmetry of the universe. There is no intrinsic orientation of a z-axis. Up and down are the same physical axis, with only a different sign convention.

A rough analogy is if you measure (classically) the z-coordinate of position. It's the same measurement, whether your z-axis is up or down. But, it is different from measuring the coordinate along any other axis.

Thank you, that’s a really nice way of putting it!