About nature of superposition of states

[HighPhy]

I was re-reading the previous posts, but a question arose.
Some posts ago, I said:

When you observe the spin, this probability distribution collapses to a defined state, and then your measurement changes that probability distribution. Depending on the case, it can collapse it to a very simple one - probability 1 for a certain value and 0 for all others.

As answers, I obtained:

Spin is a 3D vector quantity. In QM, only spin about one axis can be defined - spin about the other two axes remains undefined. If we measure about the z-axis, we get either $\pm \frac \hbar 2$ and the state collapses to z-spin-up or z-spin-down. Subsequent measurements of a free particle will always give the same outcome. But, measurements about the x or y-axis will give $\pm \frac \hbar 2$ with equal probability.

Moreover, if the particle is in a magnetic field, then the state will naturally evolve from the initial state or z-spin-up or z-spin-down. Look up Larmor Precession.

For spin measurements, this will always be the case, since spin is a discrete observable.

What is the particular connection that allows these two responses to be viewed as related?
Sorry for not grasping it.

 

[PeterDonis]

What is the particular connection that allows these two responses to be viewed as related?

My statement that you quoted was in response to your statement about collapse. Look for the word "collapse" in what you quoted from @PeroK. What does it say about that?

 

[HighPhy]

My statement that you quoted was in response to your statement about collapse. Look for the word "collapse" in what you quoted from @PeroK. What does it say about that?

It says "if we measure about the z-axis, we get either $\pm \frac{\hbar}{2}$ and the state collapses to z-spin-up or z-spin-down."

I may have understood. I focused my attention on the rest of the response, but perhaps what @PeroK says is the mathematical description of your words in the comment I quoted. Correct?

 

[PeterDonis]

what @PeroK says is the mathematical description of your words in the comment I quoted. Correct?

Yes.

 

[PeroK]

It says "if we measure about the z-axis, we get either $\pm \frac{\hbar}{2}$ and the state collapses to z-spin-up or z-spin-down."

I may have understood. I focused my attention on the rest of the response, but perhaps what @PeroK says is the mathematical description of your words in the comment I quoted. Correct?

The spin state is an abstract vector in a 2D complex vector space; and, the spin measurable is a vector in physical space - although its components cannot all be well-defined.