What Does <S> Mean in Spin Measurement?

[lonewolf219]

I am reading about spin, and I think <S$_{x}$> represents the measured value of spin along the x component. The same would follow for <S$_{y}$> and <S$_{z}$>. I also think I read that we can't measure all three values during an experiment. Once we measure one component, we can't measure the other two. So then I don't really understand what <S> means because it cannot be the total measured spin, since we can only know one component.

Does <S> mean the expectation value of each component?

 

[ShayanJ]

$<S>$ is the expectation value for the total spin.The point that it can't be known exactly doesn't mean it is not defined at all.

 

[Nugatory]

I
Does <S> mean the expectation value of each component?

No, it's the expectation value of the magnitude of the total spin vector, that is, the expectation value of $\sqrt{S_x^2+S_y^2+S_z^2}$. It is somewhat counterintuitive, but we can know the magnitude of the vector as well as the value of anyone of its three components.

 

[lonewolf219]

I'm glad I asked... thanks Shyan and Nugatory!

 

[jtbell]

So then I don't really understand what <S> means because it cannot be the total measured spin, since we can only know one component.

As noted above, it's the expectation value of the magnitude of the spin (intrinsic angular momentum) vector. For a fundamental particle this is in fact fixed for each particle: $\langle S \rangle = S = \sqrt{s(s+1)} \hbar$ where $s$ is the spin quantum number for that particle.

For an electron, $s = 1/2$ so $\langle S \rangle = S = (\sqrt{3}/2) \hbar$.

 

[lonewolf219]

jtbell, thanks for posting... that is exactly what I have just been putting together, and your post confirms it. I appreciate it!