Understanding Quantum Spin as a Vector And Spin Operators

Question

I just started learning quantum physics and there is a particular notion confusing me.
While reading McIntyre book, he suggests I find the matrix representation of the $S_n$ operator, which is the operator for the spin component allong a general direction $mathbf{hat{n}}=mathbf{hat{i}}sinthetacosphi+mathbf{hat{j}}sinthetasinphi+mathbf{hat{k}}costheta$, given that we know the matrix representations for $S_x, S_y, S_z$.
Apparently it suffices to write $S_n=vec{S}cdotmathbf{hat{n}}=S_xsinthetacosphi +S_ysinthetasinphi+ S_zcostheta$
What I don't get is: We are expressing $S_x, S_y, S_z$ as the components of the Spin vector, but those are matrices (operators). How is this right? I thought components of vectors could only be scalars.

JoshuaTS · Accepted Answer

$S_x$, $S_y$, and $S_z$ are components of a vector operator $mathbf{S}$. It's refered to as a vector operator because, when you do a rotation, the operator components of $mathbf{S}$ transform just like the components of a normal vector.

Correct answer by JoshuaTS on October 20, 2020

Superfast Jellyfish · Answer

One way to resolve the confusion is to keep in mind that the thing we ultimately want to describe is spin which is a vector in Hilbert space. It is measured by the hermitian operator $S_i$ that measures spin along the $i$th direction.
However, the measuring apparatus that we have is a vector in regular 3D space. So this measuring apparatus can measure the spin along any direction in the 3D space. And we rotate it to point along different directions in space.
So the measuring device points along the unit vector $hat{textbf{n}}$ and the corresponding operator that measures spin in this direction is given by $n_xS_x+n_yS_y+n_zS_z$
As you can see, the linear combination of the $S_i$ operators behaves exactly as a regular vector in 3D would. And that’s why its a vector.

Understanding Quantum Spin as a Vector And Spin Operators

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