Conceptual difficulties with spin

Question

I thought I had the concept of spin down (literally) but I seem to be a bit rusty. I want clarifications on a few concepts. Let's say I have a particle of spin $\vec{S}$. Consider a direction $\hat{n}$. Let us denote the wavefunction of the particle by $|\psi\rangle$. Here are my questions

1. Can the spin of a quantum particle point completely along one direction? In this case, can the spin $\vec{S}$ be along $\hat{n}$?
2. To add to the above question, does the value of the operator $\vec{S}\cdot\vec{n}|\psi \rangle = \pm|\psi\rangle$ mean that the particle's spin lies entirely parallel/antiparallel to $\hat{n}$?

Thanks.

Answer 1

Spin is a form of angular momentum. When the spin is measured, for a system without any other angular momentum, it is equivalent to asking what the angular momentum is.

Can the spin of a quantum particle point completely along one direction?

This is just like any other quantum measurement: A quantum particle can have a spin that points in one direction. This represents the particle having an angular momentum associated with spinning clockwise/anticlockwise.

As we know, if you make a measurement in one dimension, say the x-axis. This will project the quantum measurement into the x-basis. Doing so means that the spin is now in a superposition in the y and z dimensions. It is in a quantum superposition of spinning clockwise and anticlockwise as seen the "top" of each axis (just like the normal right-hand rule for angular momentum).

2. Yes. It might help to remember that the spin vector just points to the vector used for the right-hand rule. It is spinning either clockwise or anti clockwise wrt this vector, depending on the sign of this value of the spin.