What is the physical significance of projectors vs. Pauli matrices in spin measurements?

Question

I am confused about the physical significance of projectors, e.g., $|u\rangle\langle u|$ for the "up" state $|u\rangle$ of a spin.

Let $|u\rangle =\begin{pmatrix}1 \\ 0\end{pmatrix}$ and $|d\rangle =\begin{pmatrix}0 \\ 1\end{pmatrix}$ be the up and down states of a spin.
We can distinguish between these orthogonal states by measuring spin along the $z$-axis, which corresponds to the operator $\sigma_z=\begin{pmatrix}1&0\\ 0&-1\end{pmatrix}$ - the first Pauli matrix. The eigenvalues are $1$ and $-1$ : $\sigma_z|u\rangle =|u\rangle$ and $\sigma_z|d\rangle =-|d\rangle$ .

So far, I seem to understand.

Now I know that any state can be written as a projector using the outer product of the state with itself, e.g. $|u\rangle \langle u|=\begin{pmatrix}1&0\\ 0&0\end{pmatrix}$ . I think I know the basic mathematical properties of projectors, but their physical significance as Hermitian operators, which should correspond to measurements, confuses me: It seems that you can use $|u\rangle \langle u|$ to distinguish between $|u\rangle$ and $|d\rangle$ too. The eigenvalues of $|u\rangle\langle u|$ for these states are $1$ and $0$ : $|u\rangle\langle u||u\rangle =|u\rangle$ and $|u\rangle\langle u||d\rangle =0|d\rangle$ .

Is that right? Which measurement corresponds to $|u\rangle \langle u|$ ? I thought I understood that any spin measurement corresponds to a direction in 3d-space $\text{---}$ the 3 Pauli matrices standing for the $x,y,z$ directions, respectively. But what 3d-space direction does $|u\rangle \langle u|$ correspond to? If it does correspond to any direction, it should be the $z$ direction, right? But if both $\sigma_z$ and $|u\rangle\langle u|$ correspond to measurement along the $z$ -axis, what is the significance of the eigenvalue pairs $1, -1$ vs. $1, 0$?

Can anyone help? Thanks!

Answer 1

The projection operator $|u\rangle\langle u|$ corresponds to the experimental question "Does the spin point in the positive $z$-direction - yes or no?" The outcome "yes" corresponds to the eigenvalue $1$ and the outcome "no" to the eigenvalue $0$ of the projection operator.

The operator $S_z=\hbar \sigma_z/2=\frac{\hbar}{2}\left( |u \rangle \langle u|-|d\rangle \langle d |\right)$ corresponds to the experimental question "What is the value of the spin measured in $z$-direction?" with the possible outcomes $+\hbar/2$ and $-\hbar/2$ as eigenvalues of $S_z$.

Both operators correspond to an experimental set-up, where the spin in $z$-direction is measured. As $|u \rangle \langle u|+|d\rangle \langle d|=\mathbf{1}$, the operator $S_z$ can be expressed in terms of $|u \rangle \langle u|$ and the identity operator by $S_z =\frac{\hbar}{2} \left(2 |u \rangle \langle u|-\mathbf{1} \right)$ (and vice versa).