Pauli matrices as measurement operators versus spin probability

Question

Pauli matrices tell us what the spin of a particle is along a certain axis. Let's say I want to measure the spin along the z-axis then the pauli operator

$$\sigma_z = \begin{bmatrix}1&&0\\0&&-1\end{bmatrix}$$ will give me the value of the spin along the z-axis. But how do I actually measure the probability of spin being along one of its axis? For example, a spin 1/2 particle (electron) that is in a static magnetic field which can be in the up $|0\rangle$ or down $|1\rangle$ state with a probability $|\alpha|^2$ and $|\beta|^2$.

So the total state of the spin is given by: $$|\psi\rangle=\alpha|0\rangle+\beta|1\rangle$$ and $|\alpha|^2+|\beta|^2=1$. This spin can be represented by the bloch sphere.
How do I actually measure the probabilities $|\alpha|^2$ and $|\beta|^2$? How do I find the probability of the spin being along one of the axis. And how is measuring the probability related to measuring the spin?

Answer 1

Pauli matrices tell us what the spin of a particle is along a certain axis

This is not true. Your question can be understood more generally as "how do I use the operators to give me information about measuring the observable corresponding to that operator?".

In QM, the values we can measure for an observable with corresponding operator $A$ are the eigenvalues $a$ of $A$. In other words, we are looking at relations like $$A|a\rangle=a|a\rangle$$ where $|a\rangle$ is an eigenvector of $A$ with eigenvalue $a$

Now, let's say our state can be described by the abstract state vector $|\psi\rangle$. What can we say about measurement of $A$? Well, we just express $|\psi\rangle$ in the eigenbasis of $A$: $$|\psi\rangle=\sum_a|a\rangle\langle a|\psi\rangle=\sum_a\psi(a)|a\rangle$$

It is this "wavefunction" $\psi(a)$ that can tell us the probability of measuring a value $a$. More specifically, this is given by $|\psi(a)|^2$. So as you can see, operators don't tell us what an observable is for a state. All we can determine is probabilities of measuring observables. So onto your specific questions.

How do I actually measure the probabilities $\alpha$ and $\beta$?

If you want to measure the probabilities, you would need to create many systems in the same state and measure their spins. Then the ratio that are spin up gives you $\alpha$, and the ratio that are spin down gives you $\beta$ (more detail of this process can be found in Andrew Steane's answer). If you instead mean how do we mathematically determine them, you need more information about the state vector to do this.

How do I find the probability of the spin being along one of the axis?

See above.

And how is measuring the probability related to measuring the spin?

See above.

Answer 2

This is to enlarge a little on the answer given by Aaron Stevens, and deal with the practicalities of measurement.

To measure a probability, whether in quantum physics or elsewhere, you need many copies of your system, or else the ability to prepare a given system repeatedly in the same initial state. You measure each copy, and accumulate the data until you have enough to get the info you want. e.g. If 40 copies gave "+1/2" and 60 gave "-1/2" then $|\alpha|^2 = 0.4 \pm 0.1$ where the uncertainty is owing to standard statistical considerations for a finite sample. If you can repeat more times, say 10,000 trials, then if 4302 give "+1/2" then you have $|\alpha|^2 = 0.43 \pm 0.01$. In the case of spin, one usually measures a related property such as magnetic dipole moment, which can be determined using the Stern-Gerlach apparatus: the direction of travel of the particle, on leaving a known magnetic field gradient, indicates its magnetic dipole moment. With modern methods, such as 'spintronics' in solid state, or cold trapped atoms, more clever methods are used, typically involving a resonance driven by a laser or an oscillating voltage.

The measuring apparatus itself sets a direction along which to determine the spin state, and yields the outcomes according to whatever direction was thus singled out. The measurement will typically disturb the spin state. If initially the spin was pointing in some other direction, then the interaction with the measuring apparatus will "grab" the spin and project it onto the $z$ axis (if that is the axis that has been set). The spin will end up randomly up or down along this axis, with the probability of each outcome set by its initial state, i.e. $|\alpha|^2$ and $|\beta|^2$.