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I'm trying to give a less technical answer. It's not rigorous but should give you the idea how spin and the regular rotation related.

Maxwell's equations say in order to have magnetic field, you need a ring current.

This can be achieved by giving angular momentum to charged particles. This can be orbital or simply because the particle is spinning. This was the original thought hence the name 'spin'.

So in the classical picture, if you spin a tiny charged ball you'll have a spinning magnet. The axis of spinning and the north pole of the magnet pointing to the same direction.

If you put this spinning magnet into a magnetic field. The field will apply torque on it to turn it into the direction of the field (this is how compasses work).

But since the our magnet is spinning this torque cause the axis of spinning precess around the magnetic field. This means the component of the rotation axis that is parallel to the magnetic field (typically referred as the Z component) won't change while the other two components (X,Y) will circle around this axis.

On the other hand if the magnetic field inhomogeneous there will be a net force on the particle that will move it (that's why magnets can snap and repel each other). This force is proportional to the Z component. So the axis perpendicular to the magnetic field there will be no force, if it's parallel there will be maximum force (basically a dot product). This allows us measuring the Z component of magnetic field and the rotation axis.

That's the point of the Stern–Gerlach experiment. We would normally expect that particles will spin in a whole variety of random axes. So we would expect to measure random values for the Z component.

But in reality they have measured only two possible values corresponding to the Z angular momentum component: $ħ/2$ and $-ħ/2$ (for electrons). And not any other random values. Here the classical picture breaks down, angular momentum is also quantized. You can see spin is not the classical rotation vector. It's something you can dot multiply a vector to and you can only get two possible values. The positive component typically referred as the 'up' spin component while the negative is the 'down' spin.

Precession renders all axes other than the one being measured uncertain. This is how uncertainty principle plays role here: if you measure the Z component first, then measure the X component, then the Z again, you get random up/down results again, because the measurement of the X components precessed the Y and Z component. Also, you cannot cheat here: you may want to use weaker magnetic field to reduce the precession, the displacement will be too weak to distinguish between the up and down spins. If you try to use timing; you cannot cheat again because if you measure the time accurately, then the energy so the precession rate becomes uncertain.

I'm trying to give a less technical answer. It's not rigorous but should give you the idea how spin and the regular rotation related.

Maxwell's equations say in order to have magnetic field, you need a ring current.

This can be achieved by giving angular momentum to charged particles. This can be orbital or simply because the particle is spinning. This was the original thought hence the name 'spin'.

So in the classical picture, if you spin a tiny charged ball you'll have a spinning magnet. The axis of spinning and the north pole of the magnet pointing to the same direction.

If you put this spinning magnet into a magnetic field. The field will apply torque on it to turn it into the direction of the field (this is how compasses work).

But since the our magnet is spinning this torque cause the axis of spinning precess around the magnetic field. This means the component of the rotation axis that is parallel to the magnetic field (typically referred as the Z component) won't change while the other two components (X,Y) will circle around this axis.

On the other hand if the magnetic field inhomogeneous there will be a net force on the particle that will move it (that's why magnets can snap and repel each other). This force is proportional to the Z component. So the axis perpendicular to the magnetic field there will be no force, if it's parallel there will be maximum force (basically a dot product). This allows us measuring the Z component of magnetic field and the rotation axis.

That's the point of the Stern–Gerlach experiment. We would normally expect that particles will spin in a whole variety of random axes. So we would expect to measure random values for the Z component.

But in reality they have measured only two possible values corresponding to the Z angular momentum component: $ħ/2$ and $-ħ/2$ (for electrons). And not any other random values. Here the classical picture breaks down, angular momentum is also quantized. You can see spin is not the classical rotation vector. It's something you can dot multiply a vector to and you can only get two possible values. The positive component typically referred as the 'up' spin component while the negative is the 'down' spin.

Precession renders all axes other than the one being measured uncertain. This is how uncertainty principle plays role here: if you measure the Z component first, then measure the X component, then the Z again, you get random up/down results again, because the measurement of the X components precessed the Y and Z component. Also, you cannot cheat here: you may want to use weaker magnetic field to reduce the precession, the displacement will be too weak to distinguish between the up and down spins. If you try to use timing; you cannot cheat again because if you measure the time accurately, then the energy so the precession rate becomes uncertain.

I'm trying to give a less technical answer. It's not rigorous but should give you the idea how spin and the regular rotation related.

Maxwell's equations say in order to have magnetic field, you need a ring current.

This can be achieved by giving angular momentum to charged particles. This can be orbital or simply because the particle is spinning. This was the original thought hence the name 'spin'.

So in the classical picture, if you spin a tiny charged ball you'll have a spinning magnet. The axis of spinning and the north pole of the magnet pointing to the same direction.

If you put this spinning magnet into a magnetic field. The field will apply torque on it to turn it into the direction of the field (this is how compasses work).

But since the our magnet is spinning this torque cause the axis of spinning precess around the magnetic field. This means the component of the rotation axis that is parallel to the magnetic field (typically referred as the Z component) won't change while the other two components (X,Y) will circle around this axis.

On the other hand if the magnetic field inhomogeneous there will be a net force on the particle that will move it (that's why magnets can snap and repel each other). This force is proportional to the Z component. So the axis perpendicular to the magnetic field there will be no force, if it's parallel there will be maximum force (basically a dot product). This allows us measuring the Z component of the rotation axis.

That's the point of the Stern–Gerlach experiment. We would normally expect that particles will spin in a whole variety of random axes. So we would expect to measure random values for the Z component.

But in reality they have measured only two possible values corresponding to the Z angular momentum component: $ħ/2$ and $-ħ/2$ (for electrons). And not any other random values. Here the classical picture breaks down, angular momentum is also quantized. You can see spin is not the classical rotation vector. It's something you can dot multiply a vector to and you can only get two possible values. The positive component typically referred as the 'up' spin component while the negative is the 'down' spin.

Precession renders all axes other than the one being measured uncertain. This is how uncertainty principle plays role here: if you measure the Z component first, then measure the X component, then the Z again, you get random up/down results again, because the measurement of the X components precessed the Y and Z component. Also, you cannot cheat here: you may want to use weaker magnetic field to reduce the precession, the displacement will be too weak to distinguish between the up and down spins. If you try to use timing; you cannot cheat again because if you measure the time accurately, then the energy so the precession rate becomes uncertain.

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I'm trying to give a less technical answer. It's not rigorous but should give you the idea how spin and the regular rotation related.

Maxwell's equations say in order to have magnetic field, you need a ring current.

This can be achieved by giving angular momentum to charged particles. This can be orbital or simply because the particle is spinning. This was the original thought hence the name 'spin'.

So in the classical picture, if you spin a tiny charged ball you'll have a spinning magnet. The axis of spinning and the north pole of the magnet pointing to the same direction.

If you put this spinning magnet into a magnetic field. The field will apply torque on it to turn it into the direction of the field (this is how compasses work).

But since the our magnet is spinning this torque cause the axis of spinning precess around the magnetic field. This means the component of the rotation axis that is parallel to the magnetic field (typically referred as the Z component) won't change while the other two components (X,Y) will circle around this axis.

On the other hand if the magnetic field inhomogeneous there will be a net force on the particle that will move it (that's why magnets can snap and repel each other). This force is proportional to the Z component. So the axis perpendicular to the magnetic field there will be no force, if it's parallel there will be maximum force (basically a dot product). This allows us measuring the Z component of magnetic field and the rotation axis.

That's the point of the Stern–Gerlach experiment. We would normally expect that particles will spin in a whole variety of random axes. So we would expect to measure random values for the Z component.

But in reality they have measured only two possible values corresponding to the Z angular momentum component: $ħ/2$ and $-ħ/2$ (for electrons). And not any other random values. Here the classical picture breaks down, angular momentum is also quantized. You can see spin is not the classical rotation vector. It's something you can dot multiply a vector to and you can only get two possible values. The positive component typically referred as the 'up' spin component while the negative is the 'down' spin.

Precession renders all axes other than the one being measured uncertain. This is how uncertainty principle plays role here: if you measure the Z component first, then measure the X component, then the Z again, you get random up/down results again, because the measurement of the X components precessed the Y and Z component. Also, you cannot cheat here: you may want to use weaker magnetic field to reduce the precession, the displacement will be too weak to distinguish between the up and down spins. If you try to use timing; you cannot cheat again because if you measure the time accurately, then the energy so the precession rate becomes uncertain.