How Does the Pauli Exclusion Principle Limit Electron Spin Directions?

[Chemer]

Hi,
Can someone please explain this to me?

"The axis of rotation for a non-quantum-mechanical object can point any way it likes. The Earth could rotate around an axis ninety degrees from the current one, so that the North Pole always faces the sun and the South Pole always faces away from it. (This actually happens on Uranus, which has an axial tilt of almost one hundred degrees off of the plane of the rest of the solar system.) However, spin can’t possibly behave like this. The number of possible directions in which an electron can spin must be equal to the number of electrons that can fit into an orbital. Otherwise, each new electron could spin in a different direction and the Pauli exclusion principle would allow any number of electrons in a single orbital. Therefore, electrons must have a limited number of possible spin directions. And since we know from Langmuir that only two electrons can occupy a given orbital, electrons must have precisely two possible spin directions."

That's a little confusing, the number of electrons determine the spin and the spin determine the number of electrons? Isn't that only two allowed directions for spin is the reason for only two electrons occupying the single orbital? Or the presence of two electrons make the spin to have just two directions? Please explain?
Also please explain the Earth example and its relation to the topic.
Thanks.

 

[ddd123]

Isn't that only two allowed directions for spin is the reason for only two electrons occupying the single orbital?

Yes.

Or the presence of two electrons make the spin to have just two directions?

That's simply a deduction from experiments, it was trying to give a practical evidence instead of focusing on just the theoretical principles (which the text you are reading doesn't seem to focus too much on). It does sound inadvertently confusing. The logic is that, for instance, in principle you know that the possibilities must be finite: then since they are shown to be two, the spin values must be two.

Also please explain the Earth example and its relation to the topic.

Classical rotation is not quantized i.e. it can have any direction, there are infinite possibilities that correspond to the infinite points on a sphere. Spin is quantized and has only two possibilities in the case of the electron, up or down (it's not literally like that, it's actually a general direction of a cone which can point upwards or downwards with respect to an interacting environment, but the uncertainty principle imposes such a consequence for spin 1/2 particles like the electron - the details can only be shown mathematically).