Understanding Spin Basis & Representation in Modern Physics | Explained

[yeezyseason3]

So I am in an introductory modern physics class and we discussed how intrinsic spin can be a linear combination of the spin basis. I am a bit confused on the physical representation of this and whether or not there are different basis to represent spin. If it is possible, what would be the point of doing so and how would you do it? From what I am understanding, spin can be treated like a coordinate system and you can use basic mathematical methods to represent spin in another coordinate system. Another question comes with the partial superposed spin states of an eigenstate. Doesn't the Pauli exclusion principle prevent this from happening?

 

[Nugatory]

I am a bit confused on the physical representation of this and whether or not there are different basis to represent spin. If it is possible, what would be the point of doing so and how would you do it?

It's analogous to the way that I can describe the position of something as "one kilometer to the north and one kilometer to the east" or as "1.414 kilometers northeast, zero kilometers northwest". The first description is more convenient if I'm looking at a map with north up and north/south and east/west gridlines on it; the second is more convenient if I'm in a city whose streets are laid out in a grid pattern with downtown/uptown avenes running from southwest to northeast and crosstown streets at right angles to the avenues. One way, my basis vectors are north/east, the other ways they're uptown/crosstown. But it's the same point with the same physical relationship to me either way.

The key here is that spin states are mathematically a kind of vector, and a vector can always be written as the sum of other vectors in many different ways - but it's still the same vector.

From what I am understanding, spin can be treated like a coordinate system and you can use basic mathematical methods to represent spin in another coordinate system. Another question comes with the partial superposed spin states of an eigenstate. Doesn't the Pauli exclusion principle prevent this from happening?

No. The Pauli exclusion principle says that no two particles can be in the exact same state, but here we have one particle with one state, and we're just playing with the mathematical fact that the state is a vector so can be written as the sum of other vectors.

 

[PeroK]

Re your first question, spin states are essentially vectors, and any vector can be expressed in any basis. There's no more to it than that.

I don't understand why the Pauli exclusion principle would come into it.