Understanding Spin States: Theoretical Minimum and Normal Coordinate Systems

[Quarlep]

In The theoretical Minimum we shown all spins states use just two states up and down. How can we do that.?
I am confused about the directions of states and normal coordinate system Can somebody help me ?
Thanks

 

[mfb]

In The theoretical Minimum

What does that mean?

If you measure the spin direction of a particle with spin 1/2, the only possible measurements are "up" and "down". If the particle has a different spin, the options are different.
Which normal coordinate system?

 

[M Quack]

The spin value 1/2 and two spin states drop out of the Dirac equation. Spin-up and spin-down are chosen more or less arbitrarily, because they are the eigenstates of energy in a static magnetic field along the z-direction (theoretical magnetic fields are always along z...). However, they form a complete basis for spin-1/2 particles, and all spin directions can be written as a linear combination of these two basis states.

Therefore you can have spin directions in each and any direction. The theory is perfectly well developed and used e.g. in neutron diffraction.

https://www.ill.eu/en/instruments-support/instruments-groups/instruments/d3/how-it-works/spherical-polarimetry-with-cryopad/

 

[Nugatory]

In The theoretical Minimum we shown all spins states use just two states up and down. How can we do that.?

Think of spin-up as a vector pointing east and spin-down as a vector pointing north. We can write any vector as a linear combinations of those two. For example, north-east would be the vector sum of north and east, southeast would be their east minus north, and so forth.

The confusing thing is that the "directions" these vectors point in their abstract vector space isn't the same as the direction that the spin angular momentum vector points in the real world. Spin-up and spin-down are represented by orthogonal vectors in the abstract vector space, even though up and down are opposite directions.