Spin up and Spin down states can be written as .... why?

[Sara Kennedy]

Spin up and spin down states in the x direction can be written as

|Up_x> = 1/ √2 ( |Up_z> + |Down_z> )
and
|Down_x> = 1/ √2 ( - |Up_z> + |Down_z> )

My textbook just stated the above facts without referencing why and I've been going through the spin chapter for a while now and I can't see it. Why is this true?

Is it something to do with permutation rotation and orthogonality? For example I could switch the x to y and the z to x and the expressions would be correct

 

[blue_leaf77]

Find the matrix form of $S_x$ in the basis $|z;\pm\rangle$ and then find its eigenvectors. Alternatively, you can also apply the rotation operator about the y-axis by 90 degree to the $|z;\pm\rangle$ states.

 

[Sara Kennedy]

Hm okay I have the S_x matrix and a matrix in the n direction S_n. Do you know of any online notes that explain how to change from one basis to another? I found a few questions but they look like homework solutions and I was after explanations, it doesn't explain change of basis in my book. Just to clarify, |z;±> means spin up and spin down states in z direction right?

 

[blue_leaf77]

Do you know of any online notes that explain how to change from one basis to another? I found a few questions but they look like homework solutions and I was after explanations, it doesn't explain change of basis in my book.

Try searching for "change of coordinates matrix", you should find a number of resources on this matter. In general, to find the change of coordinate matrix, you have to know how the vector in one basis is expressed in terms of the basis vectors in the other basis.

Just to clarify, |z;±> means spin up and spin down states in z direction right?

Yes.

 

[forcefield]

I was after explanations

See http://feynmanlectures.caltech.edu/III_06.html for a derivation of these equations (6.32) for "amplitudes". The derivation starts from the fact that a 360 degree rotation changes the sign of both amplitudes (because otherwise a 180 degree rotation would be physically equivalent to a 360 degree rotation).