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

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

|Upx> = 1/ √2 ( |Upz> + |Downz> )
and
|Downx> = 1/ √2 ( - |Upz> + |Downz> )

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 cant 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

## Answers and Replies

blue_leaf77
Homework Helper
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.

Hm okay I have the Sx matrix and a matrix in the n direction Sn. 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 doesnt explain change of basis in my book. Just to clarify, |z;±> means spin up and spin down states in z direction right?

blue_leaf77