Represent |+x> and |-x> in the Sy basis

[TastyPastry]

Homework Statement

Determine the column vectors representing |+x> and |-x> using the states |+y> and |-y> as a basis.

Homework Equations

N/A

The Attempt at a Solution

I know that if |+y> and |-y> are used as a basis, then they are the column vectors (1,0) and (0,1) respectively. I also know that |+x> as a column vector in the Sz basis is 1/√2 (1,0) and |+y> in the Sz basis is a column vector 1/√2 (1,i). However, these were predetermined values that represented states in a Stern Gerlach experiment. I'm not sure how to approach making the |+x> and |-x> in a different basis, namely the Sy basis.

 

[nejibanana]

You want to find a transformation matrix to change basis. If we call T our transformation matrix, and we have the operator S_y in the original basis, and S_y' is in the new basis, then S_y' = T* S_y T, where the * denotes the hermitian conjugate. Since S_y' is in the y-basis, it should be diagonal and the same as S_z in the z-basis, namely ({1,0},{0 -1}). So just solve for T, then use the fact that |+x> in the new basis will be equal to T |+x>. This might be T* |+x>, but you can double check which one, since you know what the two y states will be in the new basis (i.e. (1.0) = T |+y> or T*|+y>) My copy of Sakurai is in my office right now, so I can't look the formulas up to double check.