? x |x>=+|x> ? x |-x>=-|-x> These equations also follows for ? y and ? z corresp

Question

?x|x>=+|x>
?x|-x>=-|-x>

These equations also follows for ?y and ?z corresponds states |y> and |z>.
if we measure along axis X then X state vector let it go which means up spin and opposite not go through which means down spin.

and also same for y and z axis.

But,
?x|u>=|d> ?x|d>=|u>

?y|u>=i|d> ?y|d>=-i|u>

?z|u>=|u> ?z|d>=-|d>

these 3 equations cant make any sense to me. i cant draw any physical meaning to these equations.

here 1st equation shows if up spin electron goes through along X axis then after measurement spin wilbe down.but this up electron has to be up for any specific axis??right?? without axis up down is unmeaningfull.isn't it??

Explanation / Answer

case1= (1/?2) |1/2> + (1/?2) |-1/2>

Or in other words for the first case, 1/2 of the electrons are spin up along the z-axis and 1/2 of the electrons are spin down along the z-axis.

case2 = x |1/2> + y |-1/2>

(I used x and y because it is not clear in the question)

in second case, , x2 of the electrons are spin up along the z-axis and y2 of the electrons are spin down along the z-axis.

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