Why does measuring electron spin along one axis change the spin direction?

[nurrifat]

σ_x|x>=+|x>
σ_x|-x>=-|-x>

These equations also follows for σ_y and σ_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,
σ_x|u>=|d> σ_x|d>=|u>

σ_y|u>=i|d> σ_y|d>=-i|u>

σ_z|u>=|u> σ_z|d>=-|d>

these 3 equations can't make any sense to me. i can't 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??

 

[nurrifat]

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.

opps mayb i complexed the sentence

if we measure along axis X and then if state X goes through it shows electron spin up and if not go then it means electron spin is down.

 

[wotanub]

It seems to me the u and d states are superpositions of the eigenstates of the spin operators. For example, for the first equation one you could say
$\left|u \right\rangle = \frac{1}{\sqrt{2}}\left|+x \right\rangle + \frac{1}{\sqrt{2}}\left|-x \right\rangle$
$\left|d \right\rangle = \frac{1}{\sqrt{2}}\left|+x \right\rangle - \frac{1}{\sqrt{2}}\left|-x \right\rangle$

and the equation is okay. Furthermore, I'd just change my labels like so (because what I name each axis is arbitrary)
$x → z$

$u → +x$

$d → -x$

So it looks like the expression you may have see in your textbook.

$\left|+x \right\rangle = \frac{1}{\sqrt{2}}\left|+z \right\rangle + \frac{1}{\sqrt{2}}\left|-z \right\rangle$
$\left|-x \right\rangle = \frac{1}{\sqrt{2}}\left|+z \right\rangle - \frac{1}{\sqrt{2}}\left|-z \right\rangle$

Try to see if you can find a superposition of eigenstates that work for the others to make sure you understand. Remember to make sure the states are normalized.