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Spin operator

  1. Oct 5, 2013 #1
    σx|x>=+|x>
    σx|-x>=-|-x>

    These equations also follows for σy and σz 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,
    σx|u>=|d> σx|d>=|u>

    σy|u>=i|d> σy|d>=-i|u>

    σz|u>=|u> σz|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??
     
  2. jcsd
  3. Oct 5, 2013 #2
    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.
     
  4. Oct 5, 2013 #3
    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
    [itex] \left|u \right\rangle = \frac{1}{\sqrt{2}}\left|+x \right\rangle + \frac{1}{\sqrt{2}}\left|-x \right\rangle[/itex]
    [itex]\left|d \right\rangle = \frac{1}{\sqrt{2}}\left|+x \right\rangle - \frac{1}{\sqrt{2}}\left|-x \right\rangle[/itex]

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

    [itex]u → +x[/itex]

    [itex]d → -x[/itex]

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

    [itex] \left|+x \right\rangle = \frac{1}{\sqrt{2}}\left|+z \right\rangle + \frac{1}{\sqrt{2}}\left|-z \right\rangle[/itex]
    [itex]\left|-x \right\rangle = \frac{1}{\sqrt{2}}\left|+z \right\rangle - \frac{1}{\sqrt{2}}\left|-z \right\rangle[/itex]

    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.
     
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