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Pauli matrices?

  1. Aug 26, 2014 #1
    Hey, so today for our quantum physics class we were supposed to go through these identities,

    [itex]|+_x > = \frac{1}{2^{0.5}} (|+> + |->)[/itex]

    [itex]|-_x > = \frac{1}{2^{0.5}} (|+> - |->)[/itex]

    [itex]|+_y > = \frac{1}{2^{0.5}} (|+> + i|->)[/itex]

    [itex]|-_y > = \frac{1}{2^{0.5}} (|+> - i|->)[/itex]

    where |+ (x)> would represent spin up in the x direction for example, and |+> simply denotes spin up in (I believe) the z direction.
    now I couldn't make sense of any of it, had no idea where they came from and what the proof is, I came home and googled it and noticed a strong resemblance to the pauli matrices (which I have discovered as of half an hour ago), so I'm hoping someone could enlighten me as to what this means and where it comes from.
    Thanks in advance
    Last edited: Aug 26, 2014
  2. jcsd
  3. Aug 26, 2014 #2


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    The Pauli matrices are the matrix elements of the operators
    [tex]\hat{\sigma}^j=\frac{2}{\hbar} \hat{s}^j,[/tex]
    where [itex]\hat{s}_j[/itex] are the spin components of a spin-1/2 particle.

    The Pauli matrix are the matrix elements with respect to the eigenbasis of [itex]\hat{\sigma}^3[/itex],
    [tex]\hat{\sigma}^3 |k \rangle=k |k \rangle, \quad k \in \{-1,1 \},[/tex]
    where the choice of the 3-component is conventional. The Pauli matrices are given by
    [tex]{\sigma^{j}}_{kl}=\langle{k}|\hat{\sigma}^j|l\rangle, \quad k,l \in \{-1,1 \}.[/tex]

    You can easily calculate the Pauli matrices by making use of the raising- and lowering operators
    [tex]\hat{\sigma}^{\pm} = \hat{\sigma}^{1} \pm \mathrm{i} \hat{\sigma}^{2}.[/tex]
    Just look in your textbook, how those operate on the basis vectors!
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