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Exponential of Pauli spin matrices

  1. Feb 17, 2016 #1
    How do we get (6.265)?

    Shouldn't we have

    ##exp(-i\frac{\alpha}{2}\hat{n}.\sigma)=\cos(\frac{\alpha}{2}\hat{n}.\sigma)-i\sin(\frac{\alpha}{2}\hat{n}.\sigma)##?

    Screen Shot 2016-02-18 at 3.15.30 am.png
    Screen Shot 2016-02-18 at 3.15.10 am.png

    Screen Shot 2016-02-18 at 3.15.43 am.png
    Screen Shot 2016-02-18 at 3.15.54 am.png
     
  2. jcsd
  3. Feb 17, 2016 #2

    blue_leaf77

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    By (6.241), you have ##(\mathbf{\sigma} \cdot \hat{n} )^2 = 1##. Use this identity in the Taylor expansions of ##\cos(\frac{\alpha}{2}\hat{n}.\sigma)## and ##\sin(\frac{\alpha}{2}\hat{n}.\sigma)## to reduce the higher powers.
     
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