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Prove. Pauli matrices.

  1. Jun 26, 2014 #1
    1. The problem statement, all variables and given/known data
    Prove
    [tex]\exp (\alpha \hat{\sigma}_z+\beta \hat{\sigma}_x)=\cosh \sqrt{\alpha^2+\beta^2}+\frac{\sinh \sqrt{\alpha^2+\beta^2}}{\sqrt{\alpha^2+\beta^2}}(\alpha \hat{\sigma}_z+\beta \hat{\sigma}_x)[/tex]



    2. Relevant equations
    [tex]e^{\hat{A}}=\hat{1}+\hat{A}+\frac{\hat{A}^2}{2!}+...[/tex]



    3. The attempt at a solution
    [tex]\exp (\alpha \hat{\sigma}_z+\beta \hat{\sigma}_x)=\hat{1}+\alpha \hat{\sigma}_z+\beta \hat{\sigma}_x+\frac{1}{2!}(\alpha^2\hat{\sigma}_z^2+\beta^2\hat{\sigma}_x^2+\alpha \beta \hat{\sigma}_x\hat{\sigma}_z+\alpha \beta \hat{\sigma}_z\hat{\sigma}_x)+...[/tex]
    from that
    [tex]\exp (\alpha \hat{\sigma}_z+\beta \hat{\sigma}_x)=\hat{1}+\alpha \hat{\sigma}_z+\beta \hat{\sigma}_x+\frac{1}{2!}(\alpha^2\hat{1}+\beta^2\hat{1}+\alpha \beta \hat{\sigma}_x\hat{\sigma}_z+\alpha \beta \hat{\sigma}_z\hat{\sigma}_x)+...[/tex]
    Is this way to go? I'm not sure?
     
  2. jcsd
  3. Jun 26, 2014 #2
    Have you seen an expression for ##(\vec{n}\cdot \hat{\vec{\sigma}})^k##, with ##\vec{n}## a unit vector?

    [edit: no problem]
     
    Last edited: Jun 26, 2014
  4. Jun 26, 2014 #3
    Hi LagrangeEuler!

    The series above simplifies drastically when you figure out what to do with [itex]\sigma_x \sigma_z + \sigma_z \sigma_x[/itex].

    :smile:

    [edit: sorry Bloby, posted over you by mistake]
     
  5. Jun 26, 2014 #4
    Check out the anticommuntation properties {sig_x,sig_z} = 0, so there is a lot of cancelations.
     
  6. Jun 27, 2014 #5
    I don't see connection with this problem?
     
  7. Jun 27, 2014 #6
    Tnx a lot.
     
  8. Jun 27, 2014 #7
    With ##\sqrt{\alpha^2+\beta^2}=a##, ##exp((\beta , 0 , \alpha)\cdot (\hat{\sigma}_x , \hat{\sigma}_y , \hat{\sigma}_z))=exp(\sqrt{\beta^2+\alpha^2}(\hat{n}\cdot \hat{\vec{\sigma}}))=I+a(\hat{n}\cdot\hat{\vec{\sigma}})+\frac{1}{2!}a^2(\quad)^2+...##
     
    Last edited: Jun 27, 2014
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