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Bogoliubov transform

  1. Jan 21, 2008 #1
    1. The problem statement, all variables and given/known data
    Hello. I have a question.
    I have a hamiltonian like this:
    [tex]H=\sum_{\vec{k}}[\omega_{1}(\vec{k})a^{+}(\vec{k})a(\vec{k})+\omega_{2}(\vec{k})b^{+}(\vec{k})b(\vec{k})+\phi_{1}(\vec{k})b^{+}(\vec{k})a(\vec{k})+\phi_{2}(\vec{k})a^{+}(\vec{k})b(\vec{k})][/tex]
    My purpose is making this hamiltonian diagonal by changing [tex]a,a^{+},b,b^{+}[/tex] to other operators [tex]\alpha,\alpha^{+},\beta,\beta^{+}[/tex] which also obey the commutation relations.

    2. Relevant equations
    I know the eigen-frequencies for this hamiltonian.
    [tex]\Omega_{+,-}(\vec{k})=\frac{\omega_{1}(\vec{k})+\omega_{2}(\vec{k})}{2}\pm\sqrt{\left(\frac{\omega_{1}(\vec{k})-\omega_{2}(\vec{k})}{2}\right)^{2}+\phi_{1}(\vec{k})\phi_{2}(\vec{k})}[/tex]
    and the last hamiltonian must be:
    [tex]H=\sum_{\vec{k}}[\Omega_{+}(\vec{k})\alpha^{+}(\vec{k})\alpha(\vec{k})+\Omega_{-}(\vec{k})\beta^{+}(\vec{k})\beta(\vec{k}))][/tex]

    3. The attempt at a solution
    For the case [tex]\phi_{1}=\phi_{2}[/tex], I have already reached the solution.
    I use this transform: (Bogoliubov)
    [tex]a=u.\alpha + v.\beta[/tex]
    [tex]b=-v.\alpha + u.\beta[/tex]
    They lead to some conditions such as: [tex]u^2+v^2=1[/tex],...
    I have found that [tex]u=\frac{(\Omega_{-}-\omega_{1})/\phi_{2}}{\sqrt{1+(\frac{\Omega_{-}-\omega_{1}}{\phi_2})^2}}[/tex]
    and some equation like this for [tex]v[/tex] will be the solution.

    But for the case [tex]\phi_{1}\neq\phi_{2}[/tex], I have found a condition [tex]u^2+v^2=0[/tex] which contradicts the condition [tex]u^2+v^2=1[/tex]
    So, how should I do for this case? I khow the eigen-frequencies, the eigen state but
    I can not directly change [tex]a,b\rightarrow\alpha,\beta[/tex].
    Sorry for my English. Thank you very much.
     
  2. jcsd
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