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Bogoliubov transformation / Interpretation of diagonalized Hamiltonian

  1. Jan 8, 2014 #1
    Hey,

    I consider a diagonalized Hamiltonian:

    [itex]H=\sum\limits_{k} \underbrace{ (\epsilon_{k} u_{k}^2 -\epsilon_{k} v_{k}^2 -2\Delta u_{k} v_{k} )}_{E_{k}}(d_{k \uparrow}^{\dagger}d_{k \uparrow} + d_{k \downarrow}^{\dagger}d_{k \downarrow}) +const [/itex]
    with fermionic creation and annihilation operators.

    From solution I know that: [itex]E_{k} =\sqrt{\Delta^2 +\epsilon_{k}^2}[/itex] but how can I get this result?





    Things I even know is that: [itex]u_k^2 + v_k^2 =1 [/itex] and:
    [itex]\sum\limits_k

    \underbrace{(
    -2\epsilon_k u_k v_k +\Delta v_k^2 -\Delta u_k ^2
    )}_{\stackrel{!}{=}0}

    (d_{k \uparrow}^{\dagger}d_{k \downarrow}^{\dagger} + d_{k \downarrow}d_{k \uparrow})[/itex].

    Thank you guys!!!
     
  2. jcsd
  3. Jan 8, 2014 #2

    DrDu

    User Avatar
    Science Advisor

    This are two equations for the two unknowns u and v. Solve for them and put into the defining equation for E_k!
     
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