Bogoliubov transformation / Interpretation of diagonalized Hamiltonian

  • Thread starter Abigale
  • Start date
  • #1
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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!!!
 

Answers and Replies

  • #2
DrDu
Science Advisor
6,076
787
Things I even know is that: [itex]u_k^2 + v_k^2 =1 [/itex] and:
[itex](
-2\epsilon_k u_k v_k +\Delta v_k^2 -\Delta u_k ^2
)=0
[/itex].
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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