# Bogoliubov transformation / Interpretation of diagonalized Hamiltonian

1. Jan 8, 2014

### Abigale

Hey,

I consider a diagonalized Hamiltonian:

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

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

Things I even know is that: $u_k^2 + v_k^2 =1$ and:
$\sum\limits_k \underbrace{( -2\epsilon_k u_k v_k +\Delta v_k^2 -\Delta u_k ^2 )}_{\stackrel{!}{=}0} (d_{k \uparrow}^{\dagger}d_{k \downarrow}^{\dagger} + d_{k \downarrow}d_{k \uparrow})$.

Thank you guys!!!

2. Jan 8, 2014

### DrDu

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