# Bogoliubov transform

1. Jan 21, 2008

### dexturelab

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

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

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

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