# Self consistency and mean field theory

1. Jun 24, 2015

### Hazz

1. The problem statement, all variables and given/known data
I am a little confused about the how self consistency conditions work and I was wondering if in the following case I have correctly understood the details?

2. Relevant equations

Say we have a harmonic oscillator with a perturbation
$H=\frac{\hat{p}}{2m}+\frac{1}{2}m\omega^2\hat{q}^2+\gamma \hat{q}^4$

Say we want to perform something like a Hartree-Fock approximation. We replace the $(a+a^\dagger)^4$ in $\hat{q}^4$ with $\langle (a+a^\dagger)^2 \rangle (a+a^\dagger)^2$. Say our new Hamiltonian is then

$H_{HF}=\frac{\hat{p}}{2m}+\frac{1}{2}m\omega^2\hat{q}^2+\delta (a+a^\dagger)^2$

with $\delta=\gamma(\frac{\hbar}{2m\omega})^2\langle (a+a^\dagger)^2 \rangle$

3. The attempt at a solution

This is now a quadratic Hamiltonian and can be diagonalised as

$H= \sqrt{\frac{\hbar\omega}{2}(\frac{\hbar\omega}{2}+2\delta)}\hat{b}^\dagger \hat{b}$

Using this $\langle (a+a^\dagger)^2 \rangle$ can be calculated and found to be

$\langle (a+a^\dagger)^2 \rangle = \frac{2n_B(\hbar\omega)-1}{\sqrt{\frac{\hbar\omega}{2}(\frac{\hbar\omega}{2}+2\gamma(\frac{\hbar}{2m\omega})^2\langle (a+a^\dagger)^2 \rangle)}}$

Am I correct in thinking this equation would be the self-consistency condition we are looking for? You could then plot the RHS and LHS with respect to $\langle (a+a^\dagger)^2 \rangle$ and see where the lines cross, if they meet anywhere then the approximation is consistent?

Last edited: Jun 24, 2015
2. Jun 29, 2015

### Staff: Admin

Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?

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