# 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