# Linear Chain of oscillators (Quantum Treatment)

1. Feb 17, 2014

### Henriamaa

I am reading Field Quantization by Greiner and Rienhardt and I am trying to prove that
$i\hbar \frac{d\hat{b_k}}{dt} = [\hat{b_k},H]=\hbar \omega_k \hat{b_k}$ where $$\hat{b_k} = \frac{1}{2} \sum_n u^{k*}_n(q_n(t)+ \frac{i}{\omega_km}\hat{p}_n(t))$$ and
$$H= \sum_{n=1}^{N}\frac{1}{2m}\hat{p^2_n} + \sum_{n=1}^{N} \frac{\kappa}{2}(\hat{q_{n+1}}-\hat{q_n})^2$$ lastly $$u_n^k= \frac{1}{\sqrt{n}} e^{ikan}$$ I am interested in doing it the brute force way namely using the Hamilton as specifically stated above and the definition of $\hat{b_k}$. Without showing all my work, just by directly plugging the equations into the Heisenberg equation of motion(commutator). I arrived at the following equation $$\frac{1}{2}\sum_n \frac{u^k_n}{2m}[q_n,p_n^2] + \frac{1}{2} \sum_n \frac{i\kappa u^{k*}_n}{2 \omega_k m}[p_n,(q_{n+1}-q_n)^2]$$ which according to my calculation reduces to $$\frac{1}{2}\left(\sum_n u^{k*}_n \hbar(\frac{i\hat{p_n}}{m}-\omega_k(q_{n+1}-q_n))\right)$$
The problem I am facing is I need to get rid of the $q_{n+1}$ term but I do not know how to.