I am reading Field Quantization by Greiner and Rienhardt and I am trying to prove that(adsbygoogle = window.adsbygoogle || []).push({});

[itex] i\hbar \frac{d\hat{b_k}}{dt} = [\hat{b_k},H]=\hbar \omega_k \hat{b_k}[/itex] where [tex]

\hat{b_k} = \frac{1}{2} \sum_n u^{k*}_n(q_n(t)+ \frac{i}{\omega_km}\hat{p}_n(t))[/tex] and

[tex] 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[/tex] lastly [tex] u_n^k= \frac{1}{\sqrt{n}} e^{ikan}[/tex] I am interested in doing it the brute force way namely using the Hamilton as specifically stated above and the definition of [itex] \hat{b_k}[/itex]. Without showing all my work, just by directly plugging the equations into the Heisenberg equation of motion(commutator). I arrived at the following equation [tex] \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] [/tex] which according to my calculation reduces to [tex] \frac{1}{2}\left(\sum_n u^{k*}_n \hbar(\frac{i\hat{p_n}}{m}-\omega_k(q_{n+1}-q_n))\right)[/tex]

The problem I am facing is I need to get rid of the [itex] q_{n+1} [/itex] term but I do not know how to.

**Physics Forums - The Fusion of Science and Community**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Linear Chain of oscillators (Quantum Treatment)

Can you offer guidance or do you also need help?

Draft saved
Draft deleted

Loading...

Similar Threads - Linear Chain oscillators | Date |
---|---|

A Is non-linear quantum mechanics (even) plausible? | Friday at 8:16 PM |

I Finite universe, quantized linear momentum, and the HUP... | Sep 7, 2017 |

A A question about linear response and conductivity | May 26, 2017 |

I Deriving resolution of the identity without Dirac notation | Apr 18, 2017 |

Can the Markov chain be used in qubit manipulation? | Mar 12, 2016 |

**Physics Forums - The Fusion of Science and Community**