Timeline for Proof that 1d lattice displacement by phonons is given $u_{n\pm 1}(t) = A_ke^{i\omega_k t} e^{i knd}e^{\pm i k d}$

9 events

when toggle format	what		by	license	comment
Dec 21, 2021 at 8:16	history	edited	A.G.	CC BY-SA 4.0	English
Feb 23, 2016 at 13:28	vote	accept	Mikkel Rev
Feb 22, 2016 at 19:00	history	edited	A.G.	CC BY-SA 3.0	corrected spelling
Feb 22, 2016 at 14:01	comment	added	A.G.		Ok, I have added the reply to this comment to the main answer. So, I agree whit what you have stated. This answer is meant to have you exercice with DFT and ( just Fourier transform) to understand how to fish on your own next time --- plus some reflexion about the periodic conditions of the problem. And, it has been a pleasure.
Feb 22, 2016 at 13:57	history	edited	A.G.	CC BY-SA 3.0	added 453 characters in body
Feb 21, 2016 at 15:30	comment	added	Mikkel Rev		Thanks, I want to accept this as the answer. It was fun to go through it all, and hope others do the same. Just to make sure that, I've understood correctly, we agree that $U_k(t) = A_k e^{i\omega t}$ for each wavenumber $k$. However, if we assume oscillations are sinusoidal composed of exactly one wavenumber $k$, then $u_{n\pm 1} = \sum_{j=k}^k A_j e^{i\omega_j t} e^{i (n\pm 1)j d} = A_ke^{i\omega_k t} e^{i knd}e^{\pm i k d}$? If this is correct, perhaps add it to the reply? That way we arrive at the equation in the title, and the one Kittel states. Do you agree? Thank you for your time.
Feb 21, 2016 at 11:19	history	edited	A.G.	CC BY-SA 3.0	phrasing
Feb 20, 2016 at 20:08	history	edited	A.G.	CC BY-SA 3.0	omitted pre-factors
Feb 20, 2016 at 19:57	history	answered	A.G.	CC BY-SA 3.0