A Understanding Harmonic oscillator conventions

[Djawu]

TL;DR Summary

see the equations down below

I don't quite understand how he got the line below. By using discrete time approximation, we can get the second order time expression. But i don't see how by combining terms he is able to get such expression.

 

[PeterDonis]

I don't quite understand how he got the line below.

Who is "he" and where did this come from?

 

[Djawu]

Who is "he" and where did this come from?

This is a private document shared only within the department. I have the permission to quote parts of it without releasing the rest.

 

[HomogenousCow]

Simply substitue the second Hamilton’s equation into the first.

 

[Demystifier]

Hint: $q_n$ can be written as a Fourier series with coefficients $q_k$, and vice versa.
For details see e.g. Kittel, Introduction to Solid State Physics, Appendix C: Quantization of Elastic Waves: Phonons

 

[king vitamin]

This is a private document shared only within the department. I have the permission to quote parts of it without releasing the rest.

But I recognize these notes, they are available for free on the author's public website...? But perhaps you have your own reasons for not disclosing, in any case Demystifier's hint will get you in the right direction.

 

[Djawu]

But I recognize these notes, they are available for free on the author's public website...? But perhaps you have your own reasons for not disclosing, in any case Demystifier's hint will get you in the right direction.

sharp eyes ;)