Infinite one-dimensional oscillators

[rock_pepper_scissors]

Homework Statement

Consider a series of $N$ particles in a line, with the displacement of each particle from its equilibrium position labelled by $q_{n}$ and it conjugate momentum labelled by $p_n$. Assume that the interaction between the particles is pairwise, so that the Hamiltonian is given by

$H = \Sigma_{n}\ \frac{1}{2m}\ (p_{n}^{2})+ \frac{1}{2}m\omega^{2}(q_{n}-q_{n+1})^{2}$.

Throughout the problem, assume that the boundary conditions are periodic so that $q_{N+1}=q_{1}$.

(a) Explain why the Fourier transformed variables $\tilde{q}_{n}$ and $\tilde{p}_{n}$ given by

$q_{n}=\frac{1}{\sqrt{N}}\sum\limits_{k=0}^{N-1}\tilde{q}_{k}e^{2\pi ink/N}$ and $p_{n}=\frac{1}{\sqrt{N}}\sum\limits_{k=0}^{N-1}\tilde{p}_{k}e^{2\pi ink/N}$

are not Hermitian. Show that $\tilde{q}_{N-k}^{\dagger}=\tilde{q}_{k}$ and $\tilde{p}_{N-k}^{\dagger}=\tilde{p}_{k}$.

(b) Show that the Fourier modes describe a set of $N$ decoupled harmonic oscillators (Hint: Write the Hamiltonian in terms of the Fourier modes.)

(c) Describe the Hilbert space of this theory in terms of the raising and lowering operators of these $N$ harmonic oscillators. What is the spectrum of this Hamiltonian?

This is a very simple non-relativistic quantum field theory in the large $N$ limit. Under what circumstances would it be appropriate to approximate the displacements $q_{n}(t)$ by a continuous field $\phi(x,t)$? For what observables would this be a good approximation?

Homework Equations

The Attempt at a Solution

(a) The inverse Fourier transforms are

$\tilde{q}_{k}=\frac{1}{\sqrt{N}}\sum\limits_{n=0}^{N-1}q_{n}e^{-2\pi ink/N}$ and $\tilde{p}_{k}=\frac{1}{\sqrt{N}}\sum\limits_{n=0}^{N-1}p_{n}e^{-2\pi ink/N}$.

I understand that Fourier series can be written as complex exponentials, but I don't understand why the sum is finite. Can someone please explain?

 

[mark726]

The sum is finite because we are considering a system with a fixed number of particles, N. Therefore, the sum over n runs from 0 to N-1. If we were considering an infinite system, the sum would be infinite as well.