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Infinite one-dimensional oscillators

  1. Sep 15, 2016 #1
    1. The problem statement, all variables and given/known data

    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?

    2. Relevant equations

    3. 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?
     
    Last edited: Sep 15, 2016
  2. jcsd
  3. Sep 21, 2016 #2
    Thanks for the thread! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post? The more details the better.
     
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