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For $N$ coupled oscillators(periodic BC) whose Hamiltonian is given as $H=\sum\limits_{i=1}^N (\frac{p_i}{2m} + \lambda(x_{i+1} - x_i)^2)$

decoupling can be achieved by change of variables by using a Discrete Fourier transform(DFT) for $x$ and $p$ (I might be messing up the exact formula here) $\xi_k = \sum\limits_{j=1}^N x_j e^{-i2\pi kj}$

$\chi_k = \sum\limits_{j=1}^N p_j e^{-i2\pi kj}$

Leading to the normal modes and a set of decoupled oscillators

  1. Now are there general transformations like this that can decouple a Hamiltonian(or the equivalent set of PDE's)?

  2. For what kind of Hamiltonians does the DFT work? Does DFT work for all cases with parabolic potentials(linear springs)?

  3. What is the general theory behind all this called?

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    $\begingroup$ Usually you do this when you have discrete transitional symmetry. Because in this case, the quasi-momentum is a good quantum number, your Hamiltonian is block diagonal in k-space. $\endgroup$
    – an offer can't refuse
    Commented Oct 4, 2014 at 11:49
  • $\begingroup$ Then what general transforms diagonalise the Hamiltonian? Because then I think DFT would be a subclass of that $\endgroup$
    – biryani
    Commented Oct 4, 2014 at 17:27
  • $\begingroup$ You can only diagonalise a Hamiltonian when it is a bilinear one. In most cases, the Hamiltonian can not be diagonalised. $\endgroup$
    – an offer can't refuse
    Commented Oct 5, 2014 at 3:23

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