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Can a Hamiltonian be unbounded ?

  1. May 30, 2010 #1
    the idea is can a Hamiltonian in 1-D of the form [tex] H=p^2 + V(x) [/tex] for a certain function V(x) be unbounded and have NEGATIVE energies , for example a Hamiltonian whose spectra may be [tex] E_{n} = ....,-3,-2,-1,1,2,3,..... [/tex] and so on, so we have an UNBOUNDED Hamiltonian with positive and negative energies with the property

    [tex] E_{-n}=-E_{n} [/tex]
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  3. May 30, 2010 #2


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    I think a Hamiltonian must always be bounded from below, b/c otherwiese it would be possible (e.g. due to perturbations) that there is a non-zero possibility for a state |n> to decay to |n-k>, where n is not bounded from below. So the whole system is unstable and decays into a singular state plus infinitly many photons, phonons or whatever.
  4. May 31, 2010 #3


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    In QFT, boundedness from below is a postulated for an acceptable Hamiltonian. In nonrelativistic physics, it is a derived result for each analyzed system.
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