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Infinite energy states for an harmonic oscillator?

  1. Sep 29, 2013 #1
    So, I've read conference proceedings and they appear to talk about counter-intuitive it was to create an infinite-energy state for the harmonic oscillator with a normalizable wave function (i.e. a linear combination of eigenstates). How exactly could those even exist in the first place?
     
  2. jcsd
  3. Sep 29, 2013 #2

    rubi

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    Try ##\Psi(x) = \frac{1}{1+x^2}##. It's clearly square-integrable, but if you apply the harmonic oscilllator Hamiltonian, you get a function that converges to a non-zero constant for ##x\rightarrow\infty##. The reason for why there can be such states is that the Hamiltonian of the harmonic oscillator is unbounded. The ##\Psi## I mentioned doesn't lie in the domain of ##\hat H##. Most physical observables correspond to unbounded operators.
     
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