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Resonance states and complex energies

  1. Sep 30, 2008 #1
    I'm reconsidering the problem of resonance states.
    We know that the resonances in QM are described as the complex energy poles in the scattering amplitude. In the version of QFT, the resonances are described by the complex mass poles of the scattering matrix.
    In QFT, I can understand that the masses of intermediate particles develops imaginary masses from loop corrections.
    But in the case of QM, I don't quite understand the situation. I read from a book that because the wavefunction of the unstable state extends to infinity. Hence the boundary condition changes (different from bound state's), that's why the complex energy. (The complex energy is still the eigenvalue of the Hamiltonian.)
    But I remember that we can prove that the eigenvalues of a Hermitian operators are always real. Like the following proof in the braket language,
    [tex] A|a'\rangle = a'|a'\rangle [/tex] and [tex] \langle a''|A = a''^*\langle a''| [/tex]
    where [tex] A [/tex] is an Hermitian operator and [tex] a',a'' [/tex] are its eigenvalues.
    We times the first equation with [tex] \langle a''| [/tex], the second equation with [tex] |a'\rangle [/tex], then substract,
    [tex] \Rightarrow (a' - a''^*)\langle a''|a'\rangle = 0 [/tex]
    now we select [tex] a' = a'' [/tex], then we conclude that [tex] a' [/tex] is real.
    So, eigenvalues of a Hermitian operator must be real.

    In short, my question is,
    (1) is the complex energy of resonance in QM the eigenvalue of Hamiltonian?
    (2) If (1) is true, then how to explain the breakdown of the proof I wrote above?

    Thanks for any ideas.
  2. jcsd
  3. Sep 30, 2008 #2


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    (1): No
  4. Sep 30, 2008 #3
    So...the complex energy poles are...? :shy:
  5. Sep 30, 2008 #4


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    Have you gone through the derivation of the S-matrix in non relativistic quantum mechanics?
  6. Sep 30, 2008 #5
    Actually I haven't gone though the detailed derivation of the S-matrix in non-relativistic quantum mechanics.
    I briefly glanced over the section about resonance in Sakurai's book just now.
    Do you mean that the resonance energy is the eigenvalue of the sum of the Hamiltonian and the "centrifugal potential?" So, the resonance energy is not the eigenvalue of the Hamiltonian.
    But in this way, the sum of Hamiltonian and the centrifugal potential is a Hermitian operator too. Hence the eigenvalues must be real, isn't it?
    Could you hint me the key ideas of how to develop the imaginary part of the resonance complex energy?
  7. Sep 30, 2008 #6


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    Sakurais book is not good for theory of resonance scattering, at least if you wanna do it with S-matrix etc.

    Have I implied all the things you are asking for? The resonance is a peak in the cross section.

    The Scattering chapter of Merzbacher is quite good.

    Anyway, in sakurai, you'll see that the resonance energy is obtained by doing a local taylor expansion of cot(delta_l). But you have already assumed the existance of a resonance etc, so its not so good for resonance scattering as I mentioned.
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