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Help with this semiclassical sum please.

  1. Feb 12, 2008 #1
    I know how to obtain a semiclassical expression for the sum:

    [tex] \sum_{n=0}^{\infty} e^{-a E_{n}}\approx \iint _{\Gamma}e^{-aH(x,p)} dx, dp [/tex]

    where the integral is extended over the whole phase space of particle (x,p) , my question is if E-->oo is a big (but finite) energy , then what would be the asymptotic expression to evaluate:

    [tex] \sum_{E_n \le E}e^{-aE_{n}} [/tex]

    this is just the sum of the function exp(-x) taken over all the Energies of Hamiltonian operator that are less or equal to a finite given Energy 'E' then what would be the expression for this last sum ?? , thanks in advance.

    EDIT: i am not pretty sure, however i would say that if A is a constant then .. the last sum could be approximated by the integral:

    [tex] \int_{0}^{N}exp(-E)dE [/tex] with E_n=E so 'N' is the N-th quantum

    level whose energy is precisely 'E'
    Last edited: Feb 12, 2008
  2. jcsd
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