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Dirac Delta from Continous Eigenfunctions

  1. Sep 23, 2011 #1
    In the equation for determining the coefficients of eigenfunctions of a continuous spectrum operator, I have trouble understnading the origin of the Dirac delta.

    a_f = INTEGRAL a_g ( INTEGRAL F_f F_g ) dq dg

    a is the coefficient, F = F(q) is an eigenfunction.

    From this it is shown that the first integral (ie. of the eigenfunctions with dq) must be a Dirac delta function, that is, that for f = g it is infinite. Why is this? Landau Lifgarbagez states that it is to prevent the integral with dg from vanishing, but I don't see this.

    This would mean that a_f = INTEGRAL a_f (infinity) df ....how is this?


    Cheer folks!!! :)
     
  2. jcsd
  3. Sep 23, 2011 #2

    Bill_K

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    Science Advisor

    This is pretty much the same question as before, right? Only last time it was discrete (Eq 3.5) an = Σ am ∫ ψm ψn* dq, and this time it's continuous (Eq 5.3) af = ∫ af' (∫ψf'ψf* dq) df'.

    In both cases, the reason is (like LL say): "This relation must hold for arbitrary af". The only way an = Σ am (...blah...) can hold for arbitrary an is if (...blah...) is a Kronecker delta, δmn, and the only way af = ∫ af' (...blah...) df' can hold for arbitrary af is if (...blah...) is a delta function, δ(f-f').
     
  4. Sep 23, 2011 #3
    That's all good, but it's specifically the infinity part....why must the inner product of the two eigenfunctions when f=g inner product of it with itself) be infinity to satisfy this??
     
  5. Sep 23, 2011 #4
    I can't see what those equation are. But let me say something.

    Even though
    δ(x)={infinity; x=0}={0; x≠0}

    But from the definition:
    ∫δ(x)dx=1 -----> The area under the 'curve' must be 1:
    Dirac_function_approximation.gif
    The Dirac delta function as the limit (in the sense of distributions) of the sequence of Gaussians: Wiki

    so, for every single function, there must be a value a that satisfy the equations:
    f(a)=∫f(x-a)δ(x)dx
     
  6. Sep 24, 2011 #5
    [itex]\int[/itex] [itex]\Psi[/itex] [itex]_{f}[/itex] [itex]\Psi[/itex] [itex]_{g}[/itex] dq


    Thought I might as well get used to this Latex thing :D

    My question is, why is this integral infinity when f=g ?

    Psi is an eigenfunction of a continuous spectrum operator
     
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