Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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


    User Avatar
    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:
    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:
  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
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook