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

Completeness of orthonormal functions

  1. Jul 26, 2011 #1
    In many areas (say, electrodynamics) we come across expansions of any function in terms of a series of orthonormal functions that span the space. Now the condition for completeness of a set of orthonormal functions in that space is given by (as given in Jackson)

    [tex]\sum_{n=1}^\infty U_n^*(x') U_n(x) = \delta(x'-x)[/tex]

    where x and x' are two points in the function space.

    I am not able to understand what this is intuitively due to.

    Why would the orthogonal functions not span the entire space if the summation does not go to zero ?

    Finally, is there a corresponding relation in vector space ? That will probably give me a better understanding of what is happening, if we extend it to function space..

    Thanks a ton!
     
  2. jcsd
  3. Jul 27, 2011 #2
    I strongly suggest you to study the Dirac's formalism of bras and kets, it makes all theese relations much clearer.

    Anyway, if U_n is complete and orthonormal, you must have, for every f belonging to the space:

    [itex]f=\sum_nU_n(U_n,f)[/itex]

    or, in the x-representation

    [itex]f(x)=\sum_n U_n(x)\int U^*_N(x')f(x')dx'[/itex]

    but we also have

    [itex]f(x)=\int f(x')\delta(x-x')dx'[/itex]

    so...
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook