Why can a function satisfying same boundary conditions as functions Un(x) of a complete set be expanded as Î£CnUn(x)? Answered
This is a question I encountered while reading Introduction to Quantum Theory by Hendrik F. Hameka (Please don't worry, I'm not trying to cheat on my homework, I am just asking out of curiosity). The book states the following:
"It can be shown that a function f(x) that satisfies the same boundary conditions as the functions Un(x) of a complete set can be expanded as
f(x) = ΣCnUn(x)
If we multiply by U*m(x) and integrate, we obtain
< Um | f > = Cm "
This second part is fairly straightforward because
∫ U*m f(x) dx = ∫ U*m ΣCnUn(x) dx
and it follows that
< Um | f > = Cn Σ < Um | Un > = Cn Σ δn,m = Cn
However, the first part confuses me because I can only justify it when Un(x) = (2π)^(-1/2) e^(inx) (The example of a complete set given in the book, actually)
In that case, it can easily be seen that the expression Σ < Un | f > Un(x) is just the Fourier Series expansion of f(x) and thus the result is valid for the particular complete set. However this does not explain why this is valid for all complete sets: "how can you prove that this conclusion is valid not only for this particular complete set, but for all complete sets?" is my question.
Thank you in advance for answering! Please feel free to post any questions of your own if you need clarification. Also, please be advised that you should not post responses on the order of "you didn't specify an interval of integration, so your question is impossible!" - if you know enough to answer the question, then you will understand why said interval is not specified, etc. Once again, thanks!