/sci/ - Science & Math

>>8889393
>Calculus: The Equation

>>8889482
Yes, the functions [math]e^{i \omega x}[/math] are definitely not in L^2, however the inner product I wrote in a weird sense is correct, and they are in a sense basis functions. To construct a function from L^2 out of them, you must have some spread in your spectrum, it can't be a delta function

This is something I've been trying to wrap my head around. In quantum, two famous and basic examples are the free particle (no potential) and the harmonic oscillator (quadratic potential). Applying basic QM, the free particle basis are the uncountably infinite set of functions [math]e^{i \omega x}[/math], while for the harmonic oscillator the functions are the countably infinite set of functions of exponentials raised to Hermite polynomial powers.

The mind fuck is that both can be used to describe functions in L^2. Why this is weird is because the dimensionality of the basis sets are different. It seems that vectors in infinite dimensional Hilbert spaces can be expanded w.r.t either a countably infinite set of functions in L^2, or an uncountably infinite set of functions outside of L^2 but linear combinations of them are in L^2. The choice is arbitrary.

Of course, I am hardly being mathematically rigorous, and physicist have a tendency to take vast amounts of mathematical nuance in stride, but this more or less works.