Why does wave mechanics fit so nicely into inner product spaces?

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pivoxa15
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Does it surprise you that the fundalmentals of wave mechanics fits so nicely into an inner product space. I assume this kind of algebra existed long ago but QM seem to fit perfectly into it. How amazing is that?
 
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pivoxa15 said:
Does it surprise you that the fundalmentals of wave mechanics fits so nicely into an inner product space. I assume this kind of algebra existed long ago but QM seem to fit perfectly into it. How amazing is that?
Maybe it's because waves are written with exponentials, and and so a product of two waves is still an exponential, and the derivative too?
 
pivoxa15 said:
Does it surprise you that the fundalmentals of wave mechanics fits so nicely into an inner product space. I assume this kind of algebra existed long ago but QM seem to fit perfectly into it. How amazing is that?

Once you learn about generalized vector spaces, and the fact that you can pretty much define ANY inner product for a given space of functions, this doesn't seem too extraordinary. The L^2 space of square integrable functions is by no means special.

edit: this isn't to say that all definitions of spaces with all given inner products are USEFUL to us(in the context of solving problems in QM)
 
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