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## Main Question or Discussion Point

I just realized quantum operators X and P aren't actually just generalizations of matrices in infinite dimensions that you can naively play with as if they're usual matrices. Then I learned that the space of quantum states is not actually a Hilbert space but a "rigged" Hilbert space.

It all started because I though "well, if X and P are just infinite dimensional matrixes, can't I just multiply them as I would a normal matrix, by multiplying Xxx' by Px'x and then integrating over x'?". But then I realized I'd have to integrate the product of two delta functions which... Uh... Doesn't make a ton of sense. So I asked around the web and I learned that you can't really do that, because X and P are not naive generalizations of finite dimensional matrices, and also the space of states is not a real Hilbert space. But every quantum mechanics textbook I have seen doesn't seem to say much about this and they're kind of misleading. I kinda went down a rabbit hole these last few hours trying to wrap my head around what's going on but I feel like I lack a lot of knowledge.

Generally when I try to solve a problem I feel like I am shuffling symbols around in ways that seem visually right, but I don't really understand what's going on. For this reason I'd appreciate some recommendations of books that explore the math and mathematical methods behind QM in a more rigorous way (though I'd still prefer if it wasn't entirely theoretical and also taught you useful and efficient methods you can use to solve problems), because at the moment I'm kind of overwhelmed at all the info and I don't know where to start. There is a lot of stuff I have to study so I'd prefer if it wasn't some immense 800 page bible.

It all started because I though "well, if X and P are just infinite dimensional matrixes, can't I just multiply them as I would a normal matrix, by multiplying Xxx' by Px'x and then integrating over x'?". But then I realized I'd have to integrate the product of two delta functions which... Uh... Doesn't make a ton of sense. So I asked around the web and I learned that you can't really do that, because X and P are not naive generalizations of finite dimensional matrices, and also the space of states is not a real Hilbert space. But every quantum mechanics textbook I have seen doesn't seem to say much about this and they're kind of misleading. I kinda went down a rabbit hole these last few hours trying to wrap my head around what's going on but I feel like I lack a lot of knowledge.

Generally when I try to solve a problem I feel like I am shuffling symbols around in ways that seem visually right, but I don't really understand what's going on. For this reason I'd appreciate some recommendations of books that explore the math and mathematical methods behind QM in a more rigorous way (though I'd still prefer if it wasn't entirely theoretical and also taught you useful and efficient methods you can use to solve problems), because at the moment I'm kind of overwhelmed at all the info and I don't know where to start. There is a lot of stuff I have to study so I'd prefer if it wasn't some immense 800 page bible.