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Brain Hall: Quantum Theory for Mathematicians p.87 "One peculiarity of the physics literature on quantum mechanics is a conspicuous failure of most articles to state what the Hilbert space is." As a chemist I grew up solving the Schrodinger equation to see where the atomic orbitals came from. Back in chemistry grad school in 1961 Hilbert space was never mentioned.

So given the Hamiltonian for the hydrogen atom which is Hermitian and the wave function psi and its eigenvalues, what exactly is the Hilbert space of this particular Schrodinger equation? What is its basis?

Is the wave function an element of this vector space space? Presumably it is. If so is the wave function a basis element?

I realize these are fairly trivial questions, but I'm apparently in good company with Hall himself

luysii
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NOTE: The normalization condition for the solution is a statement of the fact that $\psi(t,\cdot,\cdot,\cdot)\in L^2(\mathbb{R}^3)$ for all $t \ge 0$. $|\psi(t,\cdot,\cdot,\cdot)|^2$ is a probability distribution for every $t\ge 0$.

Disintegrating By Parts
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