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Given a single-particle system with Hamiltonian $H$, what constraints can be put on the wave function at a particular point in time $\psi(x)$? Of course $\psi(x)$ must obey boundary conditions given by $H$. However, in situations where $H$ does not yield strict boundary conditions (e.g. the harmonic oscillator) can $\psi(x)$ be any normalised function? My intuition says no for the following reason: QM textbooks say that any valid wave function can be represented as a linear combination of the eigenstates of $H$. In the case of the harmonic oscillator, the eigenstates of $H$ are a countably infinite set. However the set of all normalised functions is uncountably infinite. It seems to me that one cannot change between one complete set of basis functions to another and reduce the "dimensionality" somehow. This makes me think that the eigenstates of $H$ are not complete, that they imply some constraints on $\psi(x)$. This is not an area of mathematics that I understand well. Can somebody help me out?

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Generically, any square-integrable function is an admissible wave function, and the space of square-integrable complex functions indeed has uncountable dimension as a vector space over $\mathbb{C}$.

And it is also true that the eigenstates of the Hamiltonian span the space of states, and that they are countably many. This is the content of the spectral theorem - eigenstates of bounded self-adjoint operators form an orthonormal basis of the Hilbert space (I will ignore the subtlety of free/non-normalizable "eigenstates" here, and assume the Hamiltonian is bounded above and below).

The point is that the notion of "basis" in the context of infinite-dimensional Hilbert space is not the notion of "basis" from finite-dimensional linear algebra, where the "span" of a set is the set of the finite linear combinations (a basis that forms a basis of a vector space in this sense is sometimes called a Hamel basis). When one speaks of a basis in the context of a Hilbert space, one means a Schauder basis instead:

The "Schauder span" of a set is the set of all convergent infinite series made out of the vectors in that set. This relies on the additional topological structure a Hilbert space carries through the norm on it, and on the completeness of that norm. This span includes the usual linear span, but is larger. In particular, a countable set can span a vector space of uncountable dimension in this sense, exactly like all reals are limit points of sequences of rationals, and there are countably many rationals, and uncountably many reals.

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