I have read in a book that the set $\{ \Psi_{n}^{(\nu)} \in \mathcal{H} | \ \ \hat{H}\Psi_{n}^{(\nu)} = E_{n}\Psi_{n}^{(\nu)} \}$ (that is, the set of all eigenfunctions of the hamiltonian with the same eigenvalue) is a subspace of the Hilbert space $\mathcal{H}$. However, to it be a subspace, it has to contain the "null vector", but it is not true, right? This function has zero as eigenvalue, so is the book incorrect?
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3 Answers
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Let $\vec{O}$ be the zero vector. Then $\hat{H} \vec{O} = \vec{O} = E_n \vec{O} $, so $\vec{O}$ is an element of $\{ \Psi_{n}^{(\nu)} \in \mathcal{H} | \ \ \hat{H}\Psi_{n}^{(\nu)} = E_{n}\Psi_{n}^{(\nu)} \}$ for every $E_n$.
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The symbolic definition in your question is not quite the definition of an eigenfunction, precisely because it includes the zero vector. So that set is a linear subspace.
I don't know how your textbook actually defined the space, but in practice, people are often a bit glib about notation in this regard. I would not bat an eye at the statement that, "The set of eigenvectors form a subspace," because of the technical impediment that you mention. When that terminology is used, it is understood that the zero vector is included or not in such a way as to make the statement useful.
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Recall that, in general, if an operator $A$ has an eigenvector $\psi$ with eigenvalue $\lambda$, then $c\psi$ is also an eigenvector of $A$ with eigenvalue $\lambda$, for any $c\in\mathcal{F}$ ($\mathcal{F}$ being whatever field of scalars you are using, so $c\in\mathcal{C}$ in quantum mechanics). So the null vector is always in the the eigenspace with some particular eigenvalue... as long as we are considering the full eigenspace.
But in quantum mechanics, the situation is made a bit more complicated because we insist that state vectors be normalized, so really our state spaces are the sets of unit vectors in Hilbert spaces. So technically, our state spaces are not actually Hilbert spaces. However, we can manipulate the Hilbert spaces in which the unit vectors reside, and this turns out to provide a very convenient set of machinery for thinking and calculating in quantum mechanics, as you know.
Bottom line: the null vector is in each eigenspace of the Hamiltonian (as a full Hilbert space), but state spaces in quantum mechanics are actually sets of unit vectors within Hilbert spaces, so the null vector never actually represents a state. Another way people commonly state this is that the null vector is not normalizable.
In my experience, this subtlety has rarely come up, but it is worth having in the back of your head.
