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This is obviously a follow on question to the Phys.SE post Hilbert space of harmonic oscillator: Countable vs uncountable?

So I thought that the Hilbert space of a bound electron is countable, but the Hilbert space of a free electron is uncountable. But the arguments about smoothness and delta functions in the answers to the previous question convince me otherwise. Why is the Hilbert space of a free particle not also countable?

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The Hilbert dimension of the Hilbert space of a free particle is countable. To see this, note that

  1. The Hilbert space of a free particle in three dimensions is $L^2(\mathbb{R}^3)$.

  2. An orthonormal basis of a Hilbert space $\mathcal H$ is any subset $B\subseteq \mathcal H$ whose span is dense in $\mathcal H$.

  3. All orthornormal bases of a given non-empty Hilbert space have the same cardinality, and the cardinality of any such basis is called the Hilbert dimension of the space.

  4. The Hilbert space $L^2(\mathbb R^3)$ is separable; it admits a countable, orthonormal basis. Therefore, by the definition of the Hilbert dimension of a Hilbert space, it has countable dimension.

Addendum. 2014-10-19

There is another notion of basis that is usually not being referred to when one discusses Hilbert spaces, namely a Hamel basis (aka algebraic basis). There is a corresponding theorem called the dimension theorem which says that all Hamel bases of a vector space have the same cardinality, and the dimension of the vector space is then defined as the cardinality of any Hamel basis.

One can show that every Hamel basis of an infinite-dimensional Hilbert space is uncountable.

As a result, the dimension (in the sense of Hamel bases) of the free particle Hilbert space is uncountable, but again, this is not usually the sense in which one is using the term dimension in this context, especially in physics.

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So the total universe, or at least all of quantum mechanics, is countable? I was once told so by a very famous physicist, but I didn't think that was the majority position. –  Jim Graber May 22 '13 at 7:53
    
But then what are rigged Hilbert spaces good for? I thought their whole point was mixing countable and uncountable in a consistent way? A quick look at Wikipedia finds “They can bring together the 'bound state' (eigenvector) and 'continuous spectrum', in one place.” –  Jim Graber May 22 '13 at 7:53
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@Jim If you're using the rigged Hilbert space formalism, your state space is actually a subspace of a countable Hilbert space. See here: physics.stackexchange.com/questions/43515/… –  user1504 May 22 '13 at 11:39
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@user1504 : ok, but then it's not a Hilbert space. –  jjcale May 24 '13 at 18:02
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@jjcale Jim asked a second question (in his comment) about rigged Hilbert spaces. In this formalism, the state space is not actually a Hilbert space. Instead, it is a vector subspace of some Hilbert space. This is sufficient to answer Jim's question. For more details about the formalism, go see the answer to the question I linked to above. –  user1504 May 24 '13 at 19:19

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