quantum mechanics - Hilbert space of a free particle: Countable or Uncountable?

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?

Answer

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.