Continuum States

Usually we expand in countable orthonormal basis and

Sometimes we generalize this to continuum states

Consider ie the space of integrable functions over .
For let be a position vector.
Define operator and
Formally this wouldn’t really be allowed to be in our because they are not normalizable. We would need to include distributions but we just kinda glance over that.

Then we can expand any vector as

Note that here is the coordinate of the vector .

We extract the wavefunction by projecting on